Appendix A. Measure and integration

We suppose the reader is familiar with the basic facts concerning set theory and integration as they are presented in the introductory course of analysis. In this appendix, we review them briefly, and add some more which we shall need in the text. Basic references for proofs and a detailed exposition are, e.g., [[ H a l 1 ]] , [[ J a r 1 , 2 ]] , [[ K F 1 , 2 ]] , [[ L i L ]] , [[ R u 1 ]] , or any other textbook on analysis you might prefer.

A.1 Sets, mappings, relations

A set is a collection of objects called elements. The symbol card X denotes the cardi- nality of the set X. The subset M consisting of the elements of X which satisfy the conditions P1(x),...,Pn(x) is usually written as M = { x ∈ X : P1(x),...,Pn(x) }.A set whose elements are certain sets is called a system or family of these sets; the family of all subsystems of a given X is denoted as 2X . The operations of union, intersection, and set difference are introduced in the standard way; the first two of these are commutative, associative, and mutually distributive. In a { } system Mα of any cardinality, the de Morgan relations , X \ Mα = (X \ Mα)and X \ Mα = (X \ Mα), α α α α are valid. Another elementary property is the following: for any family {Mn} ,whichis { } at most countable, there is a disjoint family Nn of the same cardinality such that ⊂ \ ∪ \ Nn Mn and n Nn = n Mn.Theset(M N) (N M) is called the symmetric difference of the sets M,N and denoted as M #N. It is commutative, M #N = N #M, and furthermore, we have M #N =(M ∪N)\(M ∩N)andM #N =(X\M)#(X\N)for any X ⊃ M∪N. The symmetric difference is also associative, M#(N#P )=(M#N)#P , and distributive with respect to the intersection, (M # N) ∩ P =(M ∩ P ) # (N ∩ P ). A family R is called a set ring if M # N ∈R and M ∩ N ∈R holds for any pair M,N ∈R. The relation M \ N =(M # N) ∩ M also gives M \ N ∈R, and this in turn implies ∅∈R and M ∪ N ∈R. If the symmetric difference and intersection are understood as a sum and product, respectively, then a set ring is a ring in the sense of the general algebraic definition of Appendix B.1. A.1.1 Example: Let J d be the family of all bounded intervals in Rd,d≥ 1. The family Rd, which consists of all finite unions of intervals J ⊂Jd together with the empty set, is a set ring, and moreover, it is the smallest set ring containing J d. As mentioned above, any R ∈Rd can be expressed as a finite union of disjoint bounded intervals.

595 596 Appendix A Measure and integration

A set ring R⊂2X is called a set field if it contains the set X (notice that in the terminology of Appendix B.1 it is a ring with a unit element but not an algebra). A set field A⊂ X ∞ ∈A { }∞ ⊂A 2 is called a σ–field if n=1 Mn holds for any countable system Mn n=1 . ∞ ∈A A⊂ X De Morgan relations show that also n=1 Mn , and furthermore that a family 2 \ ∈A ∈A ∈A containing the set X is a σ–field iff X M for all M and n Mn for any X at most countable subsystem {Mn}⊂A. Given a family S⊂2 we consider all σ–fields A⊂2X containing S (there is at least one, A =2X ). Their intersection is again a σ–field containing S;wecallittheσ–field generated by S and denote it as A(S). A.1.2 Example: The elements of Bd := A(J d) are called Borel sets in Rd. In particular, all the open and closed sets, and thus also the compact sets, are Borel. The σ–field Bd is also generated by other systems, e.g., by the system of all open sets in Rd. In general, Borel sets in a topological space (X, τ) are defined as the elements of the σ–field A(τ). { }∞ ⊂ ⊃ A sequence Mn n=1 is nondecreasing or nonincreasing if Mn Mn+1 or Mn M Mn+1, respectively, holds for n =1, 2,.... A set family is monotonic if it contains { } the set n Mn together with any nondecreasing sequence Mn ,and n Mn together with any nonincreasing sequence {Mn}.Anyσ–field represents an example of a monotonic system. To any S there is the smallest monotonic system M(S) containingS and we M S ⊂AS R M R ∈ have ( ) ( ). If is a ring, the same is true for ( ); in addition, if M∈R M M(R), then M(R)isaσ–field and M(R)=A(R). A mapping (or map) f from a set X to Y is a rule, which associates with any x ∈ X a unique element y ≡ f(x) of the set Y ;wewrite f : X → Y and also x → f(x). If Y = R or Y = C the map f is usually called a real or a complex function, respectively. It is also often useful to consider maps which are defined on a subset Df ⊂ X only. The symbol f : X → Y must then be completed by specifying the set Df which is called the domain of f;wedenoteitalsoas D(f). If Df is not specified, it is supposed to coincide with X. The sets Ran f := { y ∈ Y : y = f(x),x∈ Df } and Ker f := { x ∈ Df : f(x)=0} are the range and kernel of the map f, respectively. A map f : X → Y is injective if f(x)=f(x) holds for any x, x ∈ X only if x = x; it is surjective if Ran f = Y . A map which is simultaneously injective and surjective is called bijective or a bijection. The sets X and Y have the same cardinality if there is a bijection f : X → Y with Df = X. The relation f = g between f : X → Y and g : X → Y means by definition Df = Dg and f(x)=g(x) for all x ∈ Df .If Df ⊃ Dg and f(x)=g(x) holds for all x ∈ Dg we say that f is an extension of g while g is a \ restriction of f to the set Dg;wewrite f ⊃ g and g = f | Dg.

A.1.3 Example: For any X ⊂ M we define a real function χM : χM (x)=1ifx ∈ M, χM (x)=0ifx ∈ X \ M; it is called the characteristic (or indicator) function of the X set M. The map M → χM is a bijection of the system 2 to the set of all functions f : X → R such that Ran f = {0, 1}. (−1) { ∈ ∈ } → ⊂ The set f (N):= x Df : f(x) N for givenf : X Y and N Y is called (−1) (−1) the pull–back of the set N by the map f. One has f α∈I Nα = α∈I f (Nα)for any family {Nα}⊂Y , and the analogous relation is valid for intersections. Furthermore, (−1) (−1) (−1) (−1) f (N1 \ N2)=f (N1) \ f (N2)andf(f (N)=Ranf ∩ N. On the other hand, f (−1)(f(M)) ⊃ M; the inclusion turns to identity if f is injective. Y A.1.4 Example: Let f : X → Y with Df = X.Toaσ–field B⊂2 we can construct the family f (−1) := { f (−1)(N): N ∈B}, which is obviously again a σ–field. Similarly, if A.1 Sets, mappings, relations 597

A⊂2X is a σ–field, then the same is true for { N ⊂ Y : f (−1)(N) ∈A}. Hence for any family S⊂2Y we can construct the σ–field f (−1)(A(S)) ⊂ 2X and the latter coincides with the σ–field generated by f (−1)(S), i.e., f (−1)(A(S)) = A(f (−1)(S)). Given f : X → Y and g : Y → Z we can define the composite map g◦f : X → Z (−1) (−1) with the domain D(g◦f):=f (Dg)=f (Dg ∩ Ran f)by(g◦f)(x):=g(f(x)). We have (g◦f)(P )(−1) = f (−1)(g(−1)(P )) for any P ⊂ Z. If f : X → Y is injective, then for any y ∈ Ran f there is just one xy ∈ Df such that y = f(xy); the prescription y → g(y):=xy defines a map g : Y → X,whichis −1 −1 −1 called the inverse of f and denoted as f .Wehave D(f )=Ranf, Ran f = Df −1 −1 and f (f(x)) = x, f(f (y)) = y for any x ∈ Df and y ∈ Ran f, respectively. These relations further imply f (−1)(N)=f −1(N) for any N ⊂ Ran f.Oftenwehaveapairof mappings f : X → Y and g : Y → X and we want to know whether f is invertible and f −1 = g; this is true if one of the following conditions is valid:

(i) Dg =Ranf and g(f(x)) = x for all x ∈ Df (ii) Ran f ⊂ Dg, g(f(x)) = x for all x ∈ Df and Ran g ⊂ Df , f(g(y)) = y for all y ∈ Dg

If f : X → Y is injective, then f −1 is also injective and (f −1)−1 = f.If g : Y → Z is also injective, then the composite map g◦f is invertible and (g◦f)−1 = f −1◦g−1. The Cartesian product M × N is the set of ordered pairs [x, y]withx ∈ M and y ∈ N; the Cartesian product of the families S and S is defined by S×S := { M × N : M ∈S,N ∈S }. For instance, the systems of bounded intervals of Example 1 satisfy J m+n = J m ×Jn.If M × N is empty, then either M = ∅ or N = ∅. On the other hand, if M × N is nonempty, then the inclusion M × N ⊂ P × R implies M ⊂ P and N ⊂ R.Wehave(M ∪ P ) × N =(M × N) ∪ (P × N) and similar simple relations for the intersection and set difference. Notice, however, that (M × N)∪(P × R) can be expressed in the form S × T only if M = P or N = R. The definition of the Cartesian product extends easily to any finite family of sets. ×···× { }→ n Alternatively, we can interpret M1 Mn as the set of maps f : 1,...,n j=1 Mj ∈ { ∈ } such that f(j) Mj. This allows us to defineXα∈I Mα for a system Mα : α I of → ∈ any cardinality as the set of maps f : I α∈I Mα which fulfil f(α) Mα for any α ∈ I. The existence of such maps is related to the axiom of choice (see below). Given f : X → C and g : Y → C, we define the function f × g on X × Y by (f × g)(x, y):=f(x)g(y). Let M ⊂ X × Y ; then to any x ∈ X we define the x–cut of the set M by Mx := { y ∈ Y :[x, y] ∈ M}; we define the y–cuts analogously. Let A⊂2X , B⊂2Y be σ–fields; then the σ–field A(A×B) is called the direct product of the fields A and B and is denoted as A⊗B. A.1.5 Example: The Borel sets in Rm and Rn in this way generate all Borel sets in Rm+n, i.e.,wehave Bm ⊗Bn = Bm+n. On the other hand, the cuts of a set M ∈A⊗B belong to the original fields: we have Mx ∈B and My ∈A for any x ∈ X and y ∈ Y , respectively. A subset Rϕ ⊂ X × X defines a relation ϕ on X:if[x, y] ∈ Rϕ we say the element x is in relation with y and write xϕy. A common example is an equivalence,whichis a relation ∼ on X that is reflexive ( x ∼ x for any x ∈ X ), symmetric ( x ∼ y implies y ∼ x ), and transitive ( x ∼ y and y ∼ z imply x ∼ z ). For any x ∈ X we define the 598 Appendix A Measure and integration

equivalence class of x as the set Tx := { y ∈ X : y ∼ x }.Wehave Tx = Ty iff x ∼ y,so the set X decomposes into a disjoint union of the equivalence classes. Another important example is a partial ordering on X, which means any relation ≺ that is reflexive, transitive, and antisymmetric, i.e., such that the conditions x ≺ y and y ≺ x imply x = y.If X is partially ordered, then a subset M ⊂ X is said to be (fully) ordered if any elements x, y ∈ M satisfy either x ≺ y or x  y. An element x ∈ X is an upper bound of a set M ⊂ X if y ≺ x holds for all y ∈ M;itisamaximal element of M if for any y ∈ M the condition y  x implies y = x. A.1.6 Theorem (Zorn’s lemma): Let any ordered subset of a partially ordered set X have an upper bound; then X contains a maximal element. Zorn’s lemma is equivalent to the so–called axiom of choice, which postulates for a system { Mα : α ∈ I } of any cardinality the existence of a map α → xα such that xα ∈ Mα for all α ∈ I — see, e.g., [[ DS 1 ]], Sec.I.2, [[ Ku ]], Sec.I.6. Notice that the maximal element in a partially ordered set is generally far from unique.

A.2 Measures and measurable functions

Let us have a pair (X, A), where X is a set and A⊂2X a σ–field. A function f : X → R is called measurable (with respect to A )if f (−1)(J) ∈A holds for any bounded interval J ⊂ R, i.e., f (−1)(J ) ⊂A. This is equivalent to any of the following statements: (i) f (−1)((c, ∞)) ∈A for all c ∈ R, (ii) f (−1)(G) ∈A for any open G ⊂ R, (iii) f (−1)(B) ⊂ A.If X is a topological space, a function f : X → R is called Borel if it is measurable w.r.t. the σ–field B of Borel sets in X. A.2.1 Example: Any continuous function f : Rd → R is Borel. Furthermore, let f : X → R be measurable (w.r.t. some A )and g : R → R be Borel; then the composite function g◦f is measurable w.r.t. A. If functions f,g : X → R are measurable, then the same is true for their linear combinations af + bg and product fg as well as for the function x → (f(x))−1 provided f(x) =0 forall x ∈ X. Even if the last condition is not valid, the function h, defined by h(x):=(f(x))−1 if f(x) =0 and h(x) := 0 otherwise, is measurable. Furthermore, if a sequence {fn} converges pointwise, then the function x → limn→∞ fn(x) is again measurable. The notion of measurability extends to complex functions: a function ϕ : X → C is measurable (w.r.t. A ) if the functions Re ϕ(·)andImϕ(·) are measurable; this is true iff ϕ(−1)(G) ∈A holds for any open set G ⊂ C. A complex linear combination of measurable functions is again measurable. Furthermore, if ϕ is measurable, then |ϕ(·)| is also measurable. In particular, the modulus of a measurable f : X → R is measurable, ± 1 | |± as are the functions f := 2 ( f f). → C ∈ C A function ϕ : X is simple (σ–simple)if ϕ = n ynχMn , where yn and the ∈A sets Mn form a finite (respectively, at most countable) disjoint system with n Mn = X. By definition, any such function is measurable; the sets of (σ–)simple functions are closed with respect to the pointwise defined operations of summation, multiplication, and scalar multiplication. The expression ϕ = n ynχMn is not unique, however, unless the numbers yn are mutually different. A.2 Measures and measurable functions 599

A.2.2 Proposition: A function f : X → R is measurable iff there is a sequence {fn} of σ–simple functions, which converges to f uniformly on X.If f is bounded, there is a sequence of simple functions with the stated property.

In fact, the approximating sequence {fn} canbechoseneventobenondecreasing.If f is not bounded it can still be approximated pointwise by a sequence of simple functions, but not uniformly. Given (X, A)and(Y,B) we can construct the pair (X × Y,A⊗B). Let ϕ : M → C be a function on M ∈A⊗B; then its x–cut is the function ϕx defined on Mx by ϕx(y):=ϕ(x, y); we define the y–cut similarly. Cuts of a measurable functions may not be measurable in general, however, it is usually important to ensure measurability a.e. – cf. Theorem A.3.13 below. A mapping λ defined on a set family S and such that λ(M) is either non–negative or λ(M)=+∞ for any M ∈S is called (a non–negative) set function.Itismonotonic if M ⊂ N implies λ(M) ≤ λ(N), additive if λ(M ∪ N)=λ(M)+λ(N) for any pair of sets such that M ∪ N ∈S and M ∩ N = ∅,andσ–additive if the last property generalizes, { }⊂S λ ( n Mn)= n λ(Mn), to any disjoint at most countable system Mn such that ∈S n Mn . A set function µ, which is defined on a certain A⊂2X ,isσ–additive, and satisfies µ(∅) = 0 is called a (non–negative) measure on X. If at least one M ∈A has µ(M) < ∞, then µ(∅) = 0 is a consequence of the σ–additivity. The triplet (X, A,µ) is called a measure space; the sets and functions measurable w.r.t. A are in this case often specified as µ–measurable.Aset M ∈A is said to be µ–zero if µ(M) = 0, a proposition–valued function defined on M ∈A is valid µ–almost everywhere if the set N ⊂ M,onwhichit is not valid, is µ–zero. A measure µ is complete if N ⊂ M implies N ∈A for any µ–zero set M; below we shall show that any measure can be extended in a standard way to a complete one. Additivity implies that any measure is monotonic, and µ(M ∪ N)=µ(M)+µ(N) − µ(M ∩ N) ≤ µ(M)+µ(N) for any sets M,N ∈Awhich satisfy µ(M ∩ N) < ∞. ∞ Using the σ–additivity, one can check that limk→∞ µ(Mk)=µ ( n=1 Mn) holds for any nondecreasing sequence {Mn}⊂A, and a similar relation with the union replaced by intersection is valid for nonincreasing sequences. A measure µ is said be finite if µ(X) < ∞ ∞ ∈A ∞ and σ–finite if X = n=1 Mn, where Mn and µ(Mn) < for n =1, 2,.... Let (X, τ) be a topological space, in which any open set can be expressed as a count- able union of compact sets (as, for instance, the space Rd; recall that an open ball there is a countable union of closed balls). Suppose that µ is a measure on X with the domain A⊃τ; then the following is true: if any point of an open set G has a µ–zero neighborhood, then µ(G)=0. Given a measure µ we can define the function µ : A×A → [0, ∞)byµ(M ×N):= µ(M # N). The condition µ(M # N) = 0 defines an equivalence relation on A and µ is a metric on the corresponding set of equivalence classes. Apoint x ∈ X such that the one–point set {x} belongs to A and µ({x}) =0 is called a discrete point of µ; the set of all such points is denoted as Pµ.If µ is σ–finite the set Pµ is at most countable. A measure µ is discrete if Pµ ∈Aand µ(M)=µ(M ∩ Pµ) for any M ∈A. 600 Appendix A Measure and integration

A measure µ is said to be concentrated on a set S ∈A if µ(M)=µ(M ∩ S) for any M ∈A. For instance, a discrete measure is concentrated on the set of its discrete points. If (X, τ) is a topological space and τ ⊂A, then the support of µ denoted as supp µ is the smallest closed set on which µ is concentrated. Next we are going to discuss some ways in which measures can be constructed. First we shall describe a construction, which starts from a given non–negative σ–additive set X functionµ ˙ defined on a ring R⊂2 ; we assume that there exists an at most countable { }⊂R ∞ disjoint system Bn such that n Bn = X andµ ˙ (Bn) < for n =1, 2,.... Let S be the system of all at most countable unions of the elements of R; it is closed w.r.t. countable unions and finite intersections, and M \R ∈Sholds for all M ∈Sand R ∈ R ∈S { }⊂R .Any M can be expressed as M = j Rj, where Rj is an at most countable S disjoint system; using it we can define the set functionµ ¨ on byµ ¨(M):= j µ˙ (Rj). It is ≤ monotonic and σ–additive. Furthermore, we haveµ ¨ ( n Mn) n µ¨(Mn); this property is called countable semiadditivity. Together with the monotonicity, it is equivalent to the ⊂ ∞ ≤ ∞ fact that M k=1 Mk impliesµ ¨(M) k=1 µ¨(Mk). The next step is to extend the functionµ ¨ to the whole system 2X by defining the outer measure by µ∗(A):=inf{ µ¨(M): M ∈S,M ⊃ A }. The outer measure is again monotonic and countably semiadditive; however, it is not additive so it is not a measure. Its importance lies in the fact that the system

∗ Aµ := {A⊂X :infµ (A # M)=0} M∈S

∗ \ is a σ–field. This finally allows us to define µ := µ | Aµ;itisacompleteσ–additive measure on the σ–field Aµ ⊃A(R), which is determined uniquely by the set functionµ ˙ in the sense that any measure ν on A(R), which is an extension ofµ ˙ , satisfies ν = µ |\ A(R). The measure µ is called the Lebesgue extension ofµ ˙ . A measure µ on a topological space (X, τ) is called Borel if it is defined on B≡A(τ) and µ(C) < ∞ holds for any compact set C. We are particularly interested in Borel measures on Rd, where the last condition is equivalent to the requirement µ(K) < ∞ for any compact interval K ⊂ Rd. Any Borel measure on Rd is therefore σ–additive and corresponds to a unique σ– additive set functionµ ˙ on Rd. The space Rd, however, has the special property that d for any bounded interval J ∈J we can find a nonincreasing sequence of open intervals ⊃ ⊂ In J and a nondecreasing sequence of compact intervals Kn J such that n In = n Kn = J. This allows us to replace the requirement of σ–additivity by the condition d d µ˙ (J)=inf{ µ˙ (I): I ∈GJ } =sup{ µ˙ (K): K ∈FJ } for any J ∈J , where GJ ⊂J is d the system of all open intervals containing J,and FJ ⊂J is the system of all compact intervals contained in J. A set functionµ ˜ on J d which is finite, additive, and fulfils the last condition is called regular. A.2.3 Theorem: There is a one–to–one correspondence between regular set functionsµ ˜ and µ := µ∗ |\ Bd on Rd. In particular, Borel measures µ and ν coincide if µ(J)=ν(J) holds for any J ∈Jd. A.2.4 Example: Let f : R → R be a nondecreasing right–continuous function. For any a, b ∈ R,a

1 J ; the corresponding Borel measure µf is called the Lebesgue–Stieltjes measure generated by the function f. In particular, if f is the identical function, f(x)=x,wespeak about the Lebesgue measure on R. Let us remark that the Lebesgue–Stieltjes measure is sometimes understood as a Lebesgue extension with a domain which is generally dependent on f; however, it contains B in any case. A.2.5 Example: Letµ ˜ andν ˜ be regular set functions on J m ⊂ Rm and J n ⊂ Rn, respectively; then the function ˜ on J m+n defined by ˜(J × L):=˜µ(J)˜ν(L) is again regular; the corresponding Borel measure is called the direct product of the measures µ and ν which correspond toµ ˜ andν ˜, respectively, and is denoted as µ ⊗ ν. In particular, repeating the procedure d times, we can in this way construct the Lebesgue measure on Rd which associates its volume with every parallelepiped. A.2.6 Proposition: Any Borel measure on Rd is regular, i.e., µ(B)=inf{µ(G): G ⊃ B, G open} =sup{µ(C): C ⊂ B, C compact}. As a consequence of this result, we can find to any B ∈Bd a nonincreasing sequence ⊃ ⊂ of open sets Gn B and a nondecreasing sequence of compact sets Cn B (both dependent generally on the measure µ ) such that µ(B) = limn→∞ µ(Gn)=µ ( n Gn)= limn→∞ µ(Cn)=µ ( n Cn). Proposition A.2.6 generalizes to Borel measures on a locally compact Hausdorff space, in which any open set is a countable union of compact sets – see [[Ru 1 ]], Sec.2.18. Let us finally remark that there are alternative ways to construct Borel measures. One can use, e.g., the Riesz representation theorem, according to which Borel measures correspond bijectively to positive linear functionals on the vector space of continuous functions with a compact support — cf. [[Ru 1 ]], Sec.2.14; [[RS 1 ]], Sec.IV.4.

A.3 Integration

Now we shall briefly review the Lebesgue integral theory on a measure space (X, A,µ). It is useful from the beginning to consider functions which may assume infinite values; this requires to define the algebraic operations a + ∞ := ∞, a ·∞:= ∞ for a>0and a ·∞ := 0 for a =0,etc., to add the requirement f (−1)(∞) ∈A to the definition of measurability, and several other simple modifications. Given a simple non–negative function s := n ynχMn on X, we define its integral by X sdµ:= n ynµ(Mn); correctness of the definition follows from the additivity of µ. In the next step, we extend it to all measurable functions f : X → [0, ∞] putting  

fdµ ≡ f(x) dµ(x):=sup sdµ : s ∈ Sf , X X X → ∞ ≤ where Sf is the set of all simple functions s : X [0, ) such that s f.Wealso ∈A define M fdµ:= X fχM dµ for any M ; in this way we associate with the function f and the set M anumberfrom[0, ∞], which is called the (Lebesgue) integral of f over M w.r.t. the measure µ.

A.3.1 Proposition: Let f,g be measurable functions X → [0, ∞]andM ∈A; then ∈ ∞ ≤ M (kf) dµ = k M fdµ holds for any k [0, ), and moreover, the inequality f g ≤ implies M fdµ M gdµ. 602 Appendix A Measure and integration

Notice that the integral of f = 0 is zero even if µ(X)=∞. On the other hand, the | | → C { ∈ relation X ϕ dµ = 0 for any measurable function ϕ : X implies that µ( x X : ϕ(x) =0 })=0,i.e., that the function ϕ is zero µ–a.e. Let us turn to limits which play the central role in the theory of integration. { } A.3.2 Theorem (monotone convergence): Let fn be a nondecreasing sequence of non- negative measurable functions; then limn→∞ X fn dµ = X (limn→∞ fn) dµ. The right side of the last relation makes sense since the limit function is measurable. However, we often need some conditions under which both sides are finite. The correspond- ing modification is also called the monotone–convergence (or Levi’s) theorem: if {fn} is a nondecreasing sequence of non–negative measurable functions and there is a k>0such ≤ → that X fn dµ k for n=1, 2,..., then the function x f(x) := limn→∞ fn(x)is ≤ µ–a.e. finite and limn→∞ X fn dµ = X fdµ k. The monotone–convergence theorem implies, in particular, that the integral of a measurable function can be approximated by a nondecreasing sequence of integrals of simple functions. ≤ A.3.3 Corollary (Fatou’s lemma): X (liminf n→∞fn) dµ liminf n→∞ X fn dµ holds for any sequence of measurable functions fn : X → [0, ∞]. { } This result has the following easy consequence: let a sequence fn of non–negative ≤ measurable functions have a limit everywhere, limn→∞ fn(x)=f(x), and X fn dµ k ≤ for n =1, 2,...; then X fdµ k. Applying the monotone–convergence theorem to a se- { } ∞ quence fn of non–negative measurable functions we get the relation X ( n=1 fn) dµ = ∞ ∞ fn dµ. In particular, if f : X → [0, ∞] is measurable and {Mn} ⊂A is n=1 X ∞ n=1 a disjoint family with Mn = M, then putting fn := fχM we get fdµ = ∞ n=1 n M fdµ. This relation is called σ–additivity of the integral; it expresses the fact n=1 Mn that the function f together with the measure µ generates another measure.

A.3.4 Proposition: Let f : X → [0, ∞] be a measurable function; then the map M → A ν(M):= M fdµ is a measure with the domain ,and X gdν = X gf dµ holds for any measurable g : X → [0, ∞].

Let us pass to integration of complex functions. A measurable function ϕ : X → C | | ∞ is integrable (over X w.r.t. µ )if X ϕ dµ < (recall that if ϕ is measurable so is |ϕ| ). The set of all integrable functions is denoted as L(X, dµ); in the same way we define L(M,dµ) for any M ∈A. Given ϕ ∈L(X, dµ)wedenotef := Re ϕ and g := Im ϕ; then f ± and g± are non–negative measurable functions belonging to L(X, dµ). This allows us to define the integral of complex functions through the positive and negative parts of the functions f,g as the mapping ϕ −→ ϕdµ := f + dµ − f − dµ + i g+ dµ − i g− dµ X X X X X

d of L(X, dµ)toC. If, in particular, µ is the Lebesgue measure on R we often use the d d d symbol L(R )orL(R ,dx) instead of L(R ,dµ), and the integral is written as ϕ(x) dx, or occasionally as ϕ(x) dx. The above definition has the following easy consequence: if M ϕdµ =0 holdsfor ∈A ≥ ∈A all M , then ϕ(x)=0 µ–a.e. in X. Similarly M ϕdµ 0 for all M implies ϕ(x) ≥ 0 µ–a.e. in X; further generalizations can be found in [[ Ru 1 ]], Sec.1.40. The integral has the following basic properties: A.3 Integration 603

L (a) linearity: (X, dµ) is a complex vector space and X (αϕ + ψ) dµ = α X ϕdµ+ ψdµ for all ϕ, ψ ∈L(X, dµ)andα ∈ C, X ≤ | | ∈L (b) X ϕdµ X ϕ dµ holds for any ϕ (X, dµ). A.3.5 Examples: A simple complex function σ= n ηnχMn on X is integrable iff | | ∞ n ηn µ(Mn) < , and in this case X σdµ = n ηnµ(Mn). The same is true for σ– simple functions. Further, let functions f : X → [0, ∞]andϕ : X → C be measurable and dν := fdµ; then ϕ ∈L(X, dν) iff ϕf ∈L(X, dµ) and Proposition A.3.4 holds again with g replaced by ϕ. For a finite measure, we have an equivalent definition based on approximation of integrable functions by sequences of σ–simple functions. A.3.6 Proposition: If µ(X) < ∞, then a measurable function ϕ : X → C belongs L { } to (X, dµ) iff there is a sequence τn of σ–simple integrable functions such that | − | limn→∞(supx∈X ϕ(x) τn(x) ) = 0; if this is the case, then X ϕdµ = limn→∞ X τn dµ. Moreover, if ϕ is bounded the assertion is valid with simple functions τn. One of the most useful tools in the theory of integral is the following theorem.

A.3.7 Theorem (dominated convergence, or Lebesgue): Let M ∈Aand {ϕn} be a sequence of complex measurable functions with the following properties: ϕ(x):= limn→∞ ϕn(x) exists for µ–almost all x ∈ M and there is a function ψ ∈L(X, dµ) such that |ϕn(x)|≤ψ(x)holdsµ–a.e. in M for n =1, 2,....Then ϕ ∈L(X, dµ)and lim |ϕ − ϕ | dµ =0, lim ϕ dµ = ϕdµ. →∞ n →∞ n n M n M M

Suppose that we have non–negative measures µ and ν on X (without loss of gener- ality, we may assume that they have the same domain) and k>0; then we can define the non–negative measure λ :=kµ + ν. We obviously have L(X, dλ)=L(X, dµ) ∩L(X, dν) ∈L and X ψdλ= k X ψdµ+ X ψdν for any ψ (X, dλ). In particular, if non–negative measures µ and λ with the same domain A satisfy µ(M) ≤ λ(M) for any M ∈A, then L ⊂L ≤ ∈L (X, dλ) (X, dµ)and X fdµ X fdλ holds for each non–negative f (X, dλ). Let µ, ν again be measures on X with the same domain A. We say that ν is absolutely continuous w.r.t. µ and write ν  µ if µ(M) = 0 implies ν(M)=0 for any M ∈A. On the other hand, if there are disjoint sets Sµ,Sν ∈A such that µ is concentrated on Sµ and ν on Sν we say that the measures are mutually singular and write µ ⊥ ν. A.3.8 Theorem: Let λ and µ be non–negative measures on A, the former being finite and the latter σ–finite; then there is a unique decomposition λ = λac + λs into the sum of non–negative mutually singular measures such that λac  µ and λs ⊥ µ. Moreover, there is a non–negative function f ∈L(X, dµ), unique up to a µ–zero measure subset of ∈A X, such that dλac = fdµ, i.e., λac(M)= M fdµ for any M .

The relation λ = λac + λs is called the Lebesgue decomposition of the measure λ.The second assertion implies the Radon–Nikod´ym theorem: let µ be σ–finite and λ finite; then λ  µ holds iff there is f ∈L(X, dµ) such that dλ = fdµ. 604 Appendix A Measure and integration

A.3.9 Remark: There is a close connection between these results (and their extensions to complex measures mentioned in the next section) and the theory of the indefinite Lebesgue integral, properties of absolutely continuous functions, etc. We refer to the literature men- tioned at the beginning; in this book, in fact, we need only the following facts: a function ϕ : R → C is absolutely continuous on a compact interval [a, b] if for any ε>0 there | − | is δ>0 such that j ϕ(βj) ϕ(αj) <ε holds for a finite disjoint system of intervals ⊂ − (αj,βj) [a, b] fulfilling j(βj αj) <δ. The function ϕ is absolutely continuous in a (noncompact) interval J if it is absolutely continuous in any compact [a, b] ⊂ J.A function ϕ : R → C is absolutely continuous in R iff its derivative ϕ exists almost L everywhere w.r.t. the Lebesgue measure and belongs to (J, dx) for any bounded interval ⊂ R ≤ − b  J with the endpoints a b; in such a case we have ϕ(b) ϕ(a)= a ϕ (x) dx. Next we shall mention integration of composite functions. Let w : X → Rd be a map such that w(−1)(Bd) ⊂A; this requirement is equivalent to measurability of the (−1) “component” functions wj : X → R, 1 ≤ j ≤ d. Suppose that µ(w (J)) < ∞ holds for any J ∈Jd; then the relation B → µ(w)(B):=µ(w(−1)(B)) defines a Borel measure µ(w) on Rd. A.3.10 Theorem: Adopt the above assumptions, and let a Borel function ϕ : Rd → C belong to L(Rd,dµ(w)); then ϕ◦w ∈L(X, dµ)and ϕdµ(w) = (ϕ◦w) dµ B w(−1)(B) holds for all b ∈Bd. In particular, if X = Rd, A = Bd,and µ(w) is the Lebesgue measure on Rd, the latter formula can under additional assumptions be brought into a convenient form. Suppose that w : Rd → Rd is injective, its domain is an open set D⊂Rd, the component functions wj : D→R have continuous partial derivatives, (∂kwj)(·)forj, k =1,...,d, and finally, the Jacobian determinant, Dw := det(∂kwj), is nonzero a.e. in D. Such a map is called regular; its range R := Ran w is an open set in Rd, and the inverse w−1 is again regular.

A.3.11 Theorem (change of variables): Let w be a regular map on Rd with the domain D and range R; then a Borel function ϕ : R→C belongs to L(R,dx) iff (ϕ◦w)Dw ∈ L(D,dx); in that case we have

ϕ(x) dx = ((ϕ◦w)|Dw|)(x) dx B w(−1)(B) for any Borel B ⊂R. In Example A.2.5 we have mentioned how the measure µ ⊗ ν can be associated with a pair of Borel µ, ν on Rd. An analogous result is valid under much more general circumstances. A.3.12 Theorem: Let (X, A,µ)and(Y,B,ν) be measure spaces with σ–finite measures; then there is just one measure λ on X × Y with the domain A⊗B such that λ(A × B) = µ(A)ν(B) holds for all A ∈A,B∈B; this measure is σ–finite and satisfies λ(M)= y ∈A⊗B X ν(Mx) dµ(x)= Y µ(M ) dν(y) for any M . A.4 Complex measures 605

The measure λ is again called the product measure of µ and ν and is denoted as µ ⊗ ν. Using it we can formulate the following important result. A.3.13 Theorem (Fubini): Suppose the assumptions of the previous theorem are valid × → C L × ⊗ ∈L and ϕ : X Y belongs to (X Y,d(µ ν)). Then the cut ϕx (Y,dν)for ∈ L µ–a.a. x M and the function Φ : Φ(x)= Y ϕx dν belongs to (X, dµ); similarly y ∈L y L ϕ (X, dµ) ν–a.e. in N and Ψ : Ψ(y)= X ϕ dµ belongs to (Y,dν). Finally, we ⊗ have X Φ dµ = Y Ψ dν = X×Y ϕd(µ ν), or in a more explicit form, ϕ(x, y) dν(y) dµ(x)= ϕ(x, y) dµ(x) dν(y)= ϕ(x, y) d(µ⊗ν)(x, y). X Y Y X X×Y

We should keep in mind that the latter identity may not be valid if at least one of the measures µ and ν is not σ–finite. It is also not sufficient that both double integrals exists — counterexamples can be found, e.g., in [[Ru 1 ]], Sec.7.9 or [[KF ]], Sec.V.6.3. However, if at least one double integral of the modulus |ϕ| is finite, then all the conclusions of the theorem are valid.

A.4 Complex measures

A σ–additive map ν : A→C corresponding to a given (X, A) is called complex measure on X;if ν(M) ∈ R for all m ∈A we speak about a real (or signed) measure. Any pair of non–negative measures µ1,µ2 with a common domain A determines a signed measure by  := µ1 − µ2; similarly a pair of real measures 1,2 defines a complex measure by ν := 1 + i2. { } Any at most countable system Mj , which is disjoint and satisfies M = j Mj, will be called decomposition of the set M; the family of all decompositions of M will be denoted by S . To a given complex measure ν and M ∈A, we define |ν|(M):=  M  | | { }∈S | | ≥| | | | · sup j ν(Mj) : Mj M . One has ν (M) ν(M) ; the set function ν ( )is called the (total) variation of the measure ν. A.4.1 Proposition: The variation of a complex measure is a non–negative measure; it is the smallest non–negative measure such that µ(M) ≥|ν(M)| holds for all M ∈A. Using the total variation, we can decompose in particular any signed measure  + − − ± ± 1 | | ± in the form  = µ µ , where µ : µ (M)= 2 [  (M) (M)]. Since in general the decomposition of a signed measure into a difference of non–negative measures is not unique, + − − one is interested in the minimal decomposition  = µ µ such that any pair of non– A − ≥ + negative measures µ1,µ2 on with the property  = µ1 µ2 satisfies µ1(M) µ (M) ≥ − ∈A and µ2(M) µ (M)foreachM . The minimality is ensured if there is a disjoint decomposition Q+ ∪Q− = X such that ± ± ∩ ± ≥ { + −} µ := (M Q ) 0; the pair Q ,Q is called Hahn decomposition of X w.r.t. the measure . The Hahn decomposition always exists but it is not unique. Nevertheless, if {Q˜+, Q˜−} is another Hahn decomposition, one has (M ∩ Q±)=(M ∩ Q˜±) for any ∈A ± M , so the measures µ depend on  only; we call them the positive and negative + − − variation of the measure . The formula  = µ µ is named the Jordan decomposition of the measure . 606 Appendix A Measure and integration

+ − | | ± {± ⊂ ∈A} One has µ (M)+µ (M)=  (M)andµ (M)=sup (A): A M, A for any M ∈A. As a consequence, the positive and negative variations of a signed measure as well as the total variation of a complex measure are finite. One can introduce also infinite signed measures; however, we shall not need them in this book. Complex Borel measures on Rd have A = Bd for the domain. Variation of a complex Borel measure is a non–negative Borel measure. As in the non–negative case, a complex Borel measure can be approximated using monotonic sequences of compact sets from inside and open sets from outside of a given M ∈Bd. Also the second part of Theorem A.2.3 can be generalized. A.4.2 Proposition: Let complex Borel measure ν andν ˜ on Rd satisfy ν(J)=˜ν(J) for all J ∈Jd; then ν =˜ν. Before proceeding further, let us mention how the notion of absolute continuity extends to complex measures. The definition is the same: a complex measure ν is absolutely continuous w.r.t. a non–negative µ if µ(M) = 0 implies ν(M) = 0 for all M ∈A. There is an alternative definition. A.4.3 Proposition: A complex measure ν satisfies ν  µ iff for any ε>0 there is a δ>0 such that µ(M) <δ implies |ν(M)| <ε. In particular, if ϕ ∈L(X, dµ)andν is the measure generated by this function, ν(M):= ϕdµ, then ν  µ, so for any ε>0 there is a δ>0 such that µ(M) <δ implies M M ϕdµ <ε; this property is called absolute continuity of the integral. Theorem A.3.8 holds for a complex measure λ as well. The measure can be even σ– finite; however, then the function f belongsnolongerto L(X, dµ), it is only measurable and integrable over any set M ∈A with λ(M) < ∞. The Radon–Nikod´ym theorem yields the polar decomposition of a complex measure: A.4.4 Proposition: For any complex measure ν there is a measurable function h such that |h(x)| =1 forall x ∈ X and dν = hd|ν|. Let us pass now to integration with respect to complex measures. We start with a + − − → C signed measure  = µ µ : a function ϕ : X is integrable w.r.t.  if it belongs to L + ∩L − L (X, dµ ) (X, dµ )=: (X, d). Its integral is then defined by + − − ϕd := ϕdµ ϕdµ ; X X X the correctness follows from the uniqueness of the Jordan decomposition. The integral w.r.t. a complex measure ν represents then a natural extension of the present definition: for any function ϕ : X → C belonging to L(X, dν):=L(X, d Re ν) ∩L(X, d Im ν)weset ϕdν := ϕdRe ν + ϕdIm ν. X X X

The set of integrable functions can be expressed alternatively as L(X, dν)=L(X, d|ν|); it → is a complex vector space and the map ϕ X ϕdν is again linear. Also other properties of the integral discussed in the previous section extend to the complex–measure case. For ≤ | | | | ∈L instance, the inequality X ϕdν X ϕ d ν holds for any ϕ (X, dµ). We shall not continue the list, restricting ourselves by quoting the appropriate generalization of Proposition A.3.4. A.5 The Bochner integral 607

A.4.5 Proposition: Let ϕ ∈L(X, dν) for a complex measure ν; then the map M → A ∈L γ(M):= M ϕdν defines a complex measure γ on . The conditions ψ (X, dγ)and ψϕ ∈L(X, dν) are equivalent for any measurable γ : X → C; if they are satisfied one has X ψdγ = X ψϕ dν.

A.5 The Bochner integral

The theory of integration recalled above can be extended to vector–valued functions F : Z →X, where X is a Banach space; they form a vector space denoted as V(Z, X ) when equipped with pointwise defined algebraic operations. Let (Z, A,µ) be a measure space with a positive measure µ. A function S ∈V(Z, X )issimple if there is a disjoint { }n ⊂A ∈X decomposition Mj j=1 of the set Z and vectors y1,...,yn such that S = n n j=1 yjχMj . The integral of such a function is defined by Z S(t) dµ(t):= j=1 yjµ(Mj); as above, it does not depend on the used representation of the function S. To any F ∈V(Z, dµ) we define the non–negative function F := F (·) . A vector– { } valued function F is integrable w.r.t. µ if there is a sequence Sn of simple functions → ∈ − → such that Sn(t) F (t) holds for µ–a.a. t Z and Z F Sn dµ 0. The set of all integrable functions F : Z →X is denoted by B(Z, dµ; X ). If F is integrable, the limit F (t) dµ(t) := lim S (t) dµ(t) →∞ n Z n Z exists and it is independent of the choice of the approximating sequence; we call it the ∈A Bochner integral of the function F . The function χM F is integrable for any set M ∈B X { } and F (Z, dµ; ), so we can also define M F (t) dµ(t):= Z χM (t)F (t) dµ(t). If Mk is a finite disjoint decomposition of M,wehave F (t) dµ(t)= F (t) dµ(t), M k Mk which means that the Bochner integral is additive. → B X ⊂ A.5.1 Proposition: The map F Z F (t) dµ(t) from the subspace (Z, dµ; ) V(Z, X )toX is linear. Suppose that for a vector–valued function F there is a sequence {Sn} of simple functions that converges to Fµ–a.e.; then F belongs to B(Z, dµ; X ) iff F ∈L(Z, dµ), and in that case F (t) dµ(t) ≤ F (t) dµ(t). Z Z

The existence of an approximating sequence of simple functions has to be checked for each particular case; it is easy in some situations, e.g.,if Z is a compact subinterval in R and F is continuous, or if Z is any interval, F is continuous and its one–sided limits at the endpoints exist. The continuity of F : R →X in an interval [a, b]also d t ∈ implies the relation dt a F (u) du = F (t) for any t (a, b). Proposition A.5.1 shows that the Bochner integral is absolutely continuous: for any ε>0 there is a δ>0 such that ∈A N F (t) dµ(t) <ε holds for any N with µ(N) <δ. Another useful result is the following. 608 Appendix A Measure and integration

A.5.2 Proposition: If B : X→Y is a bounded linear map to a Banach space Y; then the condition F ∈B(Z, dµ; X ) implies BF ∈B(Z, dµ; Y), and (BF)(t) dµ(t)=B F (t) dµ(t) . Z Z

Many properties of the Lebesgue integral can be extended to the Bochner integral. Probably the most important among them is the dominated–convergence theorem.

A.5.3 Theorem: Let {Fn}⊂B(Z, dµ; X ) be a sequence such that {Fn(t)} converges for µ–a.a. t ∈ Z and Fn(t) ≤g(t),n=1, 2,..., for some g ∈L(Z, dµ). Assume further that there is a sequence {Sn} of simple functions, which converges to the limiting function F : F (t) = limn→∞ Fn(t) µ–a.e. in Z; then F ∈B(Z, dµ; X )and lim F (t) dµ(t)= F (t) dµ(t). →∞ n n Z Z

An analogue to Theorem A.3.10 can be proven for some classes of functions, e.g.,for a monotonic w : R → R. Since B(X ) is a Banach space, the Bochner integral is also used for operator–valued functions. For instance, suppose that a map B : R →B(X ) is such that the vector–valued function t → B(t)x is continuous for any x ∈X. Further, let K ⊂ R be a compact R interval and µ a Borel measure on ; then limn→∞ K B(t)xn dµ(t)= K B(t)xdµ(t) { }⊂X holds for any sequence xn converging to a point x. Moreover, if an operator ∈C ∈ T commutes with B(t) for allt K, then K B(t)ydµ(t) belongs to D(T ) for any ∈ y D(T )andT K B(t)ydµ(t)= K B(t)Tydµ(t). Appendix B. Some algebraic notions

In this appendix we collect some algebraic definitions and results needed in the text. There are again many textbooks and monographs in which this material is set out extensively; let us name, e.g., [[BR 1 ]], [[Nai 1 ]], [[Ru 2 ]], or [[Ti ]] for associative algebras, and [[BaR ]], [[ K i r ]] , [[ P o n ]] , o r [[ Zelˇ ]] for Lie groups and algebras.

B.1 Involutive algebras

A binary operation in a set M is a map ϕ : M ×M → M;itisassociative or commutative if ϕ(ϕ(a, b),c)=ϕ(a, ϕ(b, c)) or ϕ(a, b)=ϕ(b, a) , respectively, holds for all a, b, c ∈ M . Aset G equipped with an associative binary operation is called a group if there exist the unit element e ∈ G , ϕ(g, e)=ϕ(e, g)=g for any g ∈ G , and the inverse element g−1 ∈ G to any g ∈ G , ϕ(g, g−1)=ϕ(g−1,g)=e . Consider next a set R equipped with two binary operations, which we call summation, ϕa(a, b):=a+b ,andmultiplication, ϕm(a, b):=ab . The triplet (R, ϕa,ϕm)isaring if (R, ϕa) is a commutative group and the two operations are distributive, a(b+c)=ab + ac and (a+b)c = ac + bc for all a, b, c ∈ R . If there is an e ∈ R such that ae = ea = a holds for all a ∈ R ,wecallittheunit element of R . Let A be a vector space over a field F . The vector summation gives it the structure of a commutative group; if we define a multiplication which is distributive with the sum- mation and satisfies α(ab)=(αa)b = a(αb) for any a, b ∈A,α∈ C , then A becomes a ring, which we call a linear algebra over the field F , in particular, a real or complex algebra if F = R or F = C , respectively. An algebra is said to be associative if its multiplication is associative. The term “algebra” without a further specification always means a com- plex associative algebra in what follows; we should stress, however, that many important algebras are nonassociative, e.g., the Lie algebras discussed in Sec.B.3 below. An algebra is Abelian or commutative if its multiplication is commutative. A subalgebra of an algebra A is a subset B , which is itself an algebra with respect to the same operations. If A has the unit element, which is not contained in B , then we can extend the subalgebra to B˜ := {αe + b : α ∈ C,b ∈B};inasimilarway, any algebra can be completed with the unit element by extending it to the set of pairs [α, a],α∈ C,a ∈A, with the appropriately defined operations. A proper subalgebra B⊂A is called a (two–sided) ideal in A if the products ab and ba belong to B for all a ∈A,b∈B; we define the left and right ideal analogously. A trivial example of an ideal is the zero subalgebra {0}⊂A. The algebra A itself is not regarded as an ideal;

609 610 Appendix B Some algebraic notions thus no ideal can contain the unit element. A maximal ideal in A is such that it is not a proper subalgebra of another ideal in A; any ideal in an algebra with the unit element is a subalgebra of some maximal ideal. An algebra is called simple if it contains no nontrivial two–sided ideal. The intersection of any family of subalgebras (ideals, one–sided ideals) in A is respectively a subalgebra (ideal, one–sided ideal), while the analogous assertion for the unions is not valid. Let A be an algebra with the unit element. We say that an element a ∈A is invertible if there exists an inverse element a−1 ∈A such that a−1a = aa−1 = e; we define the left and right inverse in the same way. For any a ∈A there is at most one inverse; an element is invertible iff it belongs to no one–sided ideal of the algebra A , which means, in particular, that in an algebra without one–sided ideals any nonzero element is invertible. Recall that a field is a ring with the unit element which has the last named property; the examples are R, C or the noncommutative field Q of quaternions. −1 We define the spectrum of a ∈A as the set σA(a):={λ :(a−λe) does not exist} . The complement ρA(a):=C \ σA(a) is called the resolvent set; its elements are regular −1 values for which the the resolvent ra(λ):=(a−λe) exists. B.1.1 Proposition: Let A be an algebra with the unit element; then (a) If a, ab are invertible, b is also invertible. If ab , ba are invertible, so are a and b. (b) If ab = e, the element ba is idempotent but it need not be equal to the unit element unless dim A < ∞. (c) If e−ab is invertible, the same is true for e−ba. (d) σA(ab) \{0} = σA(ba) \{0} , and moreover, σA(ab)=σA(ba) provided one of the elements a, b is invertible. −1 −1 (e) σA(a )={λ : λ ∈ σA(a)} holds for any invertible a ∈A.

For any set S⊂Awe define the algebra A0(S) generated by S as the smallest subalgebra in A containing S; it is easy to see that it consists just of all polynomials composed of the elements of S without an absolute term. We say that S is commutative if ab = ba holds for any a, b ∈S; the algebra A0(S) is then Abelian. A maximal Abelian algebra is such that it is not a proper subalgebra of an Abelian subalgebra; any Abelian subalgebra in A can be extended to a maximal Abelian subalgebra. We also define the commutant of a set S⊂Aas S := { a ∈A: ab = ba, b ∈S}; in particular, the center is the set Z := A . We define the bicommutant S := (S) and higher–order commutants in the same way. B.1.2 Proposition: Let S, T be subsets in an algebra S; then (a) S and S are subalgebras containing the center Z , and also the unit element if A has one. Moreover, S = S = ··· and S = S IV = ···. (b) The inclusion S⊂T implies S ⊃T. (c) S⊂S ,and S is commutative iff S⊂S, which is further equivalent to the condition that S is Abelian.     (d) A0(S) = S and A0(S) = S . (e) A subalgebra B⊂A is maximal Abelian iff B = B; in that case also B = B.

Let us turn to algebras with an additional unary operation. Recall that an involution a → a∗ on a vector space A is an antilinear map A→A such that (a∗)∗ = a holds for all a ∈A;aninvolution on an algebra is also required to satisfy the condition (ab)∗ = b∗a∗ for any a, b ∈A. An algebra equipped with an involution is called an involutive algebra B.1 Involutive algebras 611 or briefly a ∗–algebra. A subalgebra in A , which is itself a ∗–algebra w.r.t. the same involution, is called a ∗–subalgebra; we define the ∗–ideal in the same way. The element a∗ is said to be adjoint to a . Given a subset S⊂Awe denote S∗ := {a∗ : a ∈S}; the set S is symmetric if S∗ = S; in particular, an element a fulfilling a∗ = a is called A∗ S ∗ A S Hermitean.By 0( ) we denote the smallest –subalgebra in containing the set . B.1.3 Proposition: Let A be a ∗–algebra; then (a) Any element is a linear combination of two Hermitean elements, and e∗ = e provided A has the unit element. (b) a∗ is invertible iff a is invertible, and (a∗)−1 =(a−1)∗. ∗ (c) σA(a )=σA(a) holds for any a ∈A. (d) A subalgebra B⊂A is a ∗–subalgebra iff it is symmetric; the intersection of any family of ∗–subalgebras ( ∗–ideals) is a ∗–subalgebra ( ∗–ideal). (e) Any ∗–ideal in A is two–sided. A∗ S A S∪S∗ S⊂A S S (f) 0( )= 0( ) holds for any subset ;if is symmetric, then and S are ∗–subalgebras in A .

B.1.4 Example (bounded–operator algebras): The set B(H) with the natural algebraic operations and the involution B → B∗ is a ∗–algebra whose unit element is the operator I . Let us mention a few of its subalgebras: (a) If E is a nontrivial projection, then {EB : B ∈B(H)} is a right ideal but not a ∗–subalgebra; on the other hand, {EBE : B ∈B(H)} is a ∗–subalgebra but not an ideal. (b) If dim H = ∞ , the sets K(H) ⊃J2(H) ⊃J1(H) of compact, Hilbert–Schmidt, and trace–class operators, respectively, are ideals in B(H); similarly Jp(H)isanideal in any Jq(H),q>p, etc. (c) The algebra A0(B) generated by an operator B ∈B(H) consists of all polynomials in B without an absolute term. It is a ∗–algebra if B is Hermitean, while the opposite implication is not valid; for instance, the Fourier–Plancherel operator F 3 −1 ∗ is non–Hermitean but A0(F )isa∗–algebra because F = F = F .

The algebras of bounded operators, which represent our main topic of interest, inspire some definitions. We have already introduced the notions of spectrum and hermiticity; similarly an element a ∈A is said to be normal if aa∗ = a∗a ,aprojection if a∗ = a = a2 , and unitary if a∗ = a−1 , etc. Of course, we also employ other algebras than B(H)and its subalgebras, e.g., the Abelian ∗–algebra C(M) of continuous complex functions on a compact space M with natural summation and multiplication, and the involution given by complex conjugation, (f ∗)(x):=f(x). An ideal J in an algebra A is a subspace, so we can construct the factor space A/J . It becomes an algebra if we define on it a multiplication bya ˜˜b := ab , where a, b are any elements representing the equivalence classesa ˜ and ˜b; it is called the factor algebra (of A w.r.t. the ideal J ). If A has the unit element, then the classe ˜ := {e+c : c ∈J} is the unit element of A/J . A morphism of algebras A, B is a map ϕ : A→B which preserves the algebraic structure, ϕ(αa + b)=αϕ(a)+ϕ(b)andϕ(ab)=ϕ(a)ϕ(b) for all a, b ∈A,α∈ C . In particular, if ϕ is surjective, then the image of the unit element (an ideal, maximal ideal, maximal Abelian subalgebra) in A is respectively the unit element (an ideal, . . . ) in B .If ϕ is bijective, we call it an isomorphism; in the case A = B one uses the terms 612 Appendix B Some algebraic notions endomorphism and automorphism of A , respectively. The null–space of a morphism ϕ is −1 the pull–back ϕ (0B) of the zero element of the algebra B;itisanidealin A .If A, B are ∗–algebras and ϕ preserves the involution, ϕ(a∗)=ϕ(a)∗ , it is called ∗–morphism.

B.1.5 Example: Let J be a ( ∗–)ideal in a ( ∗–)algebra A; then the map ϕc : ϕc(a)=˜a isa(∗–)morphism of A to A/J . It is called a canonical morphism; its null–space is just the ideal J . The factor algebra A/J is simple iff the ideal J is maximal. Moreover, any ( ∗–)morphism ϕ : A→B can be expressed as a composite mapping, ϕ = π◦ϕc , −1 where ϕc is the canonical morphism corresponding to the ( ∗–)ideal J := ϕ (0B)and π : A/J→ϕ(A) is the ( ∗–)isomorphism defined by π(˜a):=ϕ(a) for any a ∈A. Let us finally recall a few notions concerning representations. This term usually means the mapping of an algebraic object onto a suitable set of operators, which preserves the algebraic structure. We shall most often (but not exclusively) use representations by bounded operators: by a representation of a ( ∗–)algebra A we understand in this case a ( ∗–)morphism π : A→B(H) , the space H is called the representation space and dim H the dimension of the representation π . If the morphism π is injective, the representation is said to be faithful. Representations πj : A→B(Hj),j=1, 2, are equivalent if there is a unitary operator U : H1 →H2 such that π2(a)U = Uπ1(a) holds for any a ∈A. A representation π : A→B(H) is called irreducible if the operator family π(A)has no nontrivial closed invariant subspace. A vector x ∈H is cyclic for the representation π if the set π(A)x := { π(a)x : a ∈A} is dense in H . Representations of groups, Lie algebras, etc., are defined in the same way.

B.2 Banach algebras Algebras can be equipped with a topological structure. Suppose that an algebra A is at the same time a locally convex topological space with a topology τ;thenwecallita topological algebra if the multiplication is separately continuous, i.e., the maps a → ab and a → ba are continuous w.r.t. the topology τ for any fixed b ∈A. A subalgebra B⊂A is closed if it is closed as a subset in A . The closed subalgebra A(S) generated by a set S⊂Ais the smallest closed subalgebra in A containing S . Isomorphisms ϕ : A→B of topological algebras are classified by their continuity: the algebras A, B are topologically isomorphic if there is a continuous isomorphism ϕ such that ϕ−1 is also continuous. B.2.1 Proposition: Let A be a topological algebra and S, B its subset and subalgebra, respectively; then (a) B is a closed subalgebra in A .If B is Abelian, the same is true for B ,andany maximal Abelian subalgebra is closed. (b) A(S)=A0(S). (c) The subalgebras S, S are closed and (S) = S. (d) If B is an ideal, B = A , then B is also an ideal in A . Any maximal ideal is closed. (e) the null–space of a continuous morphism ϕ : A→C is a closed ideal in A. There are various ways how of defining a topology on an algebra. B.2.2 Example: The strong and weak operator topologies on B(H) are both locally convex, and the operator multiplication is separately continuous with respect to them (compare with Theorem 3.1.9 and Problem 3.9); thus Bs(H)andBw(H) are topological algebras. B.2 Banach algebras 613

One of the most natural ways is to introduce a topology by means of a norm. An algebra A is called a normed algebra provided (i) A is a normed space with a norm · . (ii) ab ≤ a b for any a, b ∈A. (iii) If A has the unit element, then e =1. The last condition may be replaced by e ≤1 because the opposite inequality follows from (ii). The multiplication in a normed algebra is jointly continuous. If A is complete w.r.t. the norm · , it is called a Banach algebra. We can again assume without loss of generality that a normed algebra A has the unit element; otherwise we extend it in the above described way, defining the norm by [α, a] := |α|+ a A .If J is a closed ideal in a Banach algebra A , then A/J is a Banach algebra w.r.t. the norm a := infb∈J a−b A . A complete envelope of a normed algebra A is a Banach algebra B such that it contains A as a dense subalgebra and a A = a B holds for any a ∈A. B.2.3 Theorem: Any normed algebra A has a complete envelope, which is unique up to an isometric isomorphism preserving the elements of A . The space B(H) equipped with the operator norm provides an example of a Banach algebra; the complete envelope of any subalgebra B⊂B(H) is its closure B . By a direct generalization of the methods of Section 1.7, we can prove the following claims. B.2.4 Theorem: Let A be a Banach algebra with the unit element; then (a) Any element a ∈A fulfilling a−e < 1 is invertible. The set R of all invertible elements in A is open and the map a → a−1 is continuous in it. (b) The resolvent set ρA(a) of any element a ∈A is open in C and the resolvent ra : ρA →A is analytic. (c) The spectrum σA(a) of any element a ∈A is a nonempty compact set. (d) The spectral radius r(a):=sup{|λ| : λ ∈ σA(a) } is independent of A and equals r(a) = lim an 1/n =inf an 1/n ; n→∞ n it does not exceed the norm, r(a) ≤ a . The independence feature of part (d) is not apparent in the case of bounded operators, where the spectral quantities are related to a single algebra B(H) . To appreciate this result, notice that the spectrum, and in particular its radius, is a purely algebraic property, while the right side of the formula depends on the metric properties of the algebra A . B.2.5 Proposition: Let A, B be Banach algebras; then (a) A morphism ϕ : A→B is continuous iff there is C such that ϕ(a) B ≤ C a A for any a ∈A;if ϕ is a continuous isomorphism, then the algebras A, B are topologically isomorphic. (b) If A, B are complete envelopes of normed algebras A0, B0 , then any continuous morphism ϕ0 : A0 →B0 has just one continuous extension ϕ : A→B. (c) If J is the null–space of a continuous surjective morphism ϕ : A→B, then A/J and B are topologically isomorphic.

An isomorphism ϕ : A→B is called isometric if ϕ(a) B = a A holds for all a ∈A. B.2.6 Theorem (Gel’fand–Mazur): A Banach algebra with the unit element, in which any nonzero element is invertible, is isometrically isomorphic to the field C of complex numbers. 614 Appendix B Some algebraic notions

B.3 Lie algebras and Lie groups

A Lie algebra (real, complex, or more generally, over a field F ) is a finite–dimensional (nonassociative) linear algebra L with the multiplication which we conventionally denote as (a, b) → [a, b] . The latter is antisymmetric, [a, b]=−[b, a] , and satisfies the Jacobi identity [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for any a, b, c ∈ L .Thedimension of L is its { }n vector–space dimension; if ej j=1 isabasisin L , the product is fully determined by the i i relations [ej,ek]=cjkei , where the coefficients cjk are called the structure constants (one uses the summation convention, according to which the sum is taken over any repeated index). A complex extension LC is a complex extension of L as a vector space with the product [a1 + ib1,a2 + ib2]:=[a1,a2] − [b1,b2]+i[a1,b2]+i[b1,a2]. A Lie algebra is commutative if [a, b] = 0 for all a, b ∈ L , i.e., all the structure constants are zero; this definition differs from the associative case. In contrast, other defi- nitions like those of a subalgebra, ideal, and also morphisms, representations, etc.,modify easily for Lie algebras. B.3.1 Examples: Consider the following matrix algebras: (a) The set of all n × n real matrices forms an n2–dimensional real Lie algebra called the general linear algebra and denoted as gl(n, R); its subalgebra sl(n, R) consists of all traceless g ∈ gl(n, R) . Its complex extension is the algebra gl(n, C)ofn × n complex matrices; similarly traceless complex matrices form the algebra sl(n, C), which is often also denoted as An−1 . (b) The algebra gl(m, C) has other Lie subalgebras. A matrix g is said to be skew– symmetric if gt = −g , where gt denotes the transposed matrix of g (this property should not be confused with antihermiticity). The set of skew–symmetric m×m ma- trices forms the orthogonal Lie algebra, which is denoted as o(m, C); we alternatively speak about the algebras Bn and Dn for o(2n +1, C)ando(2n, C) , respectively. C On the other hand, consider the subset in gl(2n, ) consisting of matrices such that t 0 en g j2n + j2ng = 0 , where j2n := and en is the n × n unit matrix. The en 0 corresponding Lie algebra is called symplectic and denoted as sp(n, C)orCn .

Given a Lie algebra L , we define the subalgebras L(n) by the recursive relations (0) (n+1) (n) (n) L := L and L := [L ,L ]; similarly we define L(n) by L(0) := L and L(n+1) := (n) [L(n),L] . The algebra is solvable if L = {0} for some n;itisnilpotent if L(n) for some n . Any nilpotent algebra is solvable. A commutative Lie algebra is, of course, nilpotent; a less trivial example is the Heisenberg–Weyl algebra, which is nilpotent of order two. On the other hand, a Lie algebra L is semisimple if it has no commutative ideal; it is simple if it has no (nontrivial) ideal at all. An equivalent characterization leans on i k the notion of the Cartan tensor g : grs = crkcsi , through which one defines the Killing r s form L × L → F by (a, b):=grsa b . The algebra L is semisimple iff its Killing form is nondegenerate, i.e.,detg =0.For real Lie algebras, the Killing form may be used to introduce the following notion: L is compact if the form is positive, and noncompact otherwise. A compact L can be expressed as L = Z ⊕ S = Z ⊕ S1 ⊕···⊕Sn , where Z is its center, S is semisimple, and S1,...,Sn are simple algebras. Simple Lie algebras allow a full classification. It appears that, up to an isomorphism, complex simple algebras are almost exhausted by the types An,Bn,Cn,and Dn listed in Example 1; there are just five more simple Lie algebras called exceptional. For small B.3 Lie algebras and Lie groups 615

values of n , some of these algebras are isomorphic, namely A1 ∼ B1 ∼ C1 , B2 ∼ D2 , and A3 ∼ D3 , while D2 is semisimple and isomorphic to A1 ⊕ A1 . One can classify real forms of simple complex algebras in a similar way. The notion of a group was introduced above; for simplicity we shall here denote the group operation as a multiplication. A subgroup of a group G is a subset H ⊂ G which is itself a group w.r.t. the same operation. H is a left invariant subgroup if hg ∈ H holds for any h ∈ H and g ∈ G . We define the right invariant subgroup in a similar way; a subgroup is invariant provided it is left and right invariant at the same time (these notions play a role analogous to ideals in algebras). The notions of the direct product of groups and a factor group are easy modifications of the above discussed algebraic definitions. A topological group is a group G , which is simultaneously a T1 topological space such that the map g → g−1 is continuous and the group multiplication is jointly continuous. An isomorphism of topological groups is a map which is a group isomorphism and, at the same time, a homeomorphism of the corresponding topological spaces. One introduces various classes of topological groups according to the properties of G as a topological space, e.g., compact groups or locally compact groups. In a similar way, one defines a connected group. If G is not connected, it can be decomposed into connected components; the component containing the unit element is a closed invariant subgroup. n B.3.2 Examples: (a) The group Tn of translations of the Euclidean space R is a commutative topological group, which is locally but not globally compact. (b) The orthogonal group O(n) consists of real orthogonal n × n matrices, i.e.,such that gtg = e . It is compact and has two connected components specified by the conditions det g = ±1; the connected component of the unit element is the rotation group denoted as SO(n). (c) The group U(n) of unitary complex n×n matrices is locally compact and connected; the same is true for its subgroup SU(n) of matrices with det g =1. (d) The substitution operators Uϕ of Example 3.3.2 form a group. If we equip the set | − | of mappings ϕ with the metric (ϕ, ϕ˜):=supx∈Rn ϕ(x) ϕ˜(x) , it becomes a topological group which is not locally compact. A topological group can also be equipped with a measure. The easiest way to introduce it is through linear functionals — cf. the concluding remark in Section A.2. Consider such a functional µ on the space C0(G) of continuous functions with compact supports which is positive, i.e., µ(f) ≥ 0 holds for all f ≥ 0 . If it satisfies µ(f(g−1·)) = µ(f) for any f ∈ C0(G)andg ∈ G , it is called a left Haar measure on G . On a locally compact group G , there is always a left Haar measure and it is unique up to a multiplicative constant. We introduce the right Haar measure in a similar way; a measure on G is said to be invariant if it combines the two properties. An important class of topological groups consists of those which allow a locally Euclid- ean parametrization. To be more precise, the notion of an analytic manifold is needed. This is a Hausdorff space M together with a family of pairs (Uα,ϕα) ,α∈ I , where Uα is an → Rn open set in M and ϕα is a homeomorphism Uα for a fixed n with the following ∈ properties: α∈I Uα = M and for any α, β I the component functions of the map ◦ −1 ∩ ϕβ ϕα are real analytic on ϕα(Uα Uβ). Thenumber n is called the dimension of the manifold; replacing Rn by Cn in the definition we introduce complex analytic manifolds in the same way. Given analytic manifolds M, N , one can associate with their topological product (M × N,τM×N ) the family of pairs (Uα × Vβ,ϕα × ψβ) . The obtained structure again 616 Appendix B Some algebraic notions satisfies the above conditions; we call it the product manifold of M and N . The dimension of the product manifold is m = mM + mN . A map M → N is analytic if the component ◦ −1 functions of all the maps ψβ ϕα are analytic on their domains. Agroup G is called a (real or complex) Lie group if it is an analytic manifold (real or complex, respectively) and its multiplication and inversion as maps G × G → G and G → G , respectively, are analytic. For instance, the groups of Examples B.3.2a–c belong to this class; this is not true for the group of Example B.3.2d, where the dimension of the group manifold is infinite. Any Lie group is locally compact. A subgroup of G which is itself a Lie group with the same multiplication is called a Lie subgroup. As an analytic manifold, at the vicinity of any point a Lie group admits a description through local coordinates defined by the corresponding map ϕα . This concerns, in parti- cular, the unit element e: there is a neighborhood U of the point 0 ∈ Rn where we can parametrize the group elements by g ≡ (g1,...,gn) . The composition law of G is then n locally expressed by real analytic functions fj,j=1,...,n,from U × U to R so that (gh)j = fj(g1,...,gn,h1,...,hn); the consistency requires fj(g, 0) = gj,fj(0,h)=hj and (∂fj/∂gk)(0,0) =(∂fj/∂hk)(0,0) = δjk .Thestructure constants of the group G are i 2 i − 2 i defined by cjk := (∂ f /∂gj∂hk ∂ f /∂hj∂gk)(0,0); they satisfy the same conditions as the structure constants of Lie algebras. This is not a coincidence; there is a close connection between Lie groups and Lie algebras. Let U be the neighborhood of the parameter–space origin used above and con- ∞ −1 | sider the space C (U) . The operators Tj :(Tjφ)(g)=(∂φ/∂gj)(ggj ) g=0 are then well–defined and span a Lie algebra L which is said to be associated with G . The corre- spondence extends to subalgebras: if H is a Lie subgroup of G , then its Lie algebra M is a subalgebra of L. Moreover, if H is an invariant subgroup, then M is an ideal in L , etc.On the other hand, the association G → L is not injective. B.3.3 Examples: (a) The group of rotations (translations on a circle) SO(2) can be expressed as SO(2) = T1/Z , where T1 means translations on a line and Z is the additive group of integers. Both SO(2) and T1 have the same (one–dimensional) Lie algebra so(2) . (b) Let Z2 be the two–point group {0, 1} with the addition modulo 2 . The Lie groups SU(2) and SO(3) = SU(2)/Z2 have the same Lie algebra so(3) as was discussed in Example 10.2.3e. Basic notions concerning the representation theory of Lie groups and algebras can be readily adapted from the preceding sections. The correspondence discussed above induces a natural relation between some representations of a Lie group and those of its Lie algebra; in the simplest case of a one–dimensional G this is the content of Stone’s theorem. In general, however, the representation theory of Lie groups and algebras is a complicated subject which goes beyond the scope of the present book; we refer the reader to the literature quoted at the beginning of the appendix. References a) Monographs, textbooks, proceedings:

[[Ad ]] R.A. Adams: Sobolev Spaces, Academic, New York 1975. [[AG ]] N.I. Akhiezer, I.M. Glazman: Theory of Linear Operators in Hilbert space,3rd edition, Viˇsa Skola,ˇ Kharkov 1978 (in Russian; English translation of the 1st edition: F. Ungar Co., New York 1961, 1963). [[ACH ]] S. Albeverio et al.,eds.:Feynman Path Integrals, Lecture Notes in Physics, vol. 106, Springer, Berlin 1979. [[AFHL]] S. Albeverio et al.,eds.:Ideas and Methods in Quantum and Statistical Physics. R. Høegh–Krohn Memorial Volume, Cambridge University Press, Cambridge 1992. [[ AGHH ]] S. Albeverio, F. Gesztesy, R. Høegh–Krohn, H. Holden: Solvable Models in Quantum Mechanics, 2nd edition, with an appendix by P. Exner, AMS Chelsea Publishing, Providence, RI, 2005 [[AH]] S. Albeverio, R. Høegh–Krohn: Mathematical Theory of Feynman Path Integrals, Lecture Notes in Mathematics, vol. 523, Springer, Berlin 1976. [[ Al ]] P.S. Alexandrov: Introduction to Set Theory and General Topology, Nauka, Moscow 1977 (in Russian). [[Am ]] W.O. Amrein: Non–Relativistic Quantum Dynamics, Reidel, Dordrecht 1981. [[AJS ]] W.O. Amrein, J.M. Jauch, K.B. Sinha: Scattering Theory in Quantum Mecha- nics. Physical Principles and Mathematical Methods, Benjamin, Reading, MA 1977. [[ A ZˇS]]ˇ M.A. Antonec, G.M. Zislin,ˇ I.A. Sereˇˇ sevskii: On the discrete spectrum of N–body Hamiltonians, an appendix to the Russian translation of the monograph [[JW]], Mir, Moscow 1976 (in Russian). [[ BaR ]] A.O. Barut, R. Raczka: Theory of Group Representations and Applications, 2nd edition, World Scientific, Singapore 1986. [[ BW ]] H. Baumg¨artel, M. Wollenberg: Mathematical Scattering Theory,Akademie Verlag, Berlin 1983. [[Ber ]] F.A. Berezin: The Second Quantization Method, 2nd edition, Nauka, Moscow 1986 (in Russian; English transl. of the 1st edition: Academic, New York 1966). [[ B e Sˇ ]] F.A. Berezin, M.A. Subin:ˇ Schr¨odinger Equation, Moscow State University Publ., Moscow 1983 (in Russian; English translation: Kluwer, Dordrecht 1996). [[BL ]] L.C. Biedenharn, J.D. Louck: Angular Momentum in Quantum Theory. Theory and Applications, Addison–Wesley, Reading, MA 1981. [[ B S ]] M . S.ˇ Birman, M.Z. Solomyak: Spectral Theory of Self-Adjoint Operators in Hilbert Space, Leningrad State University Lenninguad. 1980 (in Russian; English transla- tion: Kluwer, Dordrecht 1987).

617 618 References

[[BD 1,2]] J.D. Bjorken, S.D. Drell: Relativistic Quantum Theory, I. Relativistic Quan- tum Mechanics, II. Relativistic Quantum Fields, McGraw–Hill, New York 1965. [[Boe]] H. Boerner: Darstellungen von Gruppen, 2.Ausgabe, Springer, Berlin 1967 (Eng- lish translation: North-Holland, Amsterdam 1970). [[BLOT ]] N.N. Bogolyubov, A.A. Logunov, A.I. Oksak, I.T. Todorov: General Princi- ples of Quantum Field Theory, Nauka, Moscow 1987 (in Russian; a revised edition of Foundations of the Axiomatic Approach to Quantum Field Theory by the first two and the last author, Nauka, Moscow 1969; English translation: W.A. Benjamin, Reading, MA 1975, referred to as [[BLT ]]). [[ B Sˇ ]] N.N. Bogolyubov, D.V. Sirkov:ˇ An Introduction to the Theory of Quantized Fields, 4th edition, Nauka, Moscow 1984 (in Russian; English translation of the 3rd edition: Wiley–Interscience, New York 1980). [[ B o ]] D . B o h m : Quantum Theory, Prentice-Hall, New York 1952. [[ BR 1,2 ]] O. Bratelli, D.W. Robinson: Operator Algebras and Quantum Statistical Mechanics I, II, Springer, New York 1979, 1981. [[BLM ]] P. Busch, P.J. Lahti, P. Mittelstaedt: The Quantum Theory of Measurement, 2nd revised edition, Springer LNP m2, Berlin 1996. [[ChS ]] K. Chadan, P. Sabatier: Inverse Problems in Quantum Scattering Theory,2nd edition, Springer, New York 1989. [[ CL ]] Tai–Pei Cheng, Ling–Fong Li: Gauge Theory of Elementary Particle Physics, Clarendon Press, Oxford 1984. [[Cher ]] P.R. Chernoff: Product Formulas, Nonlinear Semigroups and Addition of Un- bounded Operators, Mem. Amer. Math. Soc., Providence, RI 1974. [[CFKS ]] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon: Schr¨odinger operators, with Application to Quantum Mechanics and Global Geometry, Springer, Berlin 1987; corrected and extended 2nd printing, Springer, Berlin 2007. [[Da 1 ]] E.B. Davies: Quantum Theory of Open Systems, Academic, London 1976. [[Da 2 ]] E.B. Davies: One–Parameter Semigroups, Academic, London 1980. [[ Dav ]] A.S. Davydov: Quantum Mechanics, 2nd edition, Nauka, Moscow 1973 (in Russian; English translation of the 1st edition: Pergamon Press, Oxford 1965). [[DO ]] Yu.N. Demkov, V.N. Ostrovskii: The Zero–Range Potential Method in Atomic Physics, Leningrad University Press, Leningrad 1975 (in Russian). [[DENZ]] M. Demuth, P. Exner, H. Neidhardt, V.A. Zagrebnov, eds.: Mathematical Re- sults in Quantum Mechanics, Operator Theory: Advances and Applications, vol. 70; Birkh¨auser, Basel 1994. [[DG]] J. Derezinski, Ch. Gerard: Scattering theory of classical and quantum N–particle systems, Texts and Monographs in Physics, Springer, Berlin 1997. [[Dir ]] P.A.M. Dirac: The Principles of Quantum Mechanics, 4th edition, Clarendon Press, Oxford 1969. [[ DE ]] J. Dittrich, P. Exner, eds.: Rigorous Results in Quantum Dynamics, World Scientific, Singapore 1991. [[DET]] J. Dittrich, P. Exner, M. Tater, eds.: Mathematical Results in Quantum Mecha- nics, Operator Theory: Advances and Appl., vol. 108; Birkh¨auser, Basel 1999. [[Di 1]] J. Dixmier: Les alg`ebres des op´erateurs dans l’espace hilbertien (alg`ebres de von Neumann), 2me edition, Gauthier–Villars, Paris 1969. References 619

[[Di 2 ]] J. Dixmier: Les C∗–alg`ebras and leur repr´esentations, 2me edition, Gauthier- Vilars, Paris 1969. [[DS 1–3]] N. Dunford, J.T. Schwartz: Linear Operators, I. General Theory, II. Spectral Theory, III. Spectral Operators, Interscience Publications, New York 1958, 1962, 1971. [[Edm ]] A.R. Edmonds: Angular Momentum in Quantum Mechanics, Princeton Uni- versity Press, Princeton, NJ 1957; a revised reprint Princeton 1996. [[EG ]] S.J.L. van Eijndhoven, J. de Graaf: A Mathematical Introduction to Dirac For- malism, North–Holland, Amsterdam 1986. [[ E m ]] G . G . E m c h : Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley–Interscience, New York 1972. [[ E x ]] P. E x n e r : Open Quantum Systems and Feynman Integrals, D. Reidel, Dordrecht 1985. [[EKKST ]] P. Exner, J. Keating, P. Kuchment, T. Sunada, A. Teplyaev, eds.: Analysis on Graphs and Applications, Proceedings of the Isaac Newton Institute programme “Analysis on Graphs and Applications”, AMS “Contemporary Mathematics” Series, 2008 [[EN ]] P. Exner, J. Neidhardt, eds.: Order, Disorder and Chaos in Quantum Systems, Operator Theory: Advances and Applications, vol. 46; Birkh¨auser, Basel 1990. [[ E Sˇ 1 ]] P. Exner, P. Seba,ˇ eds.: Applications of Self–Adjoint Extensions in Quantum Physics, Lecture Notes in Physics, vol. 324; Springer, Berlin 1989. [[ E Sˇ 2 ]] P. Exner, P. Seba,ˇ eds.: Schr¨odinger Operators, Standard and Non–standard, World Scientific, Singapore 1989. [[Fel 1,2 ]] W. Feller: An Introduction to Probability Theory and Its Applications I, II, 3rd and 2nd edition, resp., Wiley, New York 1968, 1971. [[ Fey ]] R.P. Feynman: Statistical Mechanics. A Set of Lectures, W.A. Benjamin, Reading, MA. 1972. [[FH ]] R.P. Feynman, A.R. Hibbs: Quantum Mechanics and Path Integrals, McGraw– Hill, New York 1965. [[ FG ]] L. Fonda, G.C. Ghirardi: Symmetry Principles in Quantum Physics, Marcel Dekker, New York 1970. [[ G Sˇ ]] I.M. Gel’fand, G.M. Silov:ˇ Generalized Functions and Operations upon Them, vol.I, 2nd edition, Fizmatgiz, Moscow 1959 (in Russian; English translation: Acad- emic, New York 1969). [[ Gl ]] I.M. Glazman: Direct Methods of Qualitative Analysis of Singular Differential Operators, Fizmatgiz, Moscow 1963 (in Russian). [[GJ ]] J. Glimm, A. Jaffe: Quantum Physics: A Functional Integral Point of View,2nd edition, Springer, New York 1987. [[ Gr ]] A. Grothendieck: Produits tensoriels topologiques et espaces nucl´eaires,Mem. Am. Math. Soc., vol. 16, Providence, RI 1955. [[Haa ]] R. Haag: Local Quantum Physics. Fields, Particles, Algebras, 2nd revised and enlarged edition Springer, Berlin 1996. [[Hal 1 ]] P. Halmos: Measure Theory, 2nd edition, Van Nostrand, New York 1973. [[Hal 2]] P. Halmos: A Hilbert Space Problem Book, Van Nostrand, Princeton, NJ 1967. [[ Ham ]] M. Hamermesh: Group Theory and Its Applications to Physical Problems, Addison-Wesley, Reading, MA. 1964. [[ HW ]] G.H. Hardy, E.M. Wright: An Introduction to the Theory of Numbers,5th edition, Oxford University Press, oxford 1979. 620 References

[[Hel ]] B. Helffer: Semi–Classical Analysis for the Schr¨odinger Operator and Applica- tions, Lecture Notes in Mathematics, vol. 1366, Springer, Berlin 1988. [[ HP ]] E. Hille, R.S. Phillips: Functional Analysis and Semigroups, Am. Math. Soc. Colloq. Publ., vol. 31, Providence, Rhode Island 1957; 3rd printing 1974. [[Hol]] A.S. Holevo: Probabilistic and Statistical Aspects of the Quantum Theory, Nauka, Moscow 1980 (in Russian; English translation: North-Holland, Amsterdam 1982). [[ Hor]] ¨ L. H¨ormander: The Analysis of Linear Partial Differential Operators III, Sprin- ger, Berlin 1985; corrected reprint 1994. [[ Hor ]] S.S. Horuˇzii: An Introduction to Algebraic Quantum Field Theory, Nauka, Moscow 1986 (in Russian; English translation: Kluwer, Dordrecht 1990). [[ v H ]] L . va n H o v e : Sur certaines repr´esentations unitaires d’un group infini des trans- formations, Memoires Acad. Royale de Belgique XXVI/6, Bruxelles 1951. [[Hua 1 ]] K. Huang: Statistical Mechanics, Wiley, New York 1963. [[ Hua 2 ]] K. Huang: Quarks, Leptons and Gauge Fields, World Scientific, Singapore 1982. [[Hur ]] N.E. Hurt: Geometric Quantization in Action, D. Reidel, Dordrecht 1983. [[IZ ]] C. Itzykson, J.–B. Zuber: Quantum Field Theory, McGraw–Hill, New York 1980. [[ J a r 1 ]] V . J a r n´ık: Differential Calculus II, 3rd edition, Academia, Prague 1976 (in Czech). [[ J a r 2 ]] V . J a r n´ık: Integral Calculus II, 2nd edition, Academia, Prague 1976 (in Czech). [[Ja ]] J.M. Jauch: Foundations of Quantum Mechanics, Addison-Wesley, Reading, MA 1968. [[Jor ]] T.F. Jordan: Linear Operators for Quantum Mechanics, Wiley, New York 1969. [[ J W ]] K . Jorgens, ¨ J. Weidmann: Spectral Properties of Hamiltonian Operators, Lecture Notes in Mathematics, vol. 313, Springer, Berlin 1973. [[Kam]] E. Kamke: Differentialgleichungen realer Funktionen, Akademische Verlagsges- selschaft, Leipzig 1956. [[ Kas ]] D. Kastler, ed.: C∗–algebras and Their Applications to Statistical Mechanics and Quantum Field Theory, North–Holland, Amsterdam 1976. [[ K a ]] T . K a t o : Perturbation Theory for Linear Operators, Springer, 2nd edition, Berlin 1976; reprinted in 1995. [[ Kel ]] J.L. Kelley: General Topology, Van Nostrand, Toronto 1957; reprinted by Springer, Graduate Texts in Mathematics, No. 27, New York 1975. [[ Kir ]] A.A. Kirillov: Elements of the Representation Theory, 2nd edition, Nauka, Moscow 1978 (in Russian; French translation: Mir, Moscow 1974). [[KGv ]] A.A. Kirillov, A.D. Gviˇsiani: Theorems and Problems of Functional Analysis, Nauka, Moscow 1979 (in Russian; French translation: Mir, Moscow 1982). [[KS ]] J.R. Klauder, B.–S. Skagerstam, eds.: Coherent States. Applications in Physics and Mathematical Physics, World Scientific, Singapore 1985. [[Kli ]] W. Klingenberg: A Course in Differential Geometry, Springer, New York 1978. [[ KF ]] A.N. Kolmogorov, S.V. Fomin: Elements of Function Theory and Functional Analysis, 4th edition, Nauka, Moscow 1976 (in Russian; English translation of the 2nd ed.: Graylock 1961; French translation of the 3rd edition: Mir, Moscow 1974). [[Kuo ]] H.-H. Kuo: Gaussian Measures in Banach Spaces, Lecture Notes in Mathema- tics, vol. 463, Springer, Berlin 1975. [[Ku ]] A.G. Kuroˇs: Lectures on General Algebra, 2nd edition, Nauka, Moscow 1973 (in Russian). References 621

[[LL ]] L.D. Landau, E.M. Lifˇsic: Quantum Mechanics. Nonrelativistic Theory, 3rd ed., Nauka, Moscow 1974 (in Russian; English translation: Pergamon, New York 1974). [[ LP ]] P.D. Lax, R.S. Phillips: Scattering Theory, Academic, New York 1967; 2nd edition, with appendices by C.S. Morawetz and G. Schmidt, 1989. [[ LiL ]] E.H. Lieb, M. Loss: Analysis, 2nd edition, Graduate Studies in Mathematics, vol. 14, AMS, Providence, RI, 2001 [[LSW ]] E.H. Lieb, B. Simon, A.S. Wightman, eds.: Studies in Mathematical Physics. Essays in Honor of V. Bargmann, Princeton University Press, Pinceton, NJ 1976. [[Loe 1–3 ]] E.M. Loebl, ed.: Group Theory and Its Applications I–III, Academic, New York 1967. [[LCM ]] J.T. Londergan, J.P. Carini, D.P. Murdock: Binding and Scattering in Two- Dimensional Systems. Applications to Quantum Wires, Waveguides and Photonic Crystals, Springer LNP m60, Berlin 1999. [[ L u d 1 , 2 ]] G . L u d w i g : Foundations of Quantum Mechanics I, II, Springer, Berlin 1983, 1985. [[ L u d 3 ]] G . L u d w i g : An Axiomatic Basis for Quantum Mechanics, I. Derivation of Hilbert Space Structure, Springer, Berlin 1985. [[LRSS ]] S. Lundquist, A. Ranfagni, Y. Sa–yakanit, L.S. Schulman: Path Summation: Achievements and Goals, World Scientific, Singapore 1988. [[ Mac 1 ]] G. Mackey: Mathematical Foundations of Quantum Mechanics, Benjamin, New York 1963; reprinted by Dover Publications INC., Mineola, NY, 2004. [[ Mac 2 ]] G. Mackey: Induced Representations of Groups and Quantum Mechanics, W.A. Benjamin, New York 1968. [[MB ]] S. MacLane, G. Birkhoff: Algebra, 2nd edition, Macmillan, New York 1979. [[Mar ]] A.I. Markuˇseviˇc: A Short Course on the Theory of Entire Functions, 2nd edition, Fizmatgiz, Moscow 1961 (in Russian; English translation: Amer. Elsevier 1966). [[ Mas ]] V.P. Maslov: Perturbation Theory and Asymptotic Methods, Moscow State University Publ., Moscow 1965 (in Russian; French translation: Dunod, Paris 1972). [[ MF ]] V.P. Maslov, M.V. Fedoryuk: Semiclassical Approximation to Quantum Me- chanical Equations, Nauka, Moscow 1976 (in Russian). [[ Mau ]] K. Maurin: Hilbert Space Methods, PWN, Warsaw 1959 (in Polish; English translation: Polish Sci. Publ., Warsaw 1972). [[Mes]] A. Messiah: M´ecanique quantique I, II, Dunod, Paris 1959 (English translation: North–Holland, Amsterdam 1961, 1963). [[ Mul ¨ ]] C. M¨uller: Spectral Harmonics, Lecture Notes In Mathematics, vol. 17, Springer, Berlin 1966. [[Nai 1]] M.A. Naimark: Normed Rings, 2nd edition, Nauka, Moscow 1968 (in Russian; English translation: Wolters–Noordhoff, Groningen 1972). [[Nai 2]] M.A. Naimark: Linear Differential Operators, 2nd edition, Nauka, Moscow 1969 (in Russian; English translation of the 1st edition: Harrap & Co., London 1967, 1968). [[Nai 3]] M.A. Naimark: Group Representation Theory, Nauka, Moscow 1976 (in Russian; French translation: Mir, Moscow 1979). [[ v N ]] J . v o n N e u m a n n : Mathematische Grundlagen der Quantenmechanik, Springer Verlag, Berlin 1932 (English translation: Princeton University Press, Princeton, NJ 1955, reprinted 1996). [[New ]] R.G. Newton: Scattering Theory of Waves and Particles, 2nd edition, Springer Verlag, New York 1982; reprinted by Dover Publications Inc., Mineola, NY 2002. 622 References

[[OK ]] Y. Ohnuki, S. Kamefuchi: Quantum Field Theory and Parastatistics, Springer, Heidelberg 1982. [[Par ]] K.R. Parthasarathy: Introduction to Probability and Measure, New Delhi 1980. [[Pea ]] D.B. Pearson: Quantum Scattering and Spectral Theory, Techniques of Physics, vol. 9, Academic, London 1988. [[Pe ]] P. Perry; Scattering Theory by the Enss Method, Harwood, London 1983. [[Pir ]] C. Piron: Foundations of Quantum Physics, W.A. Benjamin, Reading, MA 1976. [[Pon]] L.S.Pontryagin:ContinuousGroups,3rdedition,Nauka,Moscow1973(inRussian; English translation of the 2nd edition: Gordon and Breach, New York 1966). [[ P o p ]] V . S . P o p o v : Path Integrals in Quantum Theory and Statistical Physics, Atom- izdat, Moscow 1976 (in Russian; English translation: D. Reidel, Dordrecht 1983). [[ Pru ]] E. Prugoveˇcki: Quantum Mechanics in Hilbert Space, 2nd edition, Academic, New York 1981. [[RS 1–4]] M. Reed, B. Simon: Methods of Modern Mathematical Physics, I. Functional Analysis, II. Fourier Analysis. Self–Adjointness, III. Scattering Theory, IV. Ana- lysis of Operators, Academic, New York 1972–79. [[Ri 1 ]] D.R. Richtmyer: Principles of Advanced Mathematical Physics I, Springer, New York 1978. [[ RN ]] F. Riesz, B. Sz.–Nagy: Lecons d’analyse fonctionelle, 6me edition, Akademic Kiad´o, Budapest 1972. [[Ru 1 ]] W. Rudin: Real and Complex Analysis, 3rd edition, McGraw–Hill, New York 1987. [[Ru 2 ]] W. Rudin: Functional Analysis, 2nd edition, McGraw–Hill, New York 1991. [[ Sab]]ˇ B.V. Sabat:ˇ Introduction to Complex Analysis, Nauka, Moscow 1969 (in Russ.). [[Sa ]] S. Sakai: C∗–Algebras and W ∗–Algebras, Springer, Berlin 1971. [[Sch]] R.Schatten:ATheoryofCrossSpaces,PrincetonUniversityPress,Princeton,1950. [[Sche ]] M. Schechter: Operator Methods in Quantum Mechanics, North–Holland, New York 1981; reprinted by Dover Publications Inc., Mineola, NY 2002. [[ Schm ]] K. Schm¨udgen: Unbounded Operator Algebras and Representation Theory, Akademie–Verlag, Berlin 1990. [[Schu ]] L.S. Schulman: Techniques and Applications of Path Integration, Wiley–Inter- science, New York 1981. [[Schw 1 ]] L. Schwartz: Th´eorie des distributions I, II, Hermann, Paris 1957, 1959. [[Schw 2 ]] L. Schwartz: Analyse Math´ematique I, II, Hermann, Paris 1967. [[Schwe ]] S.S. Schweber: An Introduction to Relativistic Quantum Field Theory,Row, Peterson & Co., Evanston, IL 1961. [[ Seg ]] I.E. Segal: Mathematical Problems of Relativistic Physics, with an appendix by G.W. Mackey, Lectures in Appl. Math., vol. 2, American Math. Society, Providence, RI 1963. [[Sei ]] E. Seiler: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Physics, vol. 159, Springer, Berlin 1982. [[Si 1 ]] B. Simon: Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, NJ 1971. [[Si 2]] B. Simon: The P (Φ)2 Euclidian (Quantum) Field Theory, Princeton University Press, Princeton, NJ 1974. References 623

[[ Si 3 ]] B. Simon: Trace Ideals and Their Applications, Cambridge University Press, Cambridge 1979. [[Si 4 ]] B. Simon: Functional Integration and Quantum Physics, Academic, New York 1979; 2nd edition, AMS Chelsea Publishing, Providence, RI 2005. [[ Sin ]] Ya.G. Sinai: Theory of Phase Transitions. Rigorous Results, Nauka, Moscow 1980 (in Russian). [[ Sirˇ ]] A.N. Siryaev:ˇ Probability, Nauka, Moscow 1980 (in Russian). [[ SF ]] A.A. Slavnov, L.D. Faddeev: Introduction to the Quantum Theory of Gauge Fields, Nauka, Moscow 1978 (in Russian). [[ Stock ]] H.-J. St¨ockmann: Quantum chaos. An introduction, Cambridge University Press, Cambridge, 1999. [[Stol]] P. Stollmann: Caught by Disorder: Bound States in Random Media, Birkh¨auser, Basel 2001. [[Sto ]] M.H. Stone: Linear Transformations in Hilbert Space and Their Applications to Analysis, Amer. Math. Colloq. Publ., vol. 15, New York 1932. [[ Str ]] R.F. Streater, ed.: Mathematics of Contemporary Physics, Academic, London 1972. [[SW]] R.F. Streater, A. Wightman: PCT, Spin, Statistics and All That, W.A. Benjamin, New York 1964. [[ Svˇ ]] A.S. Svarc:ˇ Mathematical Foundations of the Quantum Theory, Atomizdat, Moscow 1975 (in Russian). [[Tay ]] A.E. Taylor: Introduction to Functional Analysis, 6th edition, Wiley, New York 1967. [[Ta ]] J.R. Taylor: Scattering Theory. The Quantum Theory of Nonrelativistic Colli- sions, Wiley, New York 1972. [[ Te ]] G . Te s c h l : Jacobi operators and completely integrable nonlinear lattices, Math. Surveys and Monographs, vol. 72; AMS, Providence, RI 2000. [[Tha ]] B. Thaller: The Dirac Equation, Springer, Berlin 1992. [[ Thi 3,4 ]] W. Thirring: A Course in Mathematical Physics, 3. Quantum Mechanics of Atoms and Molecules, 4. Quantum Mechanics of Large Systems, Springer, New York 1981, 1983. [[ Ti ]] V.M. Tikhomirov: Banach Algebras, an appendix to the monograph [[ KF ]], pp. 513–528 (in Russian). [[Vai ]] B.R. Vainberg: Asymptotic methods in the Equations of Mathematical Physics, Moscow State University Publishing, Moscow 1982 (in Russian). [[ Var 1,2 ]] V.S. Varadarajan: Geometry of Quantum Theory I, II, Van Nostrand Reinhold, New York 1968, 1970. [[Vot ]] V. Votruba: Foundations of the Special Theory of Relativity, Academia, Prague 1969 (in Czech). [[ WEG ]] R. Weder, P. Exner, B. Grebert, eds.: Mathematical Results in Quantum Mechanics, Contemporary Mathematics, vol. 307, AMS, Providence, RI 2002. [[ We ]] J . We i d m a n n : Linear Operators in Hilbert Space, Springer, Heidelberg 1980 (2nd edition in German: Lineare Operatoren in Hilbertr¨aumen, I. Grundlagen, II. Anwendungen, B.G. Teubner, Stuttgart 2000, 2003). [[Wig ]] E.P. Wigner: Symmetries and Reflections, Indiana University Press, Blooming- ton, IN 1971. [[ Ya ]] D . R . Ya f a e v : Mathematical Scattering Theory: General Theory, Transl. Math. Monographs, vol. 105; AMS, Providence, RI, 1992 624 References

[[ Yo ]] K . Yo s i d a : Functional Analysis, 3rd edition, Springer, Berlin 1971. [[ Zel]]ˇ D.P. Zelobenko:ˇ Compact Lie Groups and Their Representations, Nauka, Moscow 1970 (in Russian). b) Research and review papers:

[ACFGH 1] A. Abrams, J. Cantarella, J.G. Fu, M. Ghomi, R. Howard: Circles minimize most knot energies, Topology 42 (2003), 381–394. [ Adl 1 ] S.L. Adler: Quaternionic quantum field theory, Commun. Math. Phys. 104 (1986), 611–656. [ AD 1 ] D. Aerts, I. Daubechies: Physical justification for using the tensor product to describe two quantum systems as one joint system, Helv.Phys.Acta51 (1978), 661–675. [ AD 2 ] D. Aerts, I. Daubechies: A mathematical condition for a sublattice of a propo- sitional system to represent a physical subsystem, with a physical interpretation, Lett. Math. Phys. 3 (1979), 19–27. [ AF 1 ] J. Agler, J. Froese: Existence of Stark ladder resonances, Commun. Math. Phys. 100 (1985), 161–172. [ AHS 1 ] S. Agmon, I. Herbst, E. Skibsted: Perturbation of embedded eigenvalues in the generalized N–body problem, Commun. Math. Phys. 122 (1989), 411–438. [ AC 1 ] J. Aguilar, J.–M. Combes: A class of analytic perturbations for one–body Schr¨odinger Hamiltonians, Commun. Math. Phys. 22 (1971), 269–279. [ AAAS 1 ] E. Akkermans, A. Auerbach, J. Avron, B. Shapiro: Relation between per- sistent currents and the scattering matrix, Phys. Rev. Lett. 66 (1991), 76–79. [ ADH 1 ] S. Alama, P. Deift, R. Hempel: Eigenvalue branches of Schr¨odinger operator H −λW in a gap of σ(H), Commun. Math. Phys. 121 (1989), 291–321. [ Al 1 ] S. Albeverio: On bound states in the continuum of N–body systems and the virial theorem, Ann. Phys. 71 (1972), 167–276. [ ACF 1 ] S. Albeverio, C. Cacciapuoti, D. Finco: Coupling in the singular limit of thin quantum waveguides, J. Math. Phys. 48 (2007), 032103. [ AH 1 ] S. Albeverio, R. Høegh–Krohn: Oscillatory integrals and the method of statio- nary phase, Inventiones Math. 40 (1977), 59–106. [ AN 1 ] S. Albeverio, L. Nizhnik: Approximation of general zero-range potentials, Ukrainian Math. J. 52 (2000), 582–589. [ AmC 1 ] W.O. Amrein, M.B. Cibils: Global and Eisenbud–Wigner time delay in scattering theory, Helv. Phys. Acta 60 (1987), 481–500. [ AG 1 ] W.O. Amrein, V. Georgescu: Bound states and scattering states in quantum mechanics, Helv. Phys. Acta 46 (1973), 635–658. [ And 1 ] J. Anderson: Extensions, restrictions and representations of C∗–algebras, Trans. Am. Math. Soc. 249 (1979), 303–329. [ AGS 1 ] J.–P. Antoine, F. Gesztesy, J. Shabani: Exactly solvable models of sphere interactions in quantum mechanics, J. Phys. A20 (1987), 3687–3712. [ AIT 1 ] J.–P. Antoine, A. Inoue, C. Trapani: Partial ∗–algebras of closed operators, I. Basic theory and the Abelian case, Publ. RIMS 26 (1990), 359–395. References 625

[ AK 1 ] J.–P. Antoine, W. Karwowski: Partial ∗–algebras of Hilbert space operators, in Proceedings of the 2nd Conference on Operator Algebras, Ideals and Their Appli- cations in Theoretical Physics (H. Baumg¨artel et al., eds.), Teubner, Leipzig 1984; pp. 29–39. [ AKa 1 ] A. Arai, H. Kawano: Enhanced binding in a general class of quantum field models, Rev. Math. Phys. 15 (2003), 387–423. [ Ara 1 ] H. Araki: Type of von Neumann algebra associated with free field, Progr. Theor. Phys. 32 (1964), 956–965. [ Ara 2 ] H. Araki: C∗–approach in quantum field theory, Physica Scripta 24 (1981), 981–985. [AJ 1] H. Araki, J.–P. Jurzak: On a certain class of ∗–algebras of unbounded operators, Publ. RIMS 18 (1982), 1013–1044. [ AMKG 1 ] H. Araki, Y. Munakata, M. Kawaguchi, T. Goto: Quantum Field Theory of Unstable Particles, Progr. Theor. Phys. 17 (1957), 419–442. [ArJ 1] D. Arnal, J.–P. Jurzak: Topological aspects of algebras of unbounded operators, J. Funct. Anal. 24 (1977), 397–425. [ ADE 1 ] J. Asch, P. Duclos, P. Exner: Stability of driven systems with growing gaps. Quantum rings and Wannier ladders, J. Stat. Phys. 92 (1998), 1053–1069. [AB 1] M.S. Ashbaugh, R.D. Benguria: Optimal bounds of ratios of eigenvalues of one– dimensional Schr¨odinger operators with Dirichlet boundary conditions and positive potentials, Commun. Math. Phys. 124 (1989), 403–415. [ AB 2 ] M.S. Ashbaugh, R.D. Benguria: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. Math. 135 (1992), 601–628. [AE 1] M.S. Ashbaugh, P. Exner: Lower bounds to bound state energies in bent tubes, Phys. Lett. A150 (1990), 183–186. [AsH 1] M.S. Ashbaugh, E.M. Harrell II : Perturbation theory for resonances and large barrier potentials, Commun. Math. Phys. 83 (1982), 151–170. [ AsHS 1 ] M.S. Ashbaugh, E.M. Harrell II, R. Svirsky: On the minimal and maximal eigenvalue gaps and their causes, Pacific J. Math. 147 (1991), 1–24. [ AS 1 ] M.S. Ashbaugh, C. Sundberg: An improved stability result for resonances, Trans. Am. Math. Soc. 281 (1984), 347–360. [ AJI 1 ] A. Avila, S. Jitomirskaya: Solving the ten Martini problem, Lecture Notes in Physics 690 (2006), 5–16. [ ABGM 1 ] Y. Avishai, D. Bessis, B.M. Giraud, G. Mantica: Quantum bound states in open geometries, Phys. Rev. B44 (1991), 8028–8034. [ Avr 1 ] J. Avron: The lifetime of Wannier ladder states, Ann. Phys. 143 (1982), 33–53. [ AEL 1 ] J. Avron, P. Exner, Y. Last: Periodic Schr¨odinger operators with large gaps and Wannier–Stark ladders, Phys. Rev. Lett. 72 (1994), 896–899. [ ARZ 1 ] J. Avron, A. Raveh, B. Zur: Adiabatic quantum transport in multiply con- nected systems, Rev. Mod. Phys. 60 (1988), 873–915. [ ASY 1 ] J. Avron, R. Seiler, L.G. Yaffe: Adiabatic theorem and application to the quantum Hall effect, Commun. Math. Phys. 110 (1987), 33–49. [BB 1] D. Babbitt, E. Balslev: Local distortion techniques and unitarity of the S–matrix for the 2–body problem, J. Math. Anal. Appl. 54 (1976), 316–349. [BFS1] V.Bach,J.Fr¨ohlich, I.M. Sigal: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field, Commun. Math. Phys. 207 (1999), 249–290. 626 References

[ Ba 1 ] V. Bargmann: On a Hilbert space of analytical functions and an associate integral transform I, II, Commun. Pure Appl. Math. 14 (1961), 187–214; 20 (1967), 1–101. [ Ba 2 ] V. Bargmann: Remarks on Hilbert space of analytical functions, Proc. Natl. Acad. Sci. USA 48 (1962), 199–204, 2204. [Ba 3] V. Bargmann: On unitary ray representations of continuous groups, Ann. Math. 59 (1954), 1–46. [ Ba 4 ] V. Bargmann: Note on some integral inequalities, Helv. Phys. Acta 45 (1972), 249–257. [ Ba 5 ] V. Bargmann: On the number of bound states in a central field of forces, Proc. Natl. Acad. Sci. USA 38 (1952), 961–966. [BFF1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer: Deformation theory and quantization I, II, Ann. Phys. 111 (1978), 61–110, 111–151. [Bau 1] H. Baumg¨artel: Partial resolvent and spectral concentration, Math. Nachr. 69 (1975), 107–121. [ BD 1 ] H. Baumg¨artel, M. Demuth: Perturbation of unstable eigenvalues of finite multiplicity, J. Funct. Anal. 22 (1976), 187–203. [ BDW 1 ] H. Baumg¨artel, M. Demuth, M. Wollenberg: On the equality of resonances (poles of the scattering amplitude) and virtual poles, M. Nachr. 86 (1978), 167–174. [ Beh 1 ] H. Behncke: The Dirac equation with an anomalous magnetic moment, Math. Z. 174 (1980), 213–225. [BF 1] F.A. Berezin, L.D. Faddeev: A remark on Schr¨odinger’s equation with a singular potential, Sov. Acad. Sci. Doklady 137 (1961), 1011–1014 (in Russian). [vdB 1] M. van den Berg: On the spectral counting function for the Dirichlet Laplacian, J. Funct. Anal. 107 (1992), 352–361. [ Be 1 ] M.V. Berry: Quantal phase factor accompanying adiabatic changes, Proc. Roy. Soc. London A392 (1984), 45–57. [ Be 2 ] M.V. Berry: The adiabatic limit and the semiclassical limit, J. Phys. A17 (1984), 1225–1233. [ BM 1 ] M.V. Berry, K.E. Mount: Semiclassical approximations in wave mechanics, Rep. Progr. Phys. 35 (1972), 315–389. [ BN 1 ] A. Beskow, J. Nilsson: The concept of wave function and the irreducible re- presentations of the Poincar´e group, II. Unstable systems and the exponential decay law, Arkiv f¨or Physik 34 (1967), 561–569. [ BvN 1 ] G. Birkhoff, J. von Neumann: The logic of quantum mechanics, Ann. Math. 37 (1936), 823–843. [Bir1] M.S.ˇ Birman: On the spectrum of singular boundary problems, Mat. Sbornik 55 (1961), 125–174 (in Russian). [Bir2] M.S.ˇ Birman: Existence conditions for the wave operators, Doklady Acad. Sci. USSR 143 (1962), 506–509 (in Russian). [Bir3] M.S.ˇ Birman: An existence criterion for the wave operators. Izvestiya Acad. Sci. USSR, ser. mat. 27 (1963), 883–906 (in Russian). [Bir4] M.S.ˇ Birman: A local existence criterion for the wave operators, Izvestiya Acad. Sci. USSR, ser. mat. 32 (1968), 914–942 (in Russian). References 627

[BE 1] J. Blank, P. Exner: Remarks on tensor products and their applications in quan- tum theory, I. General considerations, II. Spectral properties, Acta Univ. Carolinae, Math. Phys. 17 (1976), 75–89; 18 (1977), 3–35. [BEH 1] J. Blank, P. Exner, M. Havl´ıˇcek: Quantum–mechanical pseudo–Hamiltonians, Czech. J. Phys. B29 (1979), 1325–1341. [ BGS 1 ] R. Blankenbecler, M.L. Goldberger, B. Simon: The bound states of weakly coupled long–range one–dimensional Hamiltonians, Ann. Phys. 108 (1977), 69–78. [BChK1] D.Boll´e, K. Chadan, G. Karner: On a sufficient condition for the existence of N–particle bound states, J. Phys. A19 (1986), 2337–2343. [ Bor 1 ] H.J. Borchers: Algebras of unbounded operators in quantum field theory, Physica 124A (1984), 127–144. [ BoE 1 ] D. Borisov, P. Exner: Exponential splitting of bound states in a waveguide with a pair of distant windows, J. Phys. A37 (2004), 3411–3428. [ BEG 1 ] D. Borisov, P. Exner, R. Gadyl’shin: Geometric coupling thresholds in a two-dimensional strip, J. Math. Phys. 43 (2002), 6265–6278. [ BEGK 1 ] D. Borisov, P. Exner, R. Gadyl’shin, D. Krejˇciˇr´ık: Bound states in weakly deformed strips and layers, Ann. H. Poincar´e 2 (2001), 553–572. [ BCGH 1 ] F. Boudjeedaa, L. Chetouani, L. Guechi, T.F. Hamman: Path integral treatment for a screened potential, J. Math. Phys. 32 (1991), 441–446. [BEKSˇ 1 ] J. Brasche, P. Exner, Yu. Kuperin, P. Seba:ˇ Schr¨odinger operators with singular interactions, J. Math. Anal. Appl. 184 (1994), 112–139. [ BrT 1 ] J. Brasche, A. Teta: Spectral analysis and scattering theory for Schr¨odinger operators with an interaction supported by regular curve, in the proceedings volume [[AFHL ]], pp. 197–211. [EBG 1] J. Br¨uning, P. Exner, V. Geyler: Large gaps in point-coupled periodic systems of manifolds, J. Phys. A: Math. Gen. 36 (2003), 4875–4890. [BG1] J.Br¨uning, V.A. Geyler: Scattering on compact manifolds with infinitely thin horns, J. Math. Phys. 44 (2003), 371–405. [BGMP1] J.Br¨uning, V.A. Geyler, V.A. Margulis, M.A. Pyataev: Ballistic conduc- tance of a quantum sphere, J. Phys. A35 (2002), 4239–4247. [BGP1] J.Br¨uning, V. Geyler, K. Pankrashkin: Cantor and band spectra for periodic quantum graphs with magnetic fields, Commun. Math. Phys. 269 (2007), 87–105. [ BFS 1 ] D.C. Brydges, J. Fr¨ohlich, A. Sokal: A new proof of existence and nontrivi- 4 4 ality of the continuum Φ2 and Φ3 quantum field theories, Commun. Math. Phys. 91 (1983), 141–186. [ BEPS 1 ] E.N. Bulgakov, P. Exner, K.N. Pichugin, A.F. Sadreev: Multiple bound states in scissor-shaped waveguides, Phys. Rev. B66 (2002), 155109 [ BGRS 1 ] W. Bulla, F. Gesztesy, W. Renger, B. Simon: Weakly coupled bound states in quantum waveguides, Proc. Amer. Math. Soc. 127 (1997), 1487–1495. [ BT 1 ] W. Bulla, T. Trenkler: The free Dirac operator on compact and non–compact graphs, J. Math. Phys. 31 (1990), 1157–1163. [BW 1] L.J. Bunce, J.D.M. Wright: Quantum measures and states on Jordan algebras, Commun. Math. Phys. 98 (1985), 187–202. [ BuD 1 ] V.S. Buslaev, L. Dmitrieva: Bloch electrons in an external electric field, Leningrad Math. J. 1 (1991), 287–320. 628 References

[B¨u1] M.B¨uttiker: Small normal–metal loop coupled to an electron reservoir, Phys. Rev. B32 (1985), 1846–1849. [B¨u2] M.B¨uttiker: Absence of backscattering in the quantum Hall effect in multiprobe conductors, Phys. Rev. B38 (1988), 9375–9389. [ CaE 1 ] C. Cacciapuoti, P. Exner: Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide, J. Phys. A40 (2007), F511–F523. [Ca 1] J.W. Calkin: Two–sided ideals and congruence in the ring of bounded operators in Hilbert space, Ann. Math. 42 (1941), 839–873. [ Cal 1 ] F. Calogero: Upper and lower limits for the number of bound states in a given central potential, Commun. Math. Phys. 1 (1965), 80–88. [ Cam 1 ] R.H. Cameron: A family of integrals serving to connect the Wiener Feynman integrals, J. Math. Phys. 39 (1961), 126–141. [ Cam 2 ] R.H. Cameron: The Ilstow and Feynman integrals, J d’Analyse Math. 10 (1962–63), 287–361. [ Cam 3 ] R.H. Cameron: Approximation to certain Feynman integrals, J. d’Analyse Math. 21 (1968), 337–371. [ CLM 1 ] J.P. Carini, J.T. Londergan, K. Mullen, D.P. Murdock: Bound states and resonances in quantum wires, Phys. Rev. B46 (1992), 15538–15541. [CLM 2] J.P. Carini, J.T. Londergan, K. Mullen, D.P. Murdock: Multiple bound states in sharply bent waveguides, Phys. Rev. B48 (1993), 4503–4514. [CLMTY 1,2] J.P. Carini, J.T. Londergan, D.P. Murdock, D. Trinkle, C.S. Yung: Bound states in waveguides and bent quantum wires, I. Applications to waveguide systems, II. Electrons in quantum wires, Phys. Rev. B55 (1997), 9842–9851, 9852–9859. [ CMS 1 ] R. Carmona, W.Ch. Masters, B. Simon: Relativistic Schr¨odinger operators, J. Funct. Anal. 91 (1990), 117–142. [ CEK 1 ] G. Carron, P. Exner, D. Krejˇciˇr´ık: Topologically non-trivial quantum layers, J. Math. Phys. 45 (2004), 774–784. [ CG 1 ] G. Casati, I. Guarneri: Non–recurrent behaviour in quantum dynamics, Com- mun. Math. Phys. 95 (1984), 121–127. [CM 1] D. Castrigiano, U. Mutze: On the commutant of an irreducible set of operators in a real Hilbert space, J. Math. Phys. 26 (1985), 1107–1110. [ CEH 1 ] I. Catto, P. Exner, Ch. Hainzl: Enhanced binding revisited for a spinless particle in non-relativistic QED, J. Math. Phys. 45 (2004), 4174–4185. [ChD 1] K. Chadan, Ch. DeMol: Sufficient conditions for the existence of bound states in a potential without a spherical symmetry, Ann. Phys. 129 (1980), 466–478. [ CRWP 1 ] V. Chandrasekhar, M.J. Rooks, S. Wind, D.E. Prober: Observation of Aharonov-Bohm electron interference effects with periods h/e and h/2e in individual micron-size, normal-metal rings, Phys. Rev. Lett. 55 (1985), 1610–1613. [ CVV 1 ] T. Chen, V. Vougalter, S. Vugalter: The increase of binding energy and enhanced binding in nonrelativistic QED, J. Math. Phys. 44 (2003), 1961–1970. [ CDFK 1 ] P. Chenaud, P. Duclos, P. Freitas, D. Krejˇciˇr´ık: Geometrically induced discrete spectrum in curved tubes, Diff. Geom. Appl. 23 (2005), 95–105. [ CE 1 ] T. Cheon, P. Exner: An approximation to δ couplings on graphs, J. Phys. A37 (2004), L329–335. [ CS 1 ] T. Cheon, T. Shigehara: Realizing discontinuous wave functions with renor- malized short-range potentials, Phys. Lett. A243 (1998), 111–116. References 629

[ Che 1 ] S. Cheremshantsev: Hamiltonians with zero–range interactions supported by a Brownian path, Ann. Inst. H. Poincar´e: Phys. Th´eor. 56 (1992), 1–25. [ Cher 1 ] P.R. Chernoff: A note on product formulas for operators, J. Funct. Anal. 2 (1968), 238–242. [ ChH 1 ] P.R. Chernoff, R.Hughes: A new class of point interactions in one dimension, J. Funct. Anal. 111 (1993), 97–117. [ CH 1 ] L. Chetouani, F.F. Hamman: Coulomb’s Green function in an n–dimensional Euclidean space, J. Math. Phys. 27 (1986), 2944–2948. [ CCH 1 ] L. Chetouani, A. Chouchaoui, T.F. Hamman: Path integral solution for the Coulomb plus sector potential, Phys. Lett. A161 (1991), 87–97. [ CdV 1 ] Y. Colin de Verdi`ere: Bohr–Sommerfeld rules to all orders, Ann. H. Poincar´e 6 (2005), 925–936. [CRRS 1] Ph. Combe, G. Rideau, R. Rodriguez, M. Sirugue–Collin: On the cylindrical approximation to certain Feynman integrals, Rep. Math. Phys. 13 (1978), 279–294. [ CDS 1 ] J.–M. Combes, P. Duclos, R. Seiler: Krein’s formula and one–dimensional multiple well, J. Funct. Anal. 52 (1983), 257–301. [ CDKS 1 ] J.–M. Combes, P. Duclos, M. Klein, R. Seiler: The shape resonance, Com- mun. Math. Phys. 110 (1987), 215–236. [ CoH 1 ] J.–M. Combes, P.D. Hislop: Stark ladder resonances for small electric fields, Commun. Math. Phys. 140 (1991), 291–320. [ Com 1 ] M. Combescure: Spectral properties of periodically kicked quantum Hamil- tonians, J. Stat. Phys. 59 (1990), 679–690. [ Co 1 ] J. Conlon: The ground state of a Bose gas with Coulomb interaction I, II, Commun. Math. Phys. 100 (1985), 355–379; 108 (1987), 363–374. [ CLY 1 ] J.G. Conlon, E.H. Lieb, H.–T. Yau: The N 7/5 law for charged bosons, Commun. Math. Phys. 116 (1988), 417–488. [ Con 1 ] A. Connes: The Tomita–Takesaki theory and classification of type III factors, in the proceedings volume [[Kas ]], pp. 29–46. [ Coo 1 ] J.M. Cook: The mathematics of second quantization, Trans. Am. Math. Soc. 74 (1953), 222–245. [Coo 2] J.M. Cook: Convergence of the Møller wave matrix, J. Math. Phys. 36 (1957), 82–87. [ CJM 1 ] H. Cornean, A. Jensen, V. Moldoveanu: A rigorous proof for the Landauer- B¨uttiker formula, J. Math. Phys. 46 (2005), 042106. [ CMF 1 ] F.A.B. Coutinho, C.P. Malta, J. Fernando Perez: Sufficient conditions for the existence of bound states of N particles with attractive potentials, Phys. Lett. 100A (1984), 460–462. [Chri 1] E. Christensen: Measures on projections and physical states, Commun. Math. Phys. 86 (1982), 529–538. [ Cra 1 ] R.E. Crandall: Exact propagator for reflectionless potentials, J. Phys. A16 (1983), 3005–3011. [ Cwi 1 ] M. Cwikel: Weak type estimates for singular values and the number of bound states of Schr¨odinger operators, Ann. Math. 106 (1977), 93–100. [ Cy 1 ] H.L. Cycon: Resonances defined by modified dilations, Helv. Phys. Acta 58 (1985), 969–981. 630 References

[ DL 1 ] I. Daubechies, E.H. Lieb: One electron relativistic molecule with Coulomb interaction, Commun. Math. Phys. 90 (1983), 497–510. [ DAT 1 ] G. Dell’Antonio, L. Tenuta: Quantum graphs as holonomic constraints, J. Math. Phys. 47 (2006), 072102. [ Dem 1 ] M. Demuth: Pole approximation and spectral concentration, Math. Nachr. 73 (1976), 65–72. [ Der 1 ] J. Derezy´nski: A new proof of the propagation theorem for N-body quantum systems, Commun. Math. Phys. 122 (1989), 203–231. [ Der 2 ] J. Derezy´nski: Algebraic approach to the N–body long–range scattering, Rep. Math. Phys. 3 (1991), 1–62. [ DLSS 1 ] C. DeWitt, S.G. Low, L.S. Schulman, A.Y. Shiekh: Wedges I, Found. Phys. 16 (1986), 311–349. [ DeW 1 ] C. DeWitt–Morette: The semiclassical expansion, Ann. Phys. 97 (1976), 367–399; 101, 682–683. [ DMN 1 ] C. DeWitt–Morette, A. Maheswari, B. Nelson: Path–integration in nonrela- tivistic quantum mechanics, Phys. Rep. 50 (1979), 255–372. [ DE 1 ] J. Dittrich, P. Exner: Tunneling through a singular potential barrier, J. Math. Phys. 26 (1985), 2000–2008. [ DE 2 ] J. Dittrich, P. Exner: A non–relativistic model of two–particle decay I–IV, Czech. J. Phys. B37 (1987), 503–515, 1028–1034; B38 (1988), 591–610; B39 (1989), 121–138. [DESˇ 1 ] J. Dittrich, P. Exner, P. Seba:ˇ Dirac operators with a spherically symmetric δ–shell interaction, J. Math. Phys. 30 (1989), 2975–2982. [DESˇ 2 ] J. Dittrich, P. Exner, P. Seba:ˇ Dirac Hamiltonians with Coulombic poten- tial and spherically symmetric shell contact interaction, J. Math. Phys. 33 (1992), 2207–2214. [DK1] J.Dittrich,J.Kˇr´ıˇz: Curved planar quantum wires with Dirichlet and Neumann boundary conditions, J. Phys. A35 (2002), L269–275. [ DK 2 ] J. Dittrich, J. Kˇr´ıˇz: Bound states in straight quantum waveguides with com- bined boundary conditions, J. Math. Phys. 43 (2002), 3892–3915. [Di 1] J. Dixmier: Sur la relation i(PQ−QP )=I , Compos. Math. 13 (1956), 263–269. [ DF 1 ] J.D. Dollard, C.N. Friedmann: Existence and completeness of the Møller wave 1 | | ∞ | | ∞ operators for radial potentials satisfying 0 r v(r) dr + 1 v(r) dr < , J. Math. Phys. 21 (1980), 1336–1339. [ DS 1 ] E. Doron, U. Smilansky: Chaotic spectroscopy, Phys. Rev. Lett. 68 (1992), 1255–1258. [Dri 1] T. Drisch: Generalization of Gleason’s theorem, Int. J. Theor. Phys. 18 (1978), 239–243. [DuE 1] P. Duclos, P. Exner: Curvature–induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73–102. [ DEK 1 ] P. Duclos, P. Exner, D. Krejˇciˇr´ık: Bound states in curved quantum layers, Commun. Math. Phys. 223 (2001), 13–28. [ DEM 1 ] P. Duclos, P. Exner, B. Meller: Exponential bounds on curvature–induced res- onances in a two–dimensional Dirichlet tube, Helv. Phys. Acta 71 (1998), 133–162. References 631

[ DEM 2 ] P. Duclos, P. Exner, B. Meller: Open quantum dots: resonances from per- turbed symmetry and bound states in strong magnetic fields, Rep. Math. Phys. 47 (2001), 253–267. [DuESˇ 1 ] P. Duclos, P. Exner, P. Sˇˇtov´ıˇcek: Curvature–induced resonances in a two– dimensional Dirichlet tube, Ann.Inst. H.Poincar´e: Phys. Th´eor. 62 (1995), 81–101. [DuSˇ 1 ] P. Duclos, P. Sˇˇtov´ıˇcek: Floquet Hamiltonians with pure spectrum, Commun. Math. Phys. 177 (1996), 327–347. [DuSVˇ 1 ] P. Duclos, P. Sˇˇtov´ıˇcek, M. Vittot: Perturbations of an eigen-value from a dense point spectrum: a general Floquet Hamiltonian, Ann. Inst. H. Poincar´e 71 (1999), 241–301. [Du 1] I.H. Duru: Quantum treatment of a class of time–dependent potentials, J. Phys. A22 (1989), 4827–4833. [ EZ 1 ] J.–P. Eckmann, Ph.C. Zabey: Impossibility of quantum mechanics in a Hilbert space over a finite field, Helv. Phys. Acta 42 (1969), 420–424. [ EH 1 ] M. Eilers, M. Horst: The theorem of Gleason for nonseparable Hilbert spaces, Int. J. Theor. Phys. 13 (1975), 419–424. [ EkK 1 ] T. Ekholm, H. Kovaˇr´ık: Stability of the magnetic Schr¨odinger operator in a waveguide, Comm. PDE 30 (2005), 539–565. [ETr 1] K.D. Elworthy, A. Truman: A Cameron–Martin formula for Feynman integrals (the origin of Maslov indices), in Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, vol. 153, Springer, Berlin 1982; pp. 288–294. [Em1] G.G.Emch:M´ecanique quantique quaternionique et relativit´e restreinte I, II, Helv. Phys. Acta 36 (1963), 739–769, 770–788. [ ES 1 ] G.G. Emch, K.B. Sinha: Weak quantization in a non–perturbative model, J. Math. Phys. 20 (1979), 1336–1340. [ En 1,2 ] V. Enss: Asymptotic completeness for quantum–mechanical potential scat- tering, I. Short–range potentials, II. Singular and long–range potentials, Commun. Math. Phys. 61 (1978), 285–291; Ann. Phys. 119 (1979), 117–132. [En 3] V. Enss: Topics in scattering theory for multiparticle systems: a progress report, Physica A124 (1984), 269–292. [ EV 1 ] V. Enss, K. Veseliˇc: Bound states and propagating states for time dependent Hamiltonians, Ann. Inst. H. Poincar´e: Phys. Th´eor. 39 (1983), 159–191. [ Epi 1 ] G. Epifanio: On the matrix representation of unbounded operators, J. Math. Phys. 17 (1976), 1688–1691. [ET 1] G. Epifanio, C. Trapani: Remarks on a theorem by G. Epifanio, J. Math. Phys. 20 (1979), 1673–1675. [ ET 2 ] G. Epifanio, C. Trapani: Some spectral properties in algebras of unbounded operators, J. Math. Phys. 22 (1981), 974–978. [ET 3] G. Epifanio, C. Trapani: V ∗–algebras: a particular class of unbounded operator algebras, J. Math. Phys. 25 (1985), 2633–2637. [ ELS 1 ] W.D. Evans, R.T. Lewis, Y. Saito: Some geometric spectral properties of N–body Schr¨odinger operators, Arch. Rat. Mech. Anal. 113 (1991), 377–400. [Ex 1] P. Exner: Bounded–energy approximation to an unstable quantum system, Rep. Math. Phys. 17 (1980), 275–285. [ Ex 2 ] P. Exner: Generalized Bargmann inequalities, Rep. Math. Phys. 19 (1984), 249–255. 632 References

[ Ex 3 ] P. Exner: Representations of the Poincar´e group associated with unstable particles, Phys. Rev. D28 (1983), 3621–2627. [ Ex 4 ] P. Exner: Remark on the energy spectrum of a decaying system, Commun. Math. Phys. 50 (1976), 1–10. [Ex 5] P. Exner: One more theorem on the short–time regeneration rate, J. Math. Phys. 30 (1989), 2563–2564. [ Ex 6 ] P. Exner: A solvable model of two–channel scattering, Helv. Phys. Acta 64 (1991), 592–609. [ Ex 7 ] P. Exner: A model of resonance scattering on curved quantum wires, Ann. Physik 47 (1990), 123–138. [ Ex 8 ] P. Exner: The absence of the absolutely continuous spectrum for δ Wannier– Stark ladders, J. Math. Phys. 36 (1995), 4561–4570. [Ex 9] P. Exner: Lattice Kronig–Penney models, Phys. Rev. Lett. 74 (1995), 3503–3506. [ Ex 10 ] P. Exner: Contact interactions on graph superlattices, J. Phys. A29 (1996), 87–102. [ Ex 11 ] P. Exner: Weakly coupled states on branching graphs, Lett. Math. Phys. 38 (1996), 313–320. [ Ex 12 ] P. Exner: A duality between Schr¨odinger operators on graphs and certain Jacobi matrices, Ann. Inst. H. Poincar´e: Phys. Th´eor. 66 (1997), 359–371. [ Ex 13 ] P. Exner: Magnetoresonances on a lasso graph, Found. Phys. 27 (1997), 171–190. [ Ex 14 ] P. Exner: Point interactions in a tube, in “Stochastic Processes: Physics and Geometry: New Interplayes II” (A volume in honor of S. Albeverio; F. Gesztesy et al., eds.); CMS Conference Proceedings, vol. 29, Providence, R.I. 2000; pp. 165– 174. [Ex 15] P. Exner: Leaky quantum graphs: a review, in the proceedings volume [[EKKST]]; arXiv: 0710.5903 [math-ph] [ Ex 16 ] P. Exner: An isoperimetric problem for leaky loops and related mean-chord inequalities, J. Math. Phys. 46 (2005), 062105 [ Ex 17 ] P. Exner: Necklaces with interacting beads: isoperimetric problems, AMS “Contemporary Math” Series, vol. 412, Providence, RI, 2003; pp. 141–149. [ EFr 1 ] P. Exner, M. Fraas: The decay law can have an irregular character, J. Phys. A40 (2007), 1333–1340. [ EFK 1 ] P. Exner, P. Freitas, D. Krejˇciˇr´ık: A lower bound to the spectral threshold in curved tubes, Proc. Roy. Soc. A460 (2004), 3457–3467. [ EG 1 ] P. Exner, R. Gawlista: Band spectra of rectangular graph superlattices, Phys. Rev. B53 (1996), 7275–7286. [EGSTˇ 1] P. Exner, R. Gawlista, P. Seba,ˇ M. Tater: Point interactions in a strip, Ann. Phys. 252 (1996), 133–179. [ EHL 1 ] P. Exner, E.M. Harrell, M. Loss: Optimal eigenvalues for some Laplacians and Schr¨odinger operators depending on curvature, Proceedings of the Conference “Mathematical Results in Quantum Mechanics” (QMath7, Prague 1998); Operator Theory: Advances and Applications, Birkh¨auser, Basel; pp. 47–53. [ EHL 2 ] P. Exner, E.M. Harrell, M. Loss: Inequalities for means of chords, with appli- cation to isoperimetric problems, Lett. Math. Phys. 75 (2006), 225–233; addendum 77 (2006), 219. References 633

[ EI 1 ] P. Exner, T. Ichinose: Geometrically induced spectrum in curved leaky wires, J. Phys. A34 (2001), 1439–1450. [ EK 1 ] P. Exner, G.I. Kolerov: Uniform product formulae, with application to the Feynman–Nelson integral for open systems, Lett. Math. Phys. 6 (1982), 151–159. [ EKr 1 ] P. Exner, D. Krejˇciˇr´ık: Quantum waveguide with a lateral semitransparent barrier: spectral and scattering properties, J. Phys. A32 (1999), 4475–4494. [ EKr 2 ] P. Exner, D. Krejˇciˇr´ık: Bound states in mildly curved layers, J. Phys. A34 (2001), 5969–5985. [ ELW 1 ] P. Exner, H. Linde, T. Weidl: Lieb-Thirring inequalities for geometrically induced bound states, Lett. Math. Phys. 70 (2004), 83–95. [ ENZ 1 ] P. Exner, H. Neidhardt, V.A. Zagrebnov: Potential approximations to δ: an inverse Klauder phenomenon with norm-resolvent convergence, Commun. Math. Phys. 224 (2001), 593–612. [ EN 1 ] P. Exner, K. Nˇemcov´a: Quantum mechanics of layers with a finite number of point perturbations, J. Math. Phys. 43 (2002), 1152–1184. [EN 2] P. Exner, K. Nˇemcov´a: Magnetic layers with periodic point perturbations, Rep. Math. Phys. 52 (2003), 255–280. [ EPo 1 ] P. Exner, O. Post: Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005), 77–115. [EPo 2] P. Exner, O. Post: Convergence of resonances on thin branched quantum wave guides, J. Math. Phys. 48 (2007), 092104 [ESˇ 1] P. Exner, P. Seba:ˇ Bound states in curved quantum waveguides, J. Math. Phys. 30 (1989), 2574–2580. [ESˇ 2 ] P. Exner, P. Seba:ˇ Electrons in semiconductor microstructures: a challenge to operator theorists, in the proceedings volume [[ESˇ 2 ]], pp. 79–100. [ESˇ 3] P. Exner, P. Seba:ˇ Quantum motion in two planes connected at one point, Lett. Math. Phys. 12 (1986), 193–198. [ESˇ 4] P. Exner, P. Seba:ˇ Quantum motion on a halfline connected to a plane, J. Math. Phys. 28 (1987), 386–391, 2254. [ESˇ 5 ] P. Exner, P. Seba:ˇ Schr¨odinger operators on unusual manifolds, in the proceed- ings volume [[AFHL ]], pp. 227–253. [ESˇ 6 ] P. Exner, P. Seba:ˇ Free quantum motion on a branching graph, Rep. Math. Phys. 28 (1989), 7–26. [ESˇ 7] P. Exner, P. Seba:ˇ Trapping modes in a curved electromagnetic waveguide with perfectly conducting walls, Phys. Lett. A144 (1990), 347–350. [ESˇ 8] P. Exner, P. Seba:ˇ A “hybrid plane” with spin-orbit interaction, Russ. J. Math. Phys. 14 (2007), 401–405. [ESˇ 9 ] P. Exner, P. Seba:ˇ Resonance statistics in a microwave cavity with a thin antenna, Phys. Lett. A228 (1997), 146–150. [ESˇSˇ 1 ] P. Exner, P. Seba,ˇ P. Sˇˇtov´ıˇcek: On existence of a bound state in an L-shaped waveguide, Czech. J. Phys. B39 (1989), 1181–1191. [ESˇSˇ 2] P. Exner, P. Seba,ˇ P. Sˇˇtov´ıˇcek: Semiconductor edges can bind electrons, Phys. Lett. A150 (1990), 179–182. [ESˇSˇ 3 ] P. Exner, P. Seba,ˇ P. Sˇˇtov´ıˇcek: Quantum interference on graphs controlled by an external electric field, J. Phys. A21 (1988), 4009–4019. 634 References

[ESTVˇ 1 ] P. Exner, P. Seba,ˇ M. Tater, D. Vanˇek: Bound states and scattering in quantum waveguides coupled laterally through a boundary window, J. Math. Phys. 37 (1996), 4867–4887. [ESeˇ 1 ] P. Exner, E. Sereˇˇ sov´a: Appendix resonances on a simple graph, J. Phys. A27 (1994), 8269–8278. [ ETV 1 ] P. Exner, M. Tater, D. Vanˇek: A single-mode quantum transport in serial- structure geometric scatterers, J. Math. Phys. 42 (2001), 4050–4078. [ ETu 1 ] P. Exner, O. Turek: Approximations of singular vertex couplings in quantum graphs, Rev. Math. Phys. 19 (2007), 571–606. [ EV 1 ] P. Exner, S.A. Vugalter: Bounds states in a locally deformed waveguide: the critical case, Lett. Math. Phys. 39 (1997), 59–68. [ EV 2 ] P. Exner, S.A. Vugalter: Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window, Ann. Inst. H. Poincar´e: Phys. Th´eor. 65 (1996), 109–123. [ EV 3 ] P. Exner, S.A. Vugalter: On the number of particles that a curved quantum waveguide can bind, J. Math. Phys. 40 (1999), 4630–4638. [EY 1] P. Exner, K. Yoshitomi: Asymptotics of eigenvalues of the Schr¨odinger operator with a strong δ-interaction on a loop, J. Geom. Phys. 41 (2002), 344–358. [ EZ 1 ] P. Exner, V.A. Zagrebnov: Bose-Einstein condensation in geometrically de- formed tubes, J. Phys. A38 (2005), L463–470. [ Far 1 ] W.G. Faris: Inequalities and uncertainty principles, J. Math. Phys. 19 (1978), 461–466. [ Far 2 ] W.G. Faris: Product formulas for perturbation of linear operators, J. Funct. Anal. 1 (1967), 93–107. [ FMRS 1 ] J. Feldmann, J. Magnen, V. Rivasseau, R. Sen´eor: A renormalizable field theory: the massive Gross–Neveu model in two dimensions, Commun. Math. Phys. 103 (1986), 67–103. [Fel 1] J.G.M. Fell: The dual spaces of C∗–algebras, Trans. Am. Math. Soc. 94 (1960), 365–403. [ Fey 1 ] R.P. Feynman: Space–time approach to nonrelativistic quantum mechanics, Rev. Mod. Phys. 20 (1948), 367–387. [ FJSS 1 ] D. Finkelstein, J.M. Jauch, S. Schminovich, D. Speiser: Foundations of quaternionic quantum mechanics, J. Math. Phys. 3 (1962), 207–220. [ FJSS 2 ] D. Finkelstein, J.M. Jauch, S. Schminovich, D. Speiser: Principle of general Q covariance, J. Math. Phys. 4 (1963), 788-796. [ Fo 1 ] V.A. Fock: Konfigurationsraum und zweite Quantelung, Z. Phys. 75 (1932), 622–647. [ Fri 1 ] K.O. Friedrichs: On the perturbation of continuous spectra, Commun. Appl. Math. 1 (1948), 361–406. [FLL1] J.Fr¨ohlich, E.H. Lieb, M. Loss: Stability of Coulomb systems with magnetic fields, I. The one–electron atom, Commun. Math. Phys. 104 (1986), 251–270. [FT1] T.F¨ul¨op, I. Tsutsui: A free particle on a circle with point interaction, Phys. Lett. A264 (2000), 366–374. [Gam 1] G. Gamow: Zur Quantetheorie des Atomkernes, Z. Phys. 51 (1928), 204–212. [ GK 1 ] K. Gawedzki, A. Kupiainen: Gross–Neveu model through convergent expan- sions, Commun. Math. Phys. 102 (1985), 1–30. References 635

[ GN 1 ] I.M. Gel’fand, M.A. Naimark: On embedding of a normed ring to the ring of operators in Hilbert space, Mat. Sbornik 12 (1943), 197–213 (in Russian). [GMR 1] C. Gerard, A. Martinez, D. Robert: Breit–Wigner formulas for the scattering phase and the total cross section in the semiclassical limit, Commun. Math. Phys. 121 (1989), 323–336. [ GP 1 ] N.I. Gerasimenko, B.S. Pavlov: Scattering problem on non–compact graphs, Teor. mat. fiz. 74 (1988), 345–359 (in Russian). [GGT 1] F. Gesztesy, H. Grosse, B. Thaller: Efficient method for calculating relativistic corrections to spin 1/2 particles, Phys. Rev. Lett. 50 (1983), 625–628. [ GGT 2 ] F. Gesztesy, H. Grosse, B. Thaller: First order relativistic corrections and the spectral concentration, Adv. Appl. Math. 6 (1985), 159–176. [ GGH 1 ] F. Gesztesy, D. Gurarie, H. Holden, M. Klaus, L. Sadun, B. Simon, P. Vogel: Trapping and cascading in the large coupling limit, Commun. Math. Phys. 118 (1988), 597–634. [GSˇ 1 ] F. Gesztesy, P. Seba:ˇ New analytically solvable models of relativistic point interactions, Lett. Math. Phys. 13 (1987), 213–225. [ GST 1 ] F. Gesztesy, B. Simon, B. Thaller: On the self–adjointness of Dirac operator with anomalous magnetic moment, Proc. Am. Math. Soc. 94 (1985), 115–118. [ GGM 1 ] V. Glaser, H. Grosse, A. Martin: Bounds on the number of eigenvalues of the Schr¨odinger operator, Commun. Math. Phys. 59 (1978), 197–212. [ GMGT 1 ] V. Glaser, A. Martin, H. Grosse, W. Thirring: A family of optimal condi- tions for the absence of bound states in a potential, in [[LSW ]], pp. 169–194. [ Gla 1 ] R.J. Glauber: Photon correlations, Phys. Rev. Lett. 10 (1963), 84–86. [ Gla 2 ] R.J. Glauber: Coherent and incoherent states of the radiation fields, Phys. Rev. 131 (1963), 2766–2788. [ Gle 1 ] A.M. Gleason: Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 91–110. [ GJ 1 ] J. Glimm, A. Jaffe: Boson quantum field models, in [[Str ]], pp. 77–143. [ GoJ 1 ] J. Goldstone, R.L. Jaffe: Bound states in twisting tubes, Phys. Rev. B45 (1992), 14100–14107. [ GWW 1 ] C. Gordon, D.L. Webb, S. Wolpert: One cannot hear the shape of a drum, Bull. Am. Math. Soc. 27 (1992), 134–138. [ Gra 1 ] G.M. Graf: Asymptotic completeness for N–body short range quantum sys- tems: a new proof, Commun. Math. Phys. 132 (1990), 73–101. [ GG 1 ] S. Graffi, V. Grecchi: Resonances in Stark effect of atomic systems, Commun. Math. Phys. 79 (1981), 91–110. [GLL 1] M. Griesemer, E.H. Lieb, M. Loss: Ground states in non-relativistic quantum electrodynamics, Invent. Math. 145 (2001), 557–595. [Gri 1] D. Grieser: Spectra of graph neighborhoods and scattering, arXiv:0710.3405v1 [math.SP] [ GMa 1 ] D. Gromoll, W. Meyer, On complete open manifolds of positive curvature, Ann. Math. 90 (1969), 75–90. [Gro 1] C. Grosche: Coulomb potentials by path integration, Fortschr. Phys. 40 (1992), 695–737. [ Gr 1 ] H. Grosse: On the level order for Dirac operators, Phys. Lett. B197 (1987), 413–417. 636 References

[ Gud 1 ] S.P. Gudder: The Hilbert space axiom in quantum mechanics, in Old and New Questions in Physics, Cosmology, Philosophy and Theoretical Biology. Essays in Honor of W. Yourgrau, Plenum Press, New York 1983; pp. 109–127. [ Haa 1 ] R. Haag: Quantum field theory, in the proceedings volume [[Str ]], pp. 1–16. [ Haa 2 ] R. Haag: On quantum field theories, Danske Vid. Selsk. Mat.–Fys. Medd. 29 (1955), No. 12. [ Haa 3 ] R. Haag: Local relativistic quantum physics, Physica A124 (1984), 357–364. [ HK 1 ] R. Haag, D. Kastler: An algebraic approach to quantum field theory, J. Math. Phys. 5 (1964), 848–861. [ Ha 1 ] M. Hack: On the convergence to the Møller wave operators, Nuovo Cimento 9 (1958), 731–733. [ Hag 1 ] G. Hagedorn: Semiclassical quantum mechanics I–IV, Commun. Math. Phys. 71 (1980), 77–93; Ann. Phys. 135 (1981), 58–70; Ann. Inst. H. Poincar´e A42 (1985), 363–374. [ Hag 2 ] G. Hagedorn: Adiabatic expansions near adiabatic crossings, Ann. Phys. 196 (1989), 278–295. [ HJ 1 ] G. Hagedorn, A. Joye: Molecular propagation through small avoided crossings of electron energy levels, Rev. Math. Phys. 11 (1999), 41–101. [ HJ 2 ] G. Hagedorn, A. Joye: Time development of exponentially small non-adiabatic transitions, Commun. Math. Phys. 250 (2004), 393–413. [ HLS 1 ] G. Hagedorn, M. Loss, J. Slawny: Non–stochasticity of time–dependent quadratic Hamiltonians and the spectra of transformation, J. Phys. A19 (1986), 521–531. [ HS 1 ] Ch. Hainzl, R. Seiringer: Mass renormalization and energy level shift in non- relativistic QED, Adv. Theor. Math. Phys. 6 (2002), 847–871. [ Har 1 ] G.H. Hardy: A note on a Theorem by Hilbert, Math. Z. 6 (1920), 314–317. [ Harm 1 ] M. Harmer: Hermitian symplectic geometry and extension theory, J. Phys. A33 (2000), 9193–9203. [ Harp 1 ] P.G. Harper: Single band motion of conduction electrons in a uniform mag- netic field, Proc. Roy. Soc. London A68 (1955), 874–878. [HE1] M.Havl´ıˇcek, P. Exner: Note on the description of an unstable system, Czech. J. Phys. B19 (1973), 594–600. [ HSS 1 ] R. Hempel, L.A. Seco, B. Simon: The essential spectrum of Neumann Lapla- cians on some bounded singular domains, J. Funct. Anal. 102 (1991), 448–483. [Hep 1] K. Hepp: The classical limit for quantum correlation function, Commun. Math. Phys. 35 (1974), 265–277. [ Her 1 ] I.W. Herbst: Dilation analycity in constant electric field, I. The two–body problem, Commun. Math. Phys. 64 (1979), 279–298. [HH 1] I.W. Herbst, J.S. Howland: The Stark ladder resonances and other one–dimen- sional external field problems, Commun. Math. Phys. 80 (1981), 23–42. [Herb 1] F. Herbut: Characterization of compatibility, comparability and orthogonality of quantum propositions in terms of chains of filters, J. Phys. A18 (1985), 2901–2907. [HU 1] J. Hilgevoord, J.B.M. Uffink: Overall width, mean peak width and uncertainty principle, Phys. Lett. 95A (1983), 474–476. [ Hil 1 ] R.N. Hill: Proof that the H− ion has only one bound state, Phys. Rev. Lett. 38 (1977), 643–646. References 637

[ HSp 1 ] F. Hiroshima, H. Spohn: Enhanced binding through coupling to a quantum field, Ann. Henri Poincar´e 2 (2001), 1159–1187. [ Hof 1 ] G. Hofmann: On the existence of quantum fields in space–time dimension 4, Rep. Math. Phys. 18 (1980), 231–242. [ Hof 1 ] R. Hofstadter: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B14 (1976), 2239–2249. [H¨o1] G.H¨ohler: Uber¨ die Exponentialn¨aherung beim Teilchenzerfall, Z. Phys. 152 (1958), 546–565. [HV 1] S.S. Horuˇzii, A.V. Voronin: Field algebras do not leave field domains invariant, Commun. Math. Phys. 102 (1986), 687–692. [ HB 1 ] L.P. Horwitz, L.C. Biedenharn: Quaternion quantum mechanics: second quan- tization and gauge fields, Ann. Phys. 157 (1984), 432–488. [ HLM 1 ] L.P. Horwitz, J.A. LaVita, J.–P. Marchand: The inverse decay problem, J. Math. Phys. 12 (1971), 2537–2543. [ HL 1 ] L.P. Horwitz, J. Levitan: A soluble model for time dependent perturbation of an unstable quantum system, Phys. Lett. A153 (1991), 413–419. [ HM 1 ] L.P. Horwitz. J.–P. Marchand: The decay–scattering system, Rocky Mts. J. Math. 1 (1971), 225–253. [vH 1] L. van Hove: Sur le probleme des relations entre les transformations unitaires de la M´ecanique quantique et les transformations canoniques de la M´ecanique classique, Bull. Acad. Roy. de Belgique, Classe des Sciences 37 (1951), 610–620. [ How 1 ] J.S. Howland: Perturbations of embedded eigenvalues by operators of finite rank, J. Math. Anal. Appl. 23 (1968), 575–584. [ How 2 ] J.S. Howland: Spectral concentration and virtual poles I, II, Am.J.Math. 91 (1969), 1106–1126; Trans. Am. Math. Soc. 162 (1971), 141–156. [ How 3 ] J.S. Howland: Puiseaux series for resonances at an embedded eigenvalue, Pacific J. Math. 55 (1974), 157–176. [How 4] J.S. Howland: The Livsic matrix in perturbation theory, J. Math. Anal. Appl. 50 (1975), 415–437. [ How 5 ] J.S. Howland: Stationary theory for time–dependent Hamiltonians, Math. Ann. 207 (1974), 315–333. [ How 6 ] J.S. Howland: Floquet operator with singular spectrum I, II, Ann. Inst. H. Poincar´e: Phys.Th´eor. 49 (1989), 309–323, 325–335. [How 7] J.S. Howland: Stability of quantum oscillators, J. Phys. A25 (1992), 5177–81. [ HLT 1 ] D. Hundertmark, E.H. Lieb, L.E. Thomas: A sharp bound for an eigenvalue moment of the one-dimensional Schr¨odinger operator, Adv. Theor. Math. Phys. 2 (1998), 719–731. [ HLW 1 ] D. Hundertmark, A. Laptev, T. Weidl: New bounds on the Lieb-Thirring constants, Invent. Math. 140 (2000), 693–704. [ Hun 1 ] W. Hunziker: Resonances, metastable states and exponential decay laws in perturbation theory, Commun. Math. Phys. 132 (1990), 177–188. [Ich 1] T. Ichinose: Path integral for a hyperbolic system of the first order, Duke Math. J. 51 (1984), 1–36. [ IK 1 ] T. Ikebe, T. Kato: Uniqueness of the self–adjoint extension of singular elliptic differential operators, Arch. Rat. Mech. Anal. 9 (1962), 77–92. 638 References

[ Ito 1 ] K. Ito: Wiener integral and Feynman integral, in Proceedings of the 4th Berke- ley Symposium on Mathematical Statistics and Probability, vol. 2, University of California Press, Berkeley, CA 1961; pp. 227–238. [ Ito 2 ] K. Ito: Generalized uniform complex measures in the Hilbertian metric space with their applications to the Feynman integral, in Proceedings of the 5th Berke- ley Symposium on Mathematical Statistics and Probability, vol.2/1, University of California Press, Berkeley, CA 1967; pp. 145–167. [ IS 1 ] P.A. Ivert, T. Sj¨odin: On the impossibility of a finite proposition lattice for quantum mechanics, Helv. Phys. Acta 51 (1978), 635–636. [Ja 1,2] J.M. Jauch: Theory of the scattering operator I, II, Helv. Phys. Acta 31 (1958), 127–158, 661–684. [JP 1] J.M. Jauch, C. Piron: Can hidden variables be excluded in quantum mechanics?, Helv. Phys. Acta 36 (1963), 827–837. [ JL 1 ] H. Jauslin, J.L. Lebowitz: Spectral and stability aspects of quantum chaos, Chaos 1 (1991), 114–121. [ Jo 1 ] G.W. Johnson: The equivalence of two approaches to the Feynman integral, J. Math. Phys. 23 (1982), 2090–2096. [ Jo 2 ] G.W. Johnson: Feynman’s paper revisited, Suppl. Rend. Circ. Mat. Palermo, Ser.II, 17 (1987), 249–270. [ JS 1 ] G.W. Johnson, D.L. Skoug: A Banach algebra of Feynman integrable functions with applications to an integral equation which is formally equivalent to Schr¨odinger equation, J. Funct. Anal. 12 (1973), 129–152. [ JNW 1 ] P. Jordan, J. von Neumann, E. Wigner: On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29–64. [ JoP 1 ] R. Jost, A. Pais: On the scattering of a particle by a static potential, Phys. Rev. 82 (1951), 840–850. [Joy 1] A. Joye: Proof of the Landau–Zener formula, Asympt. Anal. 9 (1994), 209–258. [ JoyP 1 ] A. Joye, Ch.–Ed. Pfister: Exponentially small adiabatic invariant for the Schr¨odinger equation, Commun. Math. Phys. 140 (1991), 15–41. [ JoyP 2 ] A. Joye, Ch.–Ed. Pfister: Superadiabatic evolution and adiabatic transition probability between two non–degenerate levels isolated in the spectrum, J. Math. Phys. 34 (1993), 454–479. [ Kac 1 ] M. Kac: On distributions of certain Wiener functionals, Trans. Am. Math. Soc. 65 (1949), 1–13. [ Kad 1 ] R.V. Kadison: Normal states and unitary equivalence of von Neumann alge- bras, in the proceedings volume [[Kas ]], pp. 1–18. [ Kar 1 ] B. Karnarski: Generalized Dirac operators with several singularities, J. Oper- ator Theory 13 (1985), 171–188. [ Kas 1 ] D. Kastler: The C∗–algebra of a free boson field, I. Discussion of basic facts, Commun. Math. Phys. 1 (1965), 14–48. [ Ka 1 ] T. Kato: Integration of the equations of evolution in a Banach space, J. Math. Soc. Jpn 5 (1953), 208–234. [ Ka 2 ] T. Kato: Wave operators and similarity for some non–selfadjoint operators, Math. Ann. 162 (1966), 258–279. [Ka 3] T. Kato: Fundamental properties of Hamiltonian of the Schr¨odinger type, Trans. Am. Math. Soc. 70 (1951), 195–211. References 639

[ Ka 4 ] T. Kato: On the existence of solutions of the helium wave equations, Trans. Am. Math. Soc. 70 (1951), 212–218. [ Ka 5 ] T. Kato: Growth properties of solutions of the reduced wave equation with variable coefficients, Commun. Pure Appl. Math. 12 (1959), 403–425. [ Ka 6 ] T. Kato: Perturbations of continuous spectra by trace class operators, Proc. Jpn. Acad. 33 (1057), 260–264. [ Ka 7 ] T. Kato: Positive commutators i[f(P ),g(Q)] , J. Funct. Anal. 96 (1991), 117–129. [KY 1] T. Kato, K. Yajima: Dirac equations with moving nuclei, Ann. Inst. H. Poincar´e: Phys. Th´eor. 54 (1991), 209–221. [ KL 1 ] J.C. Khandekar, S.V. Lawande: Feynman path integrals: some exact results and applications, Phys. Rep. 137 (1986), 115–229. [ Kis 1 ] A. Kiselev: Some examples in one–dimensional “geometric” scattering on manifolds, J. Math. Anal. Appl. 212 (1997), 263–280. [ Kit 1 ] H. Kitada: Asymptotic completeness of N–body operators, I. Short–range systems, Rep. Math. Phys. 3 (1991), 101–124. [Kla 1] J.R. Klauder: Continuous–representation theory I, II, J. Math. Phys. 4 (1963), 1055–1058, 1058–1073. [ Kla 2 ] J.R. Klauder: The action option and Feynman quantization of spinor fields in terms of ordinary C-numbers, Ann. Phys. 11 (1960), 123–164. [ KD 1 ] J.R. Klauder, I. Daubechies: Quantum mechanical path integrals with Wiener measure for all polynomial Hamiltonians, Phys. Rev. Lett. 52 (1984), 1161–1164. [Kl 1] M. Klaus: On the bound states of Schr¨odinger operators in one dimension, Ann. Phys. 108 (1977), 288–300. [ Kl 2 ] M. Klaus: On the point spectrum of Dirac operators, Helv. Phys. Acta 53 (1980), 453–462. [ Kl 3 ] M. Klaus: Dirac operators with several Coulomb singularities, Helv. Phys. Acta 53 (1980), 463–482. [KlS 1] M. Klein, E. Schwarz: An elementary proof to formal WKB expansions in Rn , Rep. Math. Phys. 2 (1990), 441–456. [ KoS 1 ] V. Kostrykin, R. Schrader: Kirhhoff’s rule for quantum wires, J. Phys. A32 (1999), 595–630. [ KoS 2 ] V. Kostrykin, R. Schrader: Kirhhoff’s rule for quantum wires. II: The in- verse problem with possible applications to quantum computers, Fortschr. Phys. 48 (2000), 703–716. [KoS 3] V. Kostrykin, R. Schrader: The generalized star product and the factorization of scattering matrices on graphs, J. Math. Phys. 42 (2001), 1563–1598. [ KoS 4 ] V. Kostrykin, R. Schrader: Quantum wires with magnetic fluxes, Commun. Math. Phys. 237 (2003), 161–179. [Kuch 1] P. Kuchment: Quantum graphs, I. Some basic structures, Waves Rand. Media 14 (2004), S107–128. [Kuch 2] P. Kuchment: Quantum graphs, II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A38 (2005), 4887–4900. [KuZ 1] P. Kuchment, H. Zeng: Convergence of spectra of mesoscopic systems collaps- ingontoagrap,J. Math. Anal. Appl. 258 (2001), 671–700. [KuS 1] J. Kupsch, W. Sandhas: Møller operators for scattering on singular potentials, Commun. Math. Phys. 2 (1966), 147–154. 640 References

[ Kur 1,2 ] S. Kuroda: Perturbations of continuous spectra by unbounded operators I, II, J. Math. Soc. Jpn 11 (1959), 247–262; 12 (1960), 243–257. [ LF 1 ] O.A. Ladyˇzenskaya, L.D. Faddeev: On perturbations of the continuous spec- trum, Doklady Acad. Sci. USSR 120 (1958), 1187–1190 (in Russian). [ Lan 1 ] C. Lance: Tensor products of C∗–algebras, in [[Kas ]], pp. 154–166. [ La 1 ] R. Landauer: Electrical resistance of disordered one-dimensional lattices, Phil. Mag. 21 (2000), 863–867. [ Lap 1 ] M.L. Lapidus: The Feynman–Kac formula with a Lebesgue–Stieltjes measure and Feynman operational calculus, Studies Appl. Math. 76 (1987), 93–132. [ LW 1 ] A. Laptev, T. Weidl: Hardy inequalities for magnetic Dirichlet forms, Oper. Theory Adv. Appl. 108 (1999), 299–305. [LW 2] A. Laptev, T. Weidl: Sharp Lieb-Thirring inequalities in high dimensions, Acta Math. 184 (2000), 87–111. [Las 1] G. Lassner: Topological algebras of operators, Rep. Math. Phys. 3 (1972), 279–293. [ Las 2 ] G. Lassner: Topologien auf Op∗–Algebren, Wiss. Z. KMU Leipzig, Math.– Naturwiss. 24 (1975), 465–471. [Las 3] G. Lassner: Algebras of unbounded operators and quantum dynamics, Physica A124 (1984), 471–480. [ LT 1 ] G. Lassner, W. Timmermann: Normal states on algebras of unbounded opera- tors, Rep. Math. Phys. 3 (1972), 295–305. [ LT 2 ] G. Lassner, W. Timmermann: Classification of domains of operator algebras, Rep. Math. Phys. 9 (1976), 205–217. [ Lee 1 ] T.D. Lee: Some special examples in renormalizable field theory, Phys. Rev. 95 (1954), 1329–1334. [LLMRSY 1] F. Lenz, J.T. Londergan, R.J. Moniz, R. Rosenfelder, M. Stingl, K. Yazaki: Quark confinement and hadronic interactions, Ann. Phys. 170 (1986), 65–254. [LeL 1] J.–M. L´evy–Leblond: Galilei group and Galilean invariance, in the proceedings volume [[Loe 2 ]], pp. 221–299. [ LeL 2 ] J.–M. L´evy–Leblond: Galilean quantum field theories and a ghostless Lee model, Commun. Math. Phys. 4 (1967), 157–176. [ Lie 1 ] E.H. Lieb: The classical limit of quantum spin systems, Commun. Math. Phys. 31 (1973), 327–340. [Lie 2] E.H. Lieb: Bounds on the eigenvalues of the Laplace and Schr¨odinger operators, Bull. Am. Math. Soc. 82 (1976), 751–753. [Lie 3] E.H. Lieb: The number of bound states of one–body Schr¨odinger operators and the Weyl problem, Proc. Symp. Pure Math. 36 (1980), 241–252. [ Lie 4 ] E.H. Lieb: A bound on the maximal ionization of atoms and molecules, Phys. Rev. A29 (1984), 3018–3028. [ Lie 5 ] E.H. Lieb: The stability of matter, Rev. Mod. Phys. 48 (1976), 553–569. [ Lie 6 ] E.H. Lieb: The stability of matter: from atoms to stars, Bull. Am. Math. Soc. 22 (1990), 1–49. [ Lie 7 ] E.H. Lieb: A bound on maximum ionization of atoms and molecules, Phys. Rev. A29 (1984), 3018–3028. [ LL 1 ] E.H. Lieb, M. Loos: Stability of Coulomb systems with magnetic fields, II. The many–electron atom and the one–electron molecule, Commun. Math. Phys. 104 (1986), 271–282. References 641

[ LiT 1 ] E.H. Lieb, W. Thirring: Inequalities for the momenta of the eigenvalues of the Schr¨odinger Hamiltonian and their relations to Sobolev inequalities, in the proceed- ings volume [[LSW ]], pp. 269–304. [LSST 1] E.H. Lieb, I.M. Sigal, B. Simon, W. Thirring: Asymptotic neutrality of large Z ions, Phys. Rev. Lett. 52 (1984), 994–996. [ LY 1 ] E.H. Lieb, H.–T. Yau: The stability and instability of the relativistic matter, Commun. Math. Phys. 118 (1988), 177–213. [L¨oT 1 ] F. L¨offler, W. Timmermann: The Calkin representation for a certain class of algebras of unbounded operators, Rev. Roum. Math. Pures Appl. 31 (1986), 891– 903. [L¨u1] G.L¨uk˝o: On the mean lengths of the chords of a closed curve, Israel J. Math. 4 (1966), 23–32. [Mar 1] Ph. Martin: Time delay in quantum scattering processes, Acta Phys. Austriaca Suppl. XXIII (1981), 157–208. [ Mau 1 ] K. Maurin: Elementare Bemerkungenuber ¨ komutative C∗–Algebren. Beweis einer Vermutung von Dirac, Stud. Math. 16 (1957), 74–79. [MS 1] B. Misra, K.B. Sinha: A remark on the rate of regeneration in decay processes, Helv. Phys. Acta 50 (1977), 99–104. [MSˇ 1 ] B. Milek, P. Seba:ˇ Quantum instability in the kicked rotator with rank–one perturbation, Phys. Lett. A151 (1990), 289–294. [ Mol 1 ] A.M. Molˇcanov: On conditions of the spectrum discreteness of self–adjoint second–order differential equations, Trudy Mosk. mat. obˇsˇcestva 2 (1953), 169–200 (in Russian). [ MV 1 ] S. Molchanov, B. Vainberg: Scattering solutions in a network of thin fibers: small diameter asymptotics, Commun. Math. Phys. 273 (2007), 533–559. [ Nak 1 ] S. Nakamura: Shape resonances for distortion analytic Schr¨odinger operators, Commun. PDE 14 (1989), 1385–1419. [ Nar 1 ] F. Nardini: Exponential decay for the eigenfunctions of the two–body rela- tivistic Hamiltonian, J. d’Analyse Math. 47 (1986), 87–109. [ Nas 1 ] A.H. Nasr: The commutant of a multiplicative operator, J. Math. Phys. 23 (1982), 2268–2270. [Ned 1] L. Nedelec: Sur les r´esonances de l’op´erateur de Dirichlet dans un tube, Comm. PDE 22 (1997), 143–163. [ Nel 1 ] E. Nelson: Analytic vectors, Ann. Math. 70 (1959), 572–614. [ Nel 2 ] E. Nelson: Feynman integrals and the Schr¨odinger equation, J. Math. Phys. 5 (1964), 332–343. [ Nel 3 ] E. Nelson: Construction of quantum fields from Markoff fields, J. Funct. Anal. 12 (1973), 97–112. [ Nen 1 ] G. Nenciu: Adiabatic theorem and spectral concentration I, Commun. Math. Phys. 82 (1981), 121–135. [ Nen 2 ] G. Nenciu: Linear adiabatic theory. Exponential estimates, Commun. Math. Phys. 152 (1993), 479–496. [Nen 3] G. Nenciu: Distinguished self–adjoint extension for Dirac operators dominated by multicenter Coulomb potentials, Helv.Phys.Acta50 (1977), 1–3. [ vN 1 ] J. von Neumann: Mathematische Begr¨undung der Quantenmechanik, Nachr. Gessel. Wiss. G¨ottingen, Math. Phys. (1927), 1–57. 642 References

[ vN 2 ] J. von Neumann: Allgemeine Eigenwerttheorie Hermitescher Funktionalopera- toren, Math. Ann. 102 (1930), 49–131. [ vN 3 ] J. von Neumann: On infinite direct products, Compos. Math. 6 (1938), 1–77. [ New 1 ] R.G. Newton: Bounds on the number of bound states for the Schr¨odinger equations in one and two dimensions, J. Operator Theory 10 (1983), 119–125. [ NSG 1 ] M.M. Nieto, L.M. Simmons, V.P. Gutschik: Coherent states for general potentials I–VI, Phys. Rev. D20 (1979), 1321–1331, 1332–1341, 1342–1350; D22 (1980), 391–402, 403–418; D23 (1981), 927–933. [N¨o1] J.U.N¨ockel: Resonances in quantum-dot transport, Phys. Rev. B46 (1992), 15348–15356. [ Ost 1 ] K. Osterwalder: Constructive quantum field theory: goals, methods, results, Helv. Phys. Acta 59 (1986), 220–228. [ OS 1,2 ] K. Osterwalder, R. Schrader: Axioms for Euclidean Green’s functions, Com- mun. Math. Phys. 31 (1973), 83–112; 42 (1975), 281–305. [ Pav 1 ] B.S. Pavlov: A model of a zero–range potential with an internal structure, Teor. mat. fiz. 59 (1984), 345–353 (in Russian). [ Pea 1 ] D.B. Pearson: An example in potential scattering illustrating the breakdown of asymptotic completeness, Commun. Math. Phys. 40 (1975), 125–146. [ Pea 2 ] D.B. Pearson: A generalization of Birman’s trace theorem, J. Funct. Anal. 28 (1978), 182–186. [ Per 1 ] A.M. Perelomov: Coherent states for arbitrary Lie group, Commun. Math. Phys. 26 (1972), 222–236. [Pe 1] A. Peres: Separability criterion for density matrices, Phys. Rev. Lett. 77 (1996), 1413–1415. [ Pir 1 ] C. Piron: Axiomatique quantique, Helv. Phys. Acta 37 (1964), 439–468. [ Pop 1 ] I.Yu. Popov: The extension and the opening in semitransparent surface, J. Math. Phys. 33 (1982), 1685–1689. [Posi 1] A. Posilicano: Boundary triples and Weyl functions for singular perturbations of self-adjoint operator, Meth. Funct. Anal. Topol. 10 (2004), 57–63. [ Pos 1 ] O. Post: Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case, J. Phys. A38 (2005), 4917–4931. [ Pos 2 ] O. Post: Spectral convergence of non-compact quasi-one-dimensional spaces, Ann. H. Poincar´e 7 (2006), 933–973. [ Pow 1 ] R. Powers: Self–adjoint algebras of unbounded operators, Commun. Math. Phys. 21 (1971), 85–124. [ Pri 1 ] J.F. Price: Inequalities and local uncertainty principles, J. Math. Phys. 24 (1983), 1711–1714. [ Pri 2 ] J.F. Price: Position versus momentum, Phys. Lett. 105A (1984), 343–345. [Pul 1] S. Pulmannov´a: Uncertainty relations and state spaces, Ann. Inst. H. Poincar´e: Phys. Th´eor. 48 (1988), 325–332. [ Rad 1 ] J.M. Radcliffe: Some properties of coherent spin states, J. Phys. A4 (1971), 314–323. [Rel 1] F. Rellich: Die zul¨assigen Randbedingungen bei den singul¨aren Eigenwertprob- lemen der Mathematischen Physik, Math. Z. 49 (1943–44), 702–723. [RB 1] W. Renger, W. Bulla: Existence of bound states in quantum waveguides under weak conditions, Lett. Math. Phys. 35 (1995), 1–12. References 643

[ Ro 1 ] M. Rosenblum: Perturbations of continuous spectra and unitary equivalence, Pacific J. Math. 7 (1957), 997–1010. [ Ros 1 ] G.V. Rozenblium: Discrete spectrum distribution of singular differential oper- ators, Doklady Acad. Sci. USSR 202 (1972), 1012–1015. [ RSch 1 ] J. Rubinstein, M. Schatzmann: Variational problems on multiply connected thin strips, I. Basic estimates and convergence of the Laplacian spectrum, Arch. Rat. Mech. Anal. 160 (2001), 271–308. [ RuS 1 ] K. Ruedenberg, C.W. Scherr: Free–electron network model for conjugated systems, I. Theory, J. Chem. Phys. 21 (1953), 1565–1581. [ Rus 1 ] M.B. Ruskai: Absence of discrete spectrum in highly negative ions I, II, Commun. Math. Phys. 82 (1982), 457–469; 85 (1982), 325–327. [Rus 2] M.B. Ruskai: Limits of excess negative charge of a dynamic diatomic molecule, Ann. Inst. H. Poincar´e: Phys. Th´eor. 52 (1990), 397–414. [ Rus 3 ] M.B. Ruskai: Limits on stability of positive molecular ions, Lett. Math. Phys. 18 (1989), 121–132. [Sai 1] T. Saito: Convergence of the Neumann Laplacian on shrinking domains, Analysis 21 (2001), 171–204. [ SMZ 1 ] C.M. Savage, S. Marksteiner, and P. Zoller. Atomic Waveguides and Cavities from Hollow Optical Fibres. In: Fundamentals of Quantum Optics III (Ed.- F. Ehlotzky), Springer, Lecture Notes in Physics, vol. 420, 1993; p. 60 [ Schm 1 ] K. Schm¨udgen: On trace representation of linear functionals on unbounded operator algebras, Commun. Math. Phys. 63 (1978), 113–130. [ Schm 2 ] K. Schm¨udgen: On topologization of unbounded operator algebras, Rep. Math. Phys. 17 (1980), 359–371. [ Schr 1 ] E. Schr¨odinger: Der stetige Ubergang¨ von der Mikro–zur Makromechanik, Naturwissenschaften 14 (1926), 664–666. [ SRW 1 ] R.L. Schult, D.G. Ravenhall, H.W. Wyld: Quantum bound states in a clas- sically unbounded system of crossed wires, Phys. Rev. B39 (1989), 5476–5479. [ Schw 1 ] J. Schwinger: On the bound states of a given potential, Proc. Natl. Acad. Sci. USA 47 (1961), 122–129. [ Sebˇ 1 ] P. Seba:ˇ Wave chaos in singular quantum billiard, Phys. Rev. Lett. 64 (1990), 1855–1858. [ Sebˇ 2 ] P. Seba:ˇ Complex scaling method for Dirac resonances, Lett. Math. Phys. 16 (1988), 51–59. [ Sebˇ 3 ] P. Seba:ˇ Some remarks on the δ–interaction in one dimension, Rep. Math. Phys. 24 (1986), 111–120. [ Sebˇ 4 ] P. Seba:ˇ The generalized point interaction in one dimension, Czech. J. Phys. B36 (1986), 667–673. [ SSS 1 ] L.A. Seco, I.M. Sigal, J.P. Solovej: Bounds on a ionization energy of large atoms, Commun. Math. Phys. 131 (1990), 307–315. [ Seg 1 ] I.E. Segal: Irreducible representations of operator algebras, Bull. Am. Math. Soc. 53 (1947), 73–88. [ Seg 2 ] I.E. Segal: Postulates for general quantum mechanics, Ann. Math. 48 (1947), 930–948. 644 References

[ Seg 3 ] I.E. Segal: Mathematical characterization of the physical vacuum for a linear Bose–Einstein field (Foundations of the dynamics of infinite systems III), Illinois Math. J. 6 (1962), 500–523. [ Seg 4 ] I.E. Segal: Tensor algebras over Hilbert spaces I, Trans. Am. Math. Soc. 81 (1956), 106–134. [ Set 1 ] N. Seto: Bargmann inequalities in spaces of arbitrary dimension, Publ. RIMS 9 (1974), 429–461. [ Sha 1 ] J. Shabani: Finitely many δ–interactions with supports on concentric spheres, J. Math. Phys. 29 (1988), 660–664. [SMMC 1] T. Shigehara, H. Mizoguchi, T. Mishima, T. Cheon: Realization of a four pa- rameter family of generalized one-dimensional contact interactions by three nearby delta potentials with renormalized strengths, IEICE Trans. Fund. Elec. Comm. Comp. Sci. E82-A (1999), 1708–1713. [Shu 1] M.A. Shubin: Discrete magnetic Laplacian, Commun. Math. Phys. 164 (1994), 259–275. [ Sig 1 ] I.M. Sigal: On long–range scattering, Duke Math. J. 60 (1990), 473–496. [SiS 1] I.M. Sigal, A. Soffer: The N–particle scattering problem: asymptotic complete- ness for short–range potentials, Ann. Math. 126 (1987), 35–108. [SiS 2] I.M. Sigal, A. Soffer: Long–range many–body scattering. Asymptotic clustering for Coulomb–type potentials, Invent. Math. 99 (1990), 1155–143. [ Si 1 ] B. Simon: Topics in functional analysis, in the proceedings [[Str ]], pp. 17–76. [ Si 2 ] B. Simon: Coupling constant analyticity for the anharmonic oscillator, Ann. Phys. 58 (1970), 76–136. [ Si 3 ] B. Simon: Schr¨odinger semigroups, Bull. Am. Math. Soc. 7 (1982), 447–526. [ Si 4 ] B. Simon: Resonances in N–body quantum systems with dilation analytic potentials and the foundations of time–dependent perturbation theory, Ann. Math. 97 (1973), 247–272. [ Si 5 ] B. Simon: On the number of bound states of two–body Schr¨odinger operators – a review, in the proceedings volume [[LSW ]], pp. 305–326. [ Si 6 ] B. Simon: The bound state of weakly coupled Schr¨odinger operators in one and two dimensions, Ann. Phys. 97 (1976), 279–288. [ Si 7 ] B. Simon: Phase space analysis of some simple scattering systems: extensions of some work of Enss, Duke Math. J. 46 (1979), 119–168. [ Si 8 ] B. Simon: The Neumann Laplacians of a jelly roll, Proc. AMS 114 (1992), 783–785. [ SSp 1 ] B. Simon, T.Spencer: Trace class perturbations and the absence of absolutely continuous spectra, Commun. Math. Phys. 125 (1989), 111–125. [ Sin 1 ] K.B. Sinha: On the decay of an unstable particle, Helv. Phys. Acta 45 (1972), 619–628. [Solo 1] M. Solomyak: On the spectrum of the Laplacian on regular metric trees, Wave Random Media 14 (2004), S155–S171. [ Sol 1 ] J.P. Solovej: Asymptotic neutrality of diatomic molecules, Commun. Math. Phys. 130 (1990), 185–204. [ SM 1 ] F. Sols, M. Macucci: Circular bends in electron waveguides, Phys. Rev. B41 (1990), 11887–11891. [ Ste 1 ] A. Steane: Quantum computing, Rep. Progr. Phys. 61 (1998), 117–173. References 645

[ St 1 ] J. Stubbe: Bounds on the number of bound states for potentials with critical decay at infinity, J. Math. Phys. 31 (1990), 1177–1180. [ Stu 1–4 ] E.C.G. Stueckelberg, M. Guenin, C. Piron, H. Ruegg: Quantum theory in real Hilbert space I–IV, Helv.Phys.Acta33 (1960), 727–752; 34 (1961), 621–628, 675–698; 35 (1962), 673–695. [Sto 1] M.H. Stone: Linear transformations in Hilbert space, III. Operational methods and group theory, Proc. Nat. Acad. Sci. USA 16 (1930), 172–175. [ Sve 1 ] E.C. Svendsen: The effect of submanifolds upon essential self–adjointness and deficiency indices, J. Math. Anal. Appl. 80 (1980), 551–565. [ Te 1 ] A. Teta: Quadratic forms for singular perturbations of the Laplacian, Publ. RIMS 26 (1990), 803–817. [ Til 1 ] H. Tilgner: Algebraical comparison of classical and quantum polynomial ob- servables, Int. J. Theor. Phys. 7 (1973), 67–75. [ Tim 1 ] W. Timmermann: Simple properties of some ideals of compact operators in algebras of unbounded operators, Math. Nachr. 90 (1979), 189–196. [ Tim 2 ] W. Timmermann: Ideals in algebras of unbounded operators, Math. Nachr. 92 (1979), 99–110. [TBV1] G.Timp et al.: Propagation around a bend in a multichannel electron wave- guide, Phys. Rev. Lett. 60 (1988), 2081–2084. [ Tro 1 ] H. Trotter: On the product of semigroups of operators, Proc. Am. Math. Soc. 10 (1959), 545–551. [ Tru 1 ] A. Truman: The classical action in nonrelativistic quantum mechanics, J. Math. Phys. 18 (1977), 1499–1509. [ Tru 2 ] A. Truman: Feynman path integrals and quantum mechanics as → 0, J. Math. Phys. 17 (1976), 1852–1862. [ Tru 3 ] A. Truman: The Feynman maps and the Wiener integral, J. Math. Phys. 19 (1978), 1742–1750; 20 (1979), 1832–1833. [ Tru 4 ] A. Truman: The polynomial path formulation of the Feynman path integrals, in the proceedings volume [[ACH ]], pp. 73–102. [ UH 1 ] J.B.M. Uffink, J. Hilgevoord: Uncertainty principle and uncertainty relations, Found. Phys. 15 (1985), 925–944. [ UH 2 ] J.B.M. Uffink, J. Hilgevoord: New bounds for the uncertainty principle, Phys. Lett. 105A (1984), 176–178. [ Vas 1 ] A.N. Vasiliev: Algebraic aspects of Wightman axiomatics, Teor. mat. fiz. 3 (1970), 24–56 (in Russian). [ VSH 1 ] A.V. Voronin, V.N. Suˇsko, S.S. Horuˇzii: The algebra of unbounded operators and vacuum superselection in quantum field theory, 1. Some properties of Op∗– algebras and vector states on them, Teor. mat. fiz. 59 (1984), 28–48. [ Wa 1 ] X.–P. Wang: Resonances of N–body Schr¨odinger operators with Stark effect, Ann. Inst. H. Poincar´e: Phys. Th´eor. 52 (1990), 1–30. [WWU 1] R.A. Webb et al.: The Aharonov–Bohm effect in normal–metal non–ensemble averaged quantum transport, Physica A140 (1986), 175–182. [ We 1 ] J. Weidmann: The virial theorem and its application to the spectral theory of Schr¨odinger operators, Bull. Am. Math. Soc. 73 (1967), 452–456. [ We 1 ] R.F. Werner: Quantum states with Einstein–Podolsky–Rosen correlation ad- mitting a hidden–variable model, Phys. Rev. A40 (1989), 4277–4281. 646 References

[WWW 1] G.C. Wick, A.S. Wightman, E.P. Wigner: The intrinsic parity of elementary particles, Phys. Rev. 88 (1952), 101–105. [ WWW 2 ] G.C. Wick, A.S. Wightman, E.P. Wigner: Superselection rule for charge, Phys. Rev. D1 (1970), 3267–3269. [ WG 1 ] A.S. Wightman, L. G˚arding: Fields as operator–valued distributions in rela- tivistic quantum theory, Arkiv f¨or Fysik 28 (1964), 129–184. [Wig 1] E.P. Wigner: On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40 (1939), 149–204. [ Wil 1 ] D.N. Williams: New mathematical proof of the uncertainty relations, Amer. J. Phys. 47 (1979), 606–607. [ Wil 2 ] D.N. Williams: Difficulty with a kinematic concept of unstable particles: the Sz.–Nagy extension and Matthews-Salam-Zwanziger representation, Commun. Math. Phys. 21 (1971), 314–333. [ Wol 1 ] K.B. Wolf: The Heisenberg–Weyl ring in quantum mechanics, in the proceed- ings volume [[Loe 3 ]], pp. 189–247. [ Wo 1 ] M.F.K. Wong: Exact solutions of the n–dimensional Dirac–Coulomb problem, J. Math. Phys. 31 (1991), 1677–1680. [ Ya 1 ] K. Yajima: Existence of solutions for Schr¨odinger equations, Commun. Math. Phys. 110 (1987), 415–426. [Ya 2] K. Yajima: Quantum dynamics of time periodic systems, Physica A124 (1984), 613–620. [ YK 1 ] K. Yajima, H. Kitada: Bound states and scattering states for time periodic Hamiltonians, Ann. Inst. H. Poincar´e: Phys. Th´eor. 39 (1983), 145–157. ∗ [ Yea 1 ] F.J. Yeadon: Measures on projections in W –algebras of type II1 , Bull. London Math. Soc. 15 (1983), 139–145. [ Zas 1 ] T. Zastawniak: The non–existence of the path measure for the Dirac equation in four space–time dimensions, J. Math. Phys. 30 (1989), 1354–1358. [ Ziˇ 1 ] G.M. Zislin:ˇ Discussion of the spectrum of the Schr¨odinger operator for systems of many particles, Trudy Mosk. mat. obˇsˇcestva 9 (1960), 81–128. [Zy1] K. Zyczkowski:˙ Classical and quantum billiards, nonintegrable and pseudointe- grable, Acta Phys. Polon. B23 (1992), 245–269. List of symbols

A2(C) 48 a(f) annihilation operator 410 a∗(f) creation operator 410 A0(S) 610 A+ positive cone 208 (∆A)W standard deviation 293 AW mean value of an observable 256 ac(J) 126 AC(J) 97 AC2(J) 143 AE projections in an algebra 216 Bd Borel sets 596 Bε(p1,...,pn) 14 B(V1,V2) bounded operators 17 B(H) bounded operators on a Hilbert space 63 B(Z, µ; X ) 607 bd boundary of a set 5 C complex numbers Cn 1 C(H) scalar operators 217 ∞ Rn C0 ( ) 15 C(X ) bounded continuous functions 2 C∞(X) 16 C(X ) closed linear operators 25 card cardinality 595 D(T ),DT domain of an operator 24 D∞(G) 473 dim dimension 2, 44 EA(·) projection–valued measure 165 {Et} spectral decomposition 152 F(H)Fockspace 403 Fs(H), Fa(H) (anti)symmetric Fock space 404 G HD Dirichlet Laplacian 472 Hα,y point–interaction Hamiltonian 475, 478

647 648 List of symbols

Hn Hermite polynomials 46 id identical mapping 167 J d 595 J1(H) trace–class operators 84 J2(H) Hilbert–Schmidt operators 81 J∞(H), K(H) compact operators 77 Ker kernel of a mapping 596 Lj angular momentum 365 L(X, dµ) integrable functions 602 L2(X, dµ; G) 47 (α) Lk Laguerre polynomials 46 p Lloc(X),Lloc(X) 126, 446 lp 1 l∞ bounded sequences 1 Lp(M,dµ) 2 Lp(M,dµ),Lp(Rn) 2 L∞(M,dµ) 4 Lp + L∞ 444 p ∞ L + Lε 486 L∞(Rd,dE) 156 L(H) densely defined operators 93 Lb,sa(H) bounded and self–adjoint operators 228 Lc(H) closed densely defined operators 97 Lcs(H) closed symmetric operators 120 Ln(H) normal operators 100 Ls(H) symmetric operators 94 Lsa(H) self–adjoint operators 94 l.i.m. limes in medio 19 lin linear envelope 2 Mb(H) bound states 493 Ms(H) scattering states 493 N natural numbers N(V ) number of bound states 455 N (H) bounded normal operators 74 O, Ob observables, bounded observables 270 P, Pj momentum operators 97, 261 Pl Legendre polynomials 46 Pψ(·) decay law 341 P Poincar´egroup 371 Q, Qj position operators 94, 261 Q(A) form domain 115 R real numbers Rn 1 Rd 595 List of symbols 649

R+ positive semiaxis [0, ∞) −1 RT (λ)(T − λ) 27 u RH (λ) reduced resolvent 343 Ress essential range 102 Ran range 596 S(Rn) Schwartz space 15 s lim strong operator limit 66 supp support 600 sup ess essential supremum 4 T (·), Tb(·) functional calculus 156, 160 T Σ(T ), T Π(T ) second quantization 405 Uε(x) ε–neighborhood 5 u lim operator–norm limit V(X , G) vector–valued functions 47 w(∆,A; ψ) measurement outcome probability 256 Wj(W ) reduced states 384 w lim weak limit, weak operator limit 22, 67 Ylm spherical functions 395 Z integers Γ(T ) operator graph 24 − G ∆D Dirichlet Laplacian 472 Θ(T ), Θ(s) numerical range 68, 111 ρ(T ),ρA(a) resolvent set 26, 610 σ(T ),σA(a) spectrum 26, 610 σac absolutely continuous spectrum 175 σc continuous spectrum 26 σess essential spectrum 99 σp point spectrum 26 σr residual spectrum 26 σs singular spectrum 175 σsc singularly continuous spectrum 175 τs strong operator topology 66 τu operator–norm topology 66 τw weak operator topology 67 τσs σ–strong operator topology 213 τσw σ–weak operator topology 214 d ΦE(R ) 160 ΦS(f) Segal field operator 414 χM characteristic function of a set 596 ψp,q canonical coherent states 297 Ω± wave operators 492 2X subsets of set X 595 + algebraic sum 33 650 List of symbols

form sum 116 ⊕ direct, orthogonal sum ⊕ direct, orthogonal sum ⊕ direct integral 54, 140 ⊗ tensor product, product measure × algebraic tensor product 56, 108 × Cartesian product 597 / factorization 2 o interior of a set 23  commutant, dual space  bicommutant 214, 610 ∗ adjoint, involution, dual space ˆ ˇ Fourier transform 18 ⊥ orthogonal complement 4 t transposition 614 closure ◦ composite mapping 597 −1 inverse 597 (−1) pull-back 596 |\ restriction 596 ⊃ extension, inclusion ∩ intersection ∪ union \ set difference ∆ symmetric difference, Laplacian → limit, map → map (point to point) →s strong operator limit 66 →w weak limit, weak operator limit 22, 67 ⇒ implication |·| modulus, norm in Rn 3 [ · ] integer part · norm 3 · p norm 3, 88 · ∞ norm 3, 4 (·, ·) inner product 4 [·, ·] commutator, boundary form Index

accidental degeneracy 374 anticommutator 312, 411 adjoint, element of a ∗–algebra 611 approach, active 359 to a bounded operator 63 passive 359 to an unbounded operator 93 approximation, adiabatic 351 algebra, Abelian 609 pole 352 associative 609 semiclassical 307 Banach 613 asymptotic conditions 492 commutative 609 autoionization of helium 353 complex 609 automorphism of an algebra 612 discrete 218 axiom of choice 598 generated by a set 610 axioms, countability 9 Heisenberg–Weyl 364 G˚arding–Wightman 435 involutive 611 Haag–Araki 441 Jordan 288 Haag–Kastler 437 Lie 614 JNW 288 linear 609 Kuratowski 35 maximal Abelian 610 Nelson 441 normed 613 of topology 7 quasilocal 441 Osterwalder–Schrader 441 real 609 Segal 288 Segal 289 separability 9 exceptional 289 Wightman 441 special 289 simple 610 band spectrum 483 topological 612 baryon number 368 von Neumann 214 basis, of a vector space 2 algebraic sum, of subspaces 50 Hamel 29 of vector spaces 33 local 9 alternative, Fredholm 79 measurability 53 Weyl 132 occupation–number 410 analytic of a measure 57 family, Kato–type 480 of a topological space 9 type (A) 480 orthonormal 44 manifold 615 trigonometric 45 vector 183 Berry phase 351 angular momentum 365 bicommutant, extended 179 orbital 366 in an algebra 610 total 366 of a bounded-operator set 214 anomalous magnetic moment 486 bijection 596

651 652 Index binary operation 609 coherent subspace 269 Bose–Einstein condensation 420 coherent system 270 bosons 391 commutant, in an algebra 610 bound, Bargmann 455 of a bounded–operator set 214 Birman–Schwinger 457 of an operator set 179 Calogero 480 commutativity, of operators 107 Cwikel–Lieb–Roseblium 458 of self–adjoint operators 170 Ghirardi–Rimini 480 with an antilinear operator 375 GMGT 455 commutator 411 relative 104, 116 compatibility, of propositions 427 bound state 497 compatible observables 274 of subsystems 399 a complete set of 280 zero–energy 488 complete bound states, number of 455 envelope, of a normed algebra 613 boundary condition, Dirichlet 137 metric space 7 Neumann 137 boundary conditions set of commuting operators 232 free 564 completion, of a metric space 7 separated 136 of a pre–Hilbert space 41 bounds, of a Hermitean operator 68 complex bracketing 474 extension of a real Lie algebra 614 scaling 353 C∗–algebra 206 condition, Agmon 509 C∗–product of C∗–algebras 241 asymptotic 492 canonical anticommutation relations 312, 413 Enss 509 commutation relations 300, 413 Friedrichs 345 Weyl form of 302 semigroup 342 form of a self–adjoint operator 188 spectral 435 CAR 312, 413 Tang 557 Cartan tensor 614 conductance 538 cascading phenomenon 312 cone 208 Cayley transformation 120 closed future light 435 CCR 300, 412 conjecture, Dirac 288 center, of a lattice 427 conservative system 318 of an algebra 217, 610 conserved quantity 322 of mass, separation of 392 contact interaction 479 chain rule 498 continuity, global 6 classical limit 306 local 5 closure, in a metric space 5 sequential 66 of a form 113 of an operator 25 contraction, of Lie algebras 375 cluster 464 convergence, dominated 603 coherent set 269 monotone 602 maximal 269 of a net 8 coherent states 49, 310 of a sequence 5 canonical 310 strong operator 66 spin 310 weak operator 67 Index 653 convex set 29 spectral 152, 165 extremal point of 29 to partial waves 396 convolution 31 deficiency indices 118 coordinates, atomic 394 deficiency subspace 118 geodesic polar 539 degree of freedom 299, 380 Jacobi 394 dimension, of a vector space 2 coordinates, Jacobi cluster 485 Hilbertian 44 spherical 394 of a manifold 615 generalized 397 of a representation 612 core, for an operator 96 of an atomic lattice 429 for a form 113 relative 242 coupling constant 345, 512 direct integral renormalization 484 of a Hilbert–space field 54 coupling, weak 514 of an operator valued–function 140 covariance, w.r.t. transformations 374 direct product, of groups 615 covering 10 of measures 601 open 10 of projection–valued measures 155 creation operator 410 of σ–fields 597 criterion, Cook 499 direct sum, of vector spaces 2 self–adjointness 103 of Banach spaces 16 ∗ essential self–adjointness 104 of C –algebras 208 cross–norm 58 of Hilbert spaces 52 CSCO 232 of operators 140 ∗ curvature of W –algebras 218 Gauss 539 discrete point of a measure 599 mean 539 dispersion, of a random variable 287 principal 540 of a state 430 signed 528 total 431 curve 30 distribution 32 Peano 244 operator–valued 419 cut, of a function 599 tempered 31 of a set 597 distributive law, on a lattice 427 domain, of a form 111 cyclic vector, of a representation 612 of a mapping 596 of an operator set 244 of an operator 24 decay law 341 dual space 19 for a mixed state 352 algebraic 29 decomposition, Bloch 483 of a topological vector space 21 cluster 482 second 21 Floquet 483 Dyson expansion 334 Hahn 605 effect, Aharonov–Bohm 588 Jordan 605 Stark 353 Lebesgue, of a measure 603 Zeeman 448 of a set 605 eigenfunction expansion method 287 of unity 152 eigenspace 26 polar, of a complex measure 606 eigenvalue 25 of a bounded operator 74 degenerate 480 654 Index eigenvalue, continued form 3 embedded 176 below bounded 111 multiplicity of 26 bilinear 3 simple 26 closable 112 eigenvalue spacing 484 closed 112 eigenvector 25 densely defined 111 endomorphism of an algebra 612 generated by an operator 111 energy, kinetic 260 Killing 614 threshold 511 positive 3, 111 total 262 strictly 3 equation, almost Mathieu 576 quadratic 3, 111 Schr¨odinger 319 real 3 equivalence 597 relatively bounded 116 equivalence class 598 sectorial 140 sesquilinear 3 e.s.a. 96 unbounded 111 essential range 198 symmetric 3 Euler angles 365 form domain, of an operator 115 evolution operator 318 form sum, of self–adjoint operators 116 example, Nelson 300 formal operator, regular 126 Wigner–von Neumann 469 singular 126 extension, Friedrichs 115 formula, Eisenbud–Wigner 523 Lebesgue 600 Feynman–Kac 333 of a map 596 Lagrange 148 of an operator 17 Landauer 538 symmetric 94 Landauer–B¨uttiker 554 polarization 3,4 Stone 178 F –topology 32 Trotter 194 factor 217 Fourier, coefficient 43 algebra 611 expansion 44 group 615 transformation 19 space 2 function 596 family, analytic 480 absolutely continuous 604 monotonic 596 almost periodic 57 separating points 29 Borel 598 Fermi golden rule 347 characteristic 596 fermions 391 cylindrical 332 Feynman integral 332 homogeneous of order −β 470 product 333 integrable 602 field 610 locally 126 measurable Hilbert–space 54 w.r.t. a signed measure 606 field theory, constructive 440 measurable 598 filter 257 of a self–adjoint operator 177 Fock representation, of the CCR 416 of commuting s–a operators 181 Fock space 403 of positive type 196 antisymmetric 404 rapidly decreasing 15 symmetric 404 simple 598 Index 655 function, continued of unitary operators 191 spherical 395 symmetry 362 vector–valued 53 topological 615 analytic 27 velocity 327 σ–simple 598 functional 3 Hamiltonian 262 additive 3 atomic 446 antilinear 3 in the axiomatic approach 435 bounded linear 20 interaction 388 homogeneous 3 time–dependent 319 linear 3 “hedgehog” manifold 589 positive on a ∗–algebra 208 Heisenberg relations 295 ∗ normal 221 Hermitean, element of a –algebra 611 real 3 operator 67 strongly positive 240 hidden parameters 432 Hilbert space 41 gap in the spectrum 483 of analytic functions 49 generator, of a unitary group 191 of vector–valued functions 48 of an operator semigroup 197 rigged 287 GNS representation 210 Hilbertian sum of subspaces 57 GNS triplet 210 homeomorphism, linear 6 Gram determinant 34 of metric spaces 6 graph 24, 561 Hofstadter butterfly 588 boundary 570 H¨older, continuity 349 degree of a vertex 561 inequality 32 edge of 561 Hughes–Eckart term 481 finitely periodic 588 generalized 579 ideal 609 interior 571 gas 420 leaky quantum 581 maximal 610 metric 561 one–sided 545 neighboring vertices 570 identities, resolvent 40 of an operator 24 identity, Hilbert 40 star 593 Jacobi 614 vertex of 561 parallelogram 4 ground state 462 Parseval 44 group 609 of propositions 425 connected 615 implication between propositions 425 compact 615 inequalities between operators 69 Euclidean 370 inequality, Bessel 43 Galilei 370 Faber–Krahn 583 Heisenberg–Weyl 364 Herdy 311 Lie 616 H¨older 32 locally compact 615 Kato 479 Lorentz 372 Minkowski 1, 33 Poincar´e 371 Lieb–Thirring 458, 481 of transformations 360 Schwarz 4, 111 656 Index inequality, Bessel, continued lattice, atomic 427 triangle 5 Boolean 427 infimum, of a proposition set 425 complete orthocomplemented 426 of an operator set 69 countably complete 427 inner product 4 generated by a set 427 integrability 602 irreducible 427 of vector–valued functions 607 weakly modular 427 integral 601 layer, asymptotically planar 541, 543 absolute continuity of 606 curved 539 Bochner 607 locally curved 542 of a complex function 602 lemma, Riesz 42 of motion 322 Riemann-Lebesgue 19 complete system of 323 Schur 236, 244 w.r.t. a projection–valued uncertainty principle 311 measure 161 Zorn 598 interior, of a set 23 Lie algebra 614 point 5 symplectic 614 invariance w.r.t. transformations 361 commutative 614 inverse, decay problem 351 compact 614 element 610 dimension of 614 one–sided 610 exceptional 614 involution, on an algebra 610 general linear 614 involutive automorphism 429 nilpotent 614 irreducibility, algebraic 244 of a Lie group 616 of an operator set 235 orthogonal 614 topological 244 semisimple 614 isometry, linear 6 solvable 614 of metric spaces 6 Lie group 616 Lie subgroup 616 partial 73 limit, thermodynamic 422 isomorphism, isometric 613 limit–circle case 132 of algebras 611 limit–point case 132 of Hilbert spaces 41 linear, homeomorphism 13 of proposition systems 429 hull 2 of topological groups 615 independence, of vectors 2 of vector spaces 2 modulo L 135 spatial 217 linearly independent set 2 topological 13 loop 538 isospin 368 Lorentz group 372 proper orthochronous 375 Jacobi matrix 572 map 596 Kato class 479 composite 597 kernel, of a mapping 596 injective 596 Kramers degeneracy 374 inverse 597 regular 604 Laplacian, Dirichlet 472 surjective 596 Neumann 472 uniformly continuous 30 Index 657 mapping 596 method Maslov correction 349 eigenfunction expansion 287 mass operator 435 Howland 351 matrix representation of induced representations 375 of a bounded operator 66 Rayleigh–Ritz 452 matrix, skew–symmetric 614 WKB 312 maximal common part metric 5 of self–adjoint operators 125 induced by a norm 5 maximal element 598 metric space 5 mean value, of a random variable 287 completely bounded 11 mean–square deviation 287 complete 6 measurability separable 5 of a complex function 598 microcausality 436 of a vector–valued function 47 mixed state, on a C∗–algebra 242 measure second–kind 399 absolutely continuous mixture, of states 287 w.r.t. another measure 603 mode–matching technique 553 complex 606 model, Friedrichs 345 Borel 600 Kronig–Penney 490 complete 599 Lee 352 complex 605 standard 374 Borel 606 Wannier–Stark 485 concentrated on a set 600 moment, of a probability measure 287 discrete 599 momentum, canonically conjugate 414 finite 599 of a particle 259 generated by a function monodromy operator 339 and a measure 602 morphism, canonical 612 invariant 615 of algebras 611 Lebesgue 601 multiplication 609 Lebesgue–Stieltjes 601 multiplicativity left Haar 615 of operator integrals 157 non–negative 599 multiplicity, of an eigenvalue 26 outer 600 of the spectrum 514 product 605 multiplier projection–valued 151 of a projective representation 361 real 605 neighborhood 5 regular 601 in a topological space 7 signed 605 net 8 space 599 norm 3 spectral 151 dominated by another norm 35 Wiener 333 Hilbert–Schmidt 82 σ–finite 599 induced by an inner product 4 measures, mutually singular 603 normal element, of a ∗–algebra 611 measurement 251 normal form first–kind 286 of second–quantized operator 421 preparatory 280 normed space 3 second–kind 286 reflexive 21 658 Index norms, equivalent 35 bounded 17 null–space of a morphism 612 closable 25 numerical range, of a form 111 closed 25 of an operator 68 compact 77 canonical form of 81 O∗–algebra 240 densely defined 93 observable, bounded 270 dissipative 139 fundamental 283 essentially self–adjoint 96 in a broader sense 279 finite–dimensional 77 operational definition of 252 Hermitean 67 preserved by a transformation 361 Hilbert–Schmidt 81 quasilocal 437 Hilbert–Schmidt integral 89 observables, algebra of 270, 284 irreducible 106 Op∗–algebra 239 isometric 73 closed 239 linear 17 self–adjoint 239 local 370 standard 239 maximal symmetric 94 operator adjoint 63, 93 mutually dominated 504 associated with a form 115 n–particle 404 Banach adjoint 87 normal 74, 100 conjugation 148 of multiplication 101 J dual 87 of the class p 88 essentially self–adjoint 96 of the number of particles 407 Fourier–Plancherel 19 positive 68, 139 Laplace–Beltrami 400, 541 pseudo–differential 245 in a layer 540 reducible 105 left–shift 64 relatively bounded 104 Møller 522 relatively compact 173 norm 17 scalar 211 product 17 sectorial 139 right–shift 64 self–adjoint 94 Schr¨odinger 443 dominated by a s–a operator 504 splitting of 130 statistical 86 Weyl 304 strictly positive 68 operator set, commutative 228 Sturm–Liouville 141 irreducible 235 superselection 272 nondegenerate 215 symmetric 94 reduced by a subspace 235 trace–class 84 reducible 235 on a Banach space 88 symmetric 227 unitary 72 operator set, ∗–invariant 243 with a compact resolvent 173 operator sets, commuting 228 ordering, partial 598 operators, accretive 139 orthocomplementation 426 antihermitean 87 orthogonal, complement 4 antiunitary 87 projection of a vector 42 below bounded 139 set 5 Index 659

sum of operators 105 (Q2b) 256, 265 vectors 4 (Q3) 257 orthonormal, basis 44 (Q4a) 318 set 5 (Q4b) 318 (Q4c) 319 parastatistics 400 (Q5a) 359, 360 parity 73, 323, 367 (Q6a) 379 internal 367 (Q6b) 392 violation 373 (Q6c) 398 part of an operator 105 potential 443 absolutely continuous 175 centrally symmetric 394 singular 175 periodic 483 singularly continuous 175 purely attractive 456 partial isometry 73 repulsive 471 final subspace of 74 predual, of a W ∗–algebra 242 initial subspace of 74 pre–Hilbert space 4 partition 464 principle, Birman–Schwinger 456, 547 picture, Heisenberg 323 invariance of wave operators 506 interaction (Dirac) 324 minimax 80, 448 Schr¨odinger 323 of uniform boundedness 23 Poincar´egroup 371 Pauli 392 point, accumulation 5 superposition 254 in a lattice 427 uncertainty 298 interaction unique continuation 588 in one dimension 475, 477 probability density 260 in three dimensions 478 product, manifold 616 isolated 5 Cartesian 593 limit 5 symmetrized 288 Poisson bracket 308 projection 70 polarization 266 Abelian 217 formula 3, 4 in a ∗–algebra 611 pole, of a surface 539 minimal 247 polynomials, Hermite 46 orthogonal 87 Laguerre 46 projections, complete system of 72 Legendre 46 equivalent 217 position, of a particle 259 projection–valued measure 151 ∗ positive element, of a C –algebra 208 discrete 152 postulate propagator, reduced 341 (aoc) 284 unitary 317 (aos) 284 proposition 425 (aow) 284 absurd 426 (pl) 426 false 425 (plq) 429 lattice 426 (Q1a) 253, 255 modular 438 (Q1b) 253, 265 not valid 425 (Q1c) 255 system 429 (Q2a) 256 trivial 426 660 Index proposition, continued faithful 612 true 425 irreducible 612 valid 425 isometric, of a C∗–algebra 212 propositions, disjoint 428 of a group 612 simultaneously valid 425 of an algebra 612 pull–back 532 of a Lie algebra 612 projective 361 quantization, Dirac 308 space 612 practical 309 representations, equivalent 612 second 406 resolvent, first identity 40 quantum computing 399 of an operator 27 quantum field, at an instant 422 of element of an algebra 610 free 413, 419 reduced 343 generalized 440 second identity 40 Hermitean 419, 440 set 27 quantum kinematic of decays 351 of element of an algebra 610 quantum logic 438 resonance, shape 353 quantum theory, quaternionic 430 zero–energy 513 real 430 restriction, of a map 596 quantum waveguide 527 of an operator 17 a local deformation 530 ring 609 quantum wire 552 set 595 quarks 374 Rollnik class 479 quasienergy 339 rotation, in plane 365 quaternions 438 in space 365 qubit 399 of spin 366 range, essential 103 S–matrix 493 numerical 68, 111 scalar 1 of a mapping 596 product 4 ray 253 scattering, direct problem 522 reduction formulas 386 inverse problem 522 reflection 73 of subsystems 399 regular endpoint 126 operator 493 regular value 27 Schr¨odinger equation 319 of element of an algebra 610 Schr¨odinger operator 443 regularity domain 40, 75 Schr¨odinger representation relation 597 of the CCR 302 intertwining 498 Schwarz inequality 3, 4, 111 of equivalence 597 on a C∗–algebra 209 of spin to statistics 390, 440 Schwinger functions 441 relations, de Morgan 595 second quantization 406 uncertainty 294, 295 Segal field operator 414 relative bound, of a form 116 semianalytic vector 184 of an operator 104 seminorm 3 representation sequence, Cauchy 6 completely reducible 244 occupation number 409 Index 661 series, Neumann 32 space, topological vector 13 Puiseaux 516 vector 1 Rayleigh–Schr¨odinger 454 complex 1 set, arcwise connected 30 real 1 balanced 38 space–like separation 436 Borel 596 spectral, decomposition 152, 165 bounded 30 radius 28 closed 5 representation 188 compact 10 theorem 165 weakly 22 spectrum, absolutely continuous 175 completely bounded 11 continuous 26 connected 30 of a self–adjoint operator 176 dense, everywhere 5 discrete 173 in a set 5 essential 99 nowhere 5 of an operator 26 directed 8 of element of an algebra 610 field 596 point 26 function 599 pure point 76, 103 regular 600 purely discrete 173 open 5, 7 residual 26 precompact 11 simple 188 system, centered 37 singular 175 total 15 singularly continuous 175 weakly compact 22 spin 276 sets, monotonic family of 596 relation to statistics 399, 440 shape of the drum 484 splitting trick 130 simple square root, of an operator 69 algebra, type An 614 stability of matter 311, 482 type Bn 614 standard, completion procedure 7 type Cn 614 deviation of a random variable 287 type Dn 614 state function, vector–valued 607 antibound 513 eigenvalue 26 asymptotic 492 singular, endpoint 126 bound 497 value, of a compact operator 81 bounded–energy 273 space, Banach 15 completely additive 243 Bose–Fock 421 component 384 Fermi–Fock 421 conditions of preparation 252 Fr´echet 15 dispersionless 431 Hilbert 41 finite–energy 272 metric 5 Gibbs 420 normed 3 minimum–uncertainty 297 pre–Hilbert 4 mixed 265 reflection 367 normal 221 Schwartz 15 on a Banach ∗–algebra 208 Sobolev 287 on a proposition system 430 topological 7 mixed, pure 430 662 Index state, continued surface, Cartan–Hadamard 542 on an Op∗–algebra 240 complete 543 operational definition of 252 end of 545 preparation of 251 symmetric, operatorquad 94 pure 253, 265 subset of a ∗–algebra 611 on a C∗–algebra 213 symmetry reduced 384 group 362 scattering 492 commutative 362 space 253 continuous 362 Banach 289 discrete 362 n–particle 404 finite 362 one–particle 404 Lie 362 stationary 322 noncommutative 362 Werner 399 hidden 369 statistics, Bose–Einstein 391 w.r.t. transformations 362 Fermi–Dirac 391 system, composed 379 step function 16 of seminorms 14 strangeness 368 separating points 14 structure constants 614 ∗ of a Lie group 616 tensor C –product 218 subalgebra 609 tensor product, algebraic 56, 58 closed 612 of bounded operators 108 subgroup 615 of Hilbert spaces 55 invariant 615 of unbounded operators 109 ∗ sublattice 427 of W –algebras 218 subnet 30 realization of 54 ∗ subspace 2 tensor W –product 218 invariant, of an operator 26 theorem, adiabatic 351 of an operator set 235 Baire 35 Lagrangean 562 bicommutant 242 of a topological space 9 Birman 504 singlet 384 Birman–Kuroda 504 triplet 384 Bochner 197 substitution operators 72 Bolzano–Weierstrass 30 subsystem 379 Burnside 245 subsystems, noninteracting 388 closed–graph 25 summation 609 dominated–convergence 603 convention 276, 614 edge–of–the–wedge 352 superposition, of states 254 Ehrenfest 321 principle 254 embedding 288 superselection rule 272 Gleason 434 Bargmann 371 Haag 422 superselection rules Heine–Borel 10 commutativity of 272 Hellinger–Toeplitz 95 support, of a measure 600 HVZ 466 supremum, of a proposition set 426 inverse–mapping 24 of an operator set 69 Jauch–Piron 433 Index 663 theorem, adiabatic, continued locally compact 11 Jordan–Wigner 312 locally convex 14 Kaplansky density 242 normal 10 Kato, on atomic Hamiltonians 445 regular 10 on helium spectrum 461 second countable 9 on positive eigenvalues 469 σ–compact 11 Kato–Rellich 105 topological vector space 13 Kato–Rosenblum 504 complete 31 KLMN 117 weakly complete 22 Kupsch–Sandhas 500 topology 7 Levi 602 discrete 7 Molˇcanov 464 F–weak 8 monotone–convergence 602 relative 9 Noether 369 strong operator 66 open–mapping 23 trivial 7 orthogonal–decomposition 42 ultrastrong 242 Pearson 500 ultraweak 242 Piron 429 weak 14, 21 Radon–Nikod´ym 603 weak operator 67 reconstruction of fields 441 σ–strong 213 representation for forms 114 Riesz representation 601 σ–weak 214 Riesz–Markov 31 trace class 84 trace, of an operator 85 Sakai 242 ∗ SNAG 197 on a W –algebra 220 spectral 165 faithful 220 Uryson 30 finite 220 virial 469 normal 220 von Neumann, density 247 semifinite 220 on a generating operator 243 translation, n–dimensional 364 on deficiency indices 148 on line 363 on the generating operator 229 tree graph 588 Weyl 201 regular 588 W¨ust 140 rooted 588 theory, Gel’fand 241 triangle inequality 5 Kato–Birman 522 time delay 523 uncertainty principle 298 time–ordered exponential 334 local 311 topological isomorphism uncertainty relations 294 of algebras 612 uniform topology ∗ topological product 9 on an Op –algebra 240 topological space 7 unit element, of a group 609 arcwise connected 30 of an algebra 609 compact 10 unitary element, of a ∗–algebra 611 connected 30 unitary equivalence 107 first countable 9 unitary group, one–parameter 191 Hausdorff 10 strongly continuous 191 664 Index

unitary invariant 140 type In, I∞ 219 upper bound 598 type II, II1, II∞ 220 type III 220 vacuum 403 Wannier ladder 354 in the axiomatic approach 435 wave function 259 mean 436 wave operators 492 variation 408 asymptotic completeness 497 of a measure, total 605 completeness 497 negative 605 existence 497 positive 605 generalized 500 vector 1 Weyl relations 302 analytic 183 Wiener integral 333 cyclic 244, 612 Wightman functions 440 generating 188 Wronskian 149 semianalytic 184 vertex coupling yes–no experiment 256 free 564 yes–no experiments, compatible 291 Kirchhoff 587 disjoint 291 permutation–invariant 564 δ–interaction, in one dimension 475 δ 563  in three dimensions 478 δ 564   δ –interaction 477 δs 564 ε–lattice 11 δp 590 σ–additivity, of the integral 602 σ–field 596 W ∗–algebra 214 continuous 219 ∗–algebra 611 finite 220 Banach 205 homogeneous 219 normed 205 properly infinite 220 partial 245 purely infinite 220 topological 205 reduced 217 ∗–ideal 611 semifinite 220 ∗–morphism 612 type I 218 ∗–subalgebra 611