Introduction to the Spectral Theory Lecture Notes of the Course Given At

Lecture notes of the course given at the University Paris-Sud (Orsay, France) in September-December 2012

within the master programs “Partial Diﬀerential Equations and Numerical Analysis” and “Analysis, Arithmetics and Geometry”

Lecturer: Konstantin Pankrashkin

Webpage of the course: http://www.math.u-psud.fr/~pankrash/2012spec/ Contents

Notation 1

1 Unbounded operators 3 1.1 Closedoperators ...... 3 1.2 Adjointoperators...... 6

2 Operators and forms 9 2.1 Operatorsdeﬁnedbyforms...... 9 2.2 Semibounded operators and Friedrichs extensions ...... 14

3 Spectrum and resolvent 16 3.1 Deﬁnitions...... 16 3.2 Examples ...... 17 3.3 Basic facts on spectra of self-adjoint operators ...... 19

4 Spectral theory of compact operators 23 4.1 Integral and Hilbert-Schmidt operators ...... 25 4.2 Operators with compact resolvent ...... 28 4.3 Schr¨odinger operators with growing potentials ...... 29

5 Spectral theorem 31 5.1 Continuous functional calculus ...... 32 5.2 Borelian functional calculus and L2 representation ...... 37

6 Some applications of spectral theorem 44 6.1 Spectralprojections...... 44 6.2 Generalized eigenfunctions ...... 47 6.3 Tensorproducts...... 48

7 Perturbations 50 7.1 Kato-Rellichtheorem...... 50 7.2 Essential self-adjointness of Schr¨odinger operators ...... 52 7.3 Discreteandessentialspectra ...... 55 7.4 Weyl criterion and relatively compact perturbations ...... 56 7.5 Essential spectra for Schr¨odinger operators ...... 59 7.6 Perturbationsofdiscretespectra...... 60

8 Variational principle for eigenvalues 63 8.1 Max-minandmin-maxprinciples ...... 63 8.2 Negative eigenvalues of Schr¨odinger operators ...... 65

9 Laplacian eigenvalues for bounded domains 68 9.1 Dirichlet and Neumann eigenvalues ...... 68 9.2 Weylasymptotics ...... 71 9.3 Simplicity of the lowest eigenvalue ...... 74

10 Self-adjoint extensions 77 10.1 Description of self-adjoint extensions ...... 77 10.2Spectralanalysis ...... 82

References 88 Notation

Here we list some conventions used throughout the text. The symbol N denotes the sets of the natural numbers starting from 1 (contrary to the French tradition where one starts with 0). If (M,µ) is a measure space and f : M C is a measurable function, then we denote →

ess ran f := z C : µ m M : z f(m) <ε > 0 for all ε> 0 , µ ∈ ∈ − n o ess sup f := inf a R : µ m M : f(m) >a = 0 . µ | | ∈ ∈ n o If the measure µ is uniquely determined by the context, then the index µ will be sometimes omitted. In what follows the phrase “Hilbert space” should be understood as “separable complex Hilbert space”. Most propositions also work in the non-separable case if reformulated in a suitable way. If the symbol “ ” appears without explanations, it denotes a certain Hilbert space. If is a HilbertH space and x, y , then by x, y we denote the scalar product of x andH y. If there is more than one∈H Hilbet spaceh ini play, we use the more detailed notation x, y . We assume that the scalar product h iH is linear with respect to the second argument and as anti-linear with respect to the ﬁrst one, i.e. that for all α C we have x, αy = αx, y = α x, y . This means, for example, that the scalar∈ product in theh standardi h spacei L2(Rh) isi deﬁned by

f,g = f(x)g(x) dx. h i R Z If A is a ﬁnite or countable set, we denote by ℓ2(A) the vector space of the functions x : A C with 2 → ξ(a) < , ∞ a A X∈ and this is a Hilbert space with the scalar product

x, y = x(a)y(a). h i a A X∈ If and are Hilbert spaces, then by ( , ) and ( , ) we denotes the spaces of theH linearG operators and the one of the compactL H G operatorsK H froG m and , respectively. Furtheremore, ( ) := ( , ) and ( ) := ( , ). H G L H L H H K H K H H If Ω Rd is an open set and k N, then Hk(Ω) denotes the kth Sobolev space, i.e. the⊂ space of L2 functions whose∈ partial derivatives up to order k are also in 2 k L (Ω), and by H0 (Ω) we denote the completion of Cc∞(Ω) with respect to the norm of Hk(Ω). The symbol Ck(Ω) denotes the space of functions on Ω whose partial derivatives up to order k are continuous; i.e. the set of the continuous functions 0 d is denoted as C (Ω). This should not be confused with C0(R ) which is the set

1 d of the continuous functions f on R vanishing at inﬁnity: lim x f(x) = 0. The | |→∞ subindex comp means that we only consider the functions with compact support in 1 Rd 1 Rd the respective space. E.g. Hcomp( ) is the set of the functions from H ( ) having compact supports.

2 1 Unbounded operators

1.1 Closed operators

A linear operator T in is a linear map from a subspace (the domain of deﬁnition H of T ) dom A to . The range of T is the set ran T := T x : x dom T . We say that a linear⊂H operatorH T is bounded if the quantity { ∈ }

T x µ(T ) := sup k k x dom T x ∈x=0 k k 6 is ﬁnite. In what follows, the word combination “an unbounded operator” should be understood as “an operator which is not assumed to be bounded”. If dom T = and T is bounded, we arrive at the notion of a continuous linear operator in ; theH space of such operators is denoted by ( ). This is a Banach space equippedH with the norm T := µ(T ). L H k k During the whole course, by introducing a linear operator we always assume that its domain is dense, if the contrary is not stated explicitly. If T is a bounded operator in , it can be uniquely extended to a continuous linear H operator. Let us discuss the question on the continuation of unbounded operators. The graph of a linear operator T in is the set H gr T := (x, T x) : x dom T . ∈ ⊂H×H For two linear operators T1 and T2 in we write T1 T2 if gr T1 gr T2. I.e. T T means that dom T dom T andH that T x = T⊂x for all x dom⊂ T ; T is 1 ⊂ 2 1 ⊂ 2 2 1 ∈ 1 2 then called an extension of T1 and T1 is called a restriction of T2. Deﬁnition 1.1 (Closed operator, closable operator).

A linear operator T in is called closed if its graph is a closed subspace in • . H H×H A linear operator T in is called closable, if the closure gr T of the graph of • T in is still theH graph of a certain operator T . This operator T with gr T =H×Hgr T is called the closure of T .

The following propositions are obvious

Proposition 1.2. A linear operator T in is closed if and only if the three conditions H

x dom T , • n ∈ x converge to x in , • n H 3 T x converge to y in • n H imply the inclusion x dom T and the equality y = T x. ∈ Proposition 1.3. Let T be a linear operator in . Equip dom T with the scalar H product u, v T = u, v, + Tu,Tv . h i h iH h iH T is closed if and only if (dom T, , ) is a Hilbert space. • h· ·iT If T is closable, then dom T is exactly the completion of dom T with respect to • , . h· ·iT Consider some examples.

Example 1.4 (Bounded linear operators are closed). By the closed graph theorem, a linear operator T in with dom T = is closed if and only if it is bounded. In this course we considerH mostly unboundedH closed operators.

Example 1.5 (Multiplication operator). Take again = L2(Rd) and pick f d H ∈ L∞ (R ). Introduce a linear operator M in as follows: loc f H dom M = u L2(Rd) : fu L2(Rd) and M u = fu for u dom M . f { ∈ ∈ } f ∈ f Let us show that this operator is closed.

Note ﬁrst that one can construct an orthonormal basis (em)m N of consisting of ∈ H functions with compact support, and then one has clearly em dom Mf . 2 ∈ 2 Let (u ) dom M be a sequence such that u L u and M u = fu L v. For n ⊂ f n −→ f n n −→ any ﬁxed basis vector em we have then:

n n e , u →∞ e , u and e , M u →∞ e ,v . h m ni −−−→h m i h m f ni −−−→h n i On the other hand, using the integral expression for the scalar product, n e , M u = fe , u →∞ fe , u , which implies the equalities h m f ni h m ni −−−→h m i

f(x)em(x)u(x)dx = em(x)v(x)dx m N. Rd Rd ∀ ∈ Z Z By the Parseval equality,

2 2 f(x)em(x)u(x)dx em(x)v(x)dx < , Rd Rd ∞ m N Z m N Z X∈ X∈ which shows that fu L2(Rd). This gives u dom M and, ﬁnally, v = fu = M u. ∈ ∈ f f An interested reader can generalize this example by considering multiplications operators in measure spaces.

4 d 2 d Example 1.6 (Laplacians in R ). Take = L (R ) and denote by T0 the operator H Rd acting as T0u = ∆u with the domain dom T0 = Cc∞( ). We are going to show − Rd that this operator is not closed. To see this we can use the density of Cc∞( ) in the 2 Rd 2 Rd Rd Sobolev space H ( ). Take u H ( ) Cc∞( ), then one can find a sequence Rd ∈ \ un Cc∞( ) such that un converge to u and T0un = ∆un converge to ∆u in the L2-norm.∈ As u / dom T , T is not a closed operator.− − ∈ 0 0 Introduce another operator T1 in acting as T1u = ∆u but now with the domain 2 d H − dom T1 = H (R ). We are going to show that this operator is now closed. Denote by F : the Fourier transform and consider the following operator T in : H→H H dom T = f L2 : p p2f(p) L2 , T f(p)= p2f(p). { ∈ 7→ ∈ } b Indeed, T is closed operator, as this is just a multiplication operator, see Example b b 1.5. On the other hand for f one has the following equivalence: f dom T ∈ H ∈ 1 iff F f bdom T , and in that case F T1f(p) = T F f(p). In other words, one can represent∈ 1 1 b gr T = (F − u, F − T u) : u domb T = K(gr T ), 1 { ∈ } 1 1 where K is the linear operator in defined by K(x, y)=(F − x, F − y). As F is a unitary operator, so is K, whichH×Hb means, in particular,b that bK maps closed sets to closed sets. As gr T is closed, the graph gr T1 is also closed.

Let us show that T0 and T1 are related by T0 = T1. Take u dom T 0. Be definition, b Rd ∈ there exists a sequence of un Cc∞( ) such that un converge to u in and that ∈ Rd H ∆un converge to some v . This means that for any φ Cc∞( )) the scalar products− u ,φ converge to∈ Hu, φ , and ∆u ,φ converge to∈ v,φ . On the other h n i h i h− n i h i hand, using the theory of distributions we can write ∆un,φ = un, ∆φ and the h− i h − i Rd term on the right-hand side converge to u, ∆φ . Therefore, for any φ Cc∞( ) we have the equality v,φ = u, ∆φ , whichh − meansi that v = ∆u in the∈ sense of distributions. Passingh to thei Fourierh − transform,i this means that−u and v ara related by v(p) = p2u(p). So the function p p2u(p) belongs to L2, and then also all the α 7→ 2 Rd products p p u(p), with any multiindex α 2, belong to Lb( ).b This means 27→Rd | | ≤ thatb u H (b ) and that T 0u = ∆u. Web have just shown the inclusion T 0 T1. The inclusion∈ T T is shown when− discussing the operator T (see above). ⊂ 1 b⊂ 0 0 Example 1.7 (Non-closable operator). Take = L2(R) and pick a g with g = 0. Consider the operator L defined on dom L =HC0(R = L2(R) by Lf∈H= f(0)g. 6 ∩ Assume that there exists the closure L and let f dom L. One can find two ∈ 2 sequences (fn), (gn) in dom L such that both converge in the L norm to f but such that fn(0) = 0 and gn(0) = 1 for all n. Then Lfn = 0, Lgn = g for all n, and both sequences Lfn and Lgn converge, but to different limits. This contradicts to the closedness of L. Hence L is not closed. Example 1.8 (Partial differential operators). Let Ω be an open subset of Rd and P (x, Dx) be a partial differential expression with C∞ coefficients. Introduce in 2 := L (Ω) a linear operator P by: dom P = Cc∞(Ω), P u(x)= P (x, Dx)u(x). Like inH the previous example one shows the inclusion

gr P (u, f) : P (x, D )u = f in ′(Ω) , ⊂ ∈H×H x D 5 hence gr P is still a graph, and P is closable.

So we see that we naturally associate with the differential expression P (x, Dx) several linear operators (besides P ), in particular, the minimal operator Pmin := P , which is always closed, and the maximal operator Pmax defined by Pmaxu = P (x, Dx)u on the domain dom P := u : P (x, D )u , max ∈H x ∈H where P (x, Dx)u is understood in the sense of distributions. Clearly, one always has the inclusion P P , and we saw in example 1.6 that one can have P = min ⊂ max min Pmax. But one can easily find examples where this equality does not hold. For 1 example, for P (x, Dx) = d/dx and Ω = (0, + ) we have dom Pmin = H0 (0, ) 1 ∞ ∞ and dom Pmax = H (0, ). In general, one expects that Pmin = Pmax if Ω has a boundary. ∞ 6 Such questions become more involved if one studies the partial differential operators with more singular coefficients (e.g. with coefficients which are not smooth but just belong to some Lp).

1.2 Adjoint operators

Recall that for T ( ) its adjoint T ∗ is deﬁned by the relation ∈ L H

x, T y = T ∗x, y for all x, y . h i h i ∈H The proof of the existence comes from the Riesz representation theorem: for each x the map y x, T y C is a continuous linear functional, which means ∈H H∋ 7→ h i∈ that there exists a unique vector, denoted by T ∗x with x, T y = T ∗x, y for all h i h i y . One can then show that the map x T ∗x is linear, and by estimating the ∈H 7→ scalar product one shows that T ∗ is also continuous. Let us use the same idea for unbounded operators.

Definition 1.9 (Adjoint operator). If T be a linear operator in , then its adjoint H T ∗ is defined as follows. The domain dom T ∗ consists of the vectors u for which the map dom T v u,Tv C is bounded with respect to the∈H-norm. For ∋ 7→ h i ∈ H such u there exists, by the Riesz theorem, a unique vector, to be denoted by T ∗u such that u,Tv = T ∗u, v for all v dom T . h i h i ∈ We note that the implicit assumption dom T = is important here: if it is not H satisfied, then the value T ∗u is not uniquely determined, one can add to T ∗u an arbitrary vector which is orthogonal to dom T in . H Let us give a geometric interpretation of the adjoint operator. Consider a unitary linear operator J : , J(x, y)=(y, x) H×H→H×H − and note that J commutes with the operator of the orthogonal complement. This will be used several times during the course.

6 Proposition 1.10 (Geometric interpretation of the adjoint). Let T be a linear operator in . A vector u belongs to dom T ∗ and f = T ∗u if and only if one H ∈ H has (u, T ∗u),J(v,Tv) = 0 for all v dom T . In other words, h iH×H ∈ gr T ∗ = J(gr T )⊥. (1.1)

As a simple application we obtain

Proposition 1.11. One has ( T )∗ = T ∗, and T ∗ is a closed operator.

Proof. Follows from (1.1): the orthogonal complement is always closed, and J(gr T )⊥ = J(gr T )⊥.

Up to now we do not know if the domain of the adjoint contains non-zero vectors. This is discussed in the following proposition. Proposition 1.12 (Domain of the adjoint). Let T be a closable operator , then: H

(i) dom T ∗ is dense in , H (ii) T ∗∗ := (T ∗)∗ = T .

Proof. The item (ii) easily follows from (i) and (1.1): one applies the same operations again and remark that J 2 = 1 and that taking twice the orthogonal complement results in taking the closure.−

Now let us prove the item (i). Let a vector w be orthogonal to dom T ∗: u, w = ∈H h i 0 for all u dom T ∗. Then one has J(u, T ∗u), (0,w) u, w + T ∗u, 0 = 0 ∈ h iH×H ≡ h i h i for all u dom T ∗, which means that (0,w) J(gr T ∗)⊥ = gr T . As the operator T is closed,∈ the set gr T must be a graph, which∈ means that w = 0.

Let us look at some examples. Example 1.13 (Adjoint for bounded operators). The general deﬁnition of the adjoint operator is compatible with the one for continuous linear operators.

d Example 1.14 (Laplacians in R ). Let us look again at the operators T0 and T1 from example 1.6. We claim that T0∗ = T1. To see this, let us describe the adjoint 2 Rd T0∗ using the deﬁnition. The domain dom T0∗ consists of the functions u L ( ) for which there exists a vector f L2(Rd) such that the equality ∈ ∈ u(x)( ∆v)(x)dx = f(x)v(x)dx Rd − Rd Z Z Rd holds for all v dom T0 Cc∞( ). This means that one should have f = ∆u ∈ ≡ − 2 in the sense of distributions. Therefore, dom T0∗ consists of the functions u L such that ∆u L2. Passing to the Fourier transform, the conditions u L2∈and p p2u(p−) L2∈imply p pαu(p) L2 for any multiindex α 2, which∈ means exactly7→ that∈u H2(Rd) =7→ dom T . ∈ | | ≤ ∈ 1 b b b 7 Example 1.15. As an exercise, one can show that for the multiplication operator

Mf from example 1.5 one has (Mf )∗ = Mf .

The following deﬁnition introduces two classes of linear operator that will be studied throughout the course.

Deﬁnition 1.16 (Symmetric and self-adjoint operators). We say that a linear operator T in is symmetric (or Hermitian) if H u,Tv = Tu,v for all u, v dom T. h i h i ∈

Furthermore, T is called self-adjoint if T = T ∗.

An important feature of symmetric operators is their closability:

Proposition 1.17. Symmetric operators are closable.

Proof. Indeed for a symmetric operator T we have gr T gr T ∗ and, due to the ⊂ closedness of T ∗, gr T gr T ∗. ⊂ d Example 1.18 (Self-adjoint Laplacian in R ). For the laplacian T1 from example 1.6 one has T1 = T1∗. Indeed, T1 = T0∗, then T1∗ = T0∗∗ = T0 = T1. Example 1.19 (Bounded symmetric operators are self-adjoint). Note that for T ( ) the fact to be symmetric is equivalent to the fact to be self-adjoint, but it∈ is L notH the case for unbounded operators!

Example 1.20 (Self-adjoint multiplication operators). As follows from example 1.5, the multiplication operator Mf from example (1.5) is self-adjoint iﬀ f(x) R for a.e. x Rd. ∈ ∈ A large class of self-adjoint operators comes from the following proposition.

Proposition 1.21. Let T be an injective self-adjoint operator, then its inverse is also self-adjoint.

1 Proof. We show ﬁrst that dom T − := ran T is dense in . Let u ran T , then u,Tv = 0 for all v dom T . This can be rewritten asHu,Tv =⊥ 0,v for all h i ∈ h i h i v dom T , which shows that u dom T ∗ and T ∗u = 0. As T ∗ = T , we have u ∈ dom T and T u = 0. As T in injective,∈ one has u = 0 ∈ Now consider the operator S : given by S(x, y)=(y, x). One has 1 H×H→H×H then gr T − = S(gr T ). We conclude the proof by noting that S commutes with J and with the operation of the orthogonal complement in . H×H

8 2 Operators and forms

2.1 Operators deﬁned by forms

A sesquilinear form a in a Hilbert space is a map a : dom a dom a C H H×H ⊃ × → which is linear with respect to the second argument and is antilinear with respect to the ﬁrst one. By default we assume that dom a is a dense subset of . H Now let be a Hilbert space and let a be a bounded sesquilinear form in , i.e. assume thatV dom a = and that there exists some C > 0 with V V a(u, v) C u v for all u, v . | | ≤ k kV k kV ∈ V It is known that there is a uniquely determined operator A (V ) such that a(u, v)= u, Av for all u, v . Let us try to recover some additional∈ L properties h iV ∈ V of A from some information on the form a

Deﬁnition 2.1 ( -ellipticity). We say that a bounded sesqulinear form a is - elliptic if there existsV some α> 0 such that a(u, u) α u 2 for all u . V ≥ k kV ∈ V

Theorem 2.2 (Lax-Milgram theorem). Let a be bounded and -elliptic, then the associated operator A is an isomorphism of . V V Proof. By -ellipticity we have, for all v , V ∈ V α v 2 a(v,v) = v, Av v Av . k kV ≤ h iV ≤ k kV · k kV

Hence Av α v for all v . (2.1) k kV ≥ k kV ∈ V Step 1. We can see that A is injective, because Av = 0 implies v = 0 by (2.1).

Step 2. Let us show that ran A is closed. Assume that fn ran A and that fn converge to f in the norm of . By the result of step 1 there are∈ uniquely determined vectors v with f = AvV. The sequence (f )=(Av ) is convergent and is then n ∈ V n n n n a Cauchy one. By (2.1), the sequence (vn) is also a Cauchy one and, due to the completeness of , converges to some v . As A is continuous, Avn converges to Av. Hence, f = VAv, which shows that f∈ Vran A ∈ Step 3. Let us show ﬁnally that ran A = . As we showed already that ran A is V closed, it is suﬃcient to show that (ran A)⊥ = 0 . Let u ran A, then a(u, v) = u, Av = 0 for all v V . Taking v = u we{ obtain} a(u,⊥ u) = 0, and u = 0 by V h -ellipticity.i ∈ V Let us extend the above construction to unbounded operators and forms.

Definition 2.3 (Operator defined by a form, ( , ) construction). Let and a be as in Theorem 2.2. Moreover, assumeV H that is a dense subset of anotherV Hilbert space and that there exists a constant c >V 0 such that u H c u for all v . IntroduceH an operator T defined by a as follows. Thek domaink ≤ k kV ∈ V 9 dom T consists of the vectors v for which the map V u a(u, v) can be extended to a continuous antilinear∈ V map from to C. By the∋ Riesz7→ theorem, for H such a v there exists a uniquely defined fv such that a(u, v) = u, fv for all H u , and we set Tv := f . ∈ H h i ∈ V v Theorem 2.4. In the situation of definition 2.3 one has

the domain of T is dense in , • H T : dom T is bijective, • →H 1 T − ( ). • ∈ L H Proof. Let v dom T . Using the -ellipticity we have the following inequalities: ∈ V α v 2 αc2 v 2 αc2 a(v,v) αc2 v,Tv αc2 v Tv , k kH ≤ k kV ≤ ≤ h iH ≤ k kH · k kH showing that 1 Tv v . (2.2) k kH ≥ αc2 k kH We see immediately that T in injective. Let us show that T is surjective. Let h and let A ( ) be the operator associated with a. The map u u,∈ h H is a continuous∈ L antilinearV map from H to C, so one can ﬁnd w V ∋such7→ that h u,i h = u, w for all u . Denote V 1 ∈ V h iH h iV ∈ V v := A− w, then u, h = u, Av = a(u, v). By deﬁnition this means that H V v dom T and h =hTv.i h i ∈ Hence, T is surjective and injective, and the inverse is bounded by (2.2). It remains to show that the domain of T is dense in . Let h with u, h = 0 for all H u dom T . As T is surjective, there existsHv dom∈T Hwith h =h Tvi. Taking now u =∈ v we obtain v,Tv = 0, and the -ellipticity∈ gives v = 0 and h = 0. h iH V If the form a has some additional properties, then the associated operators T also enjoys some additional properties as well.

Definition 2.5 (Symmetric form). We say that a sesqulinear form a is symmetric (or Hermitian) if a(u, v)= a(v, u) for all u, v dom a. ∈ Theorem 2.6 (Self-adjoint operators defined by forms). Let T be the operator associated with a symmetric sesqulinear form a in the sense of definition 2.3, then

1. T is a self-adjoint operator in , H 2. dom T is dense in . V Proof. For any u, v dom T we have: ∈ u,Tv = a(u, v)= a(v, u)= v,Tu = Tu,v . h iH h iH h iH

10 Therefore, T is at least symmetric and T T ∗. Let v dom T ∗. We know from the previous theorem that T is surjective. This⊂ means that∈ we can ﬁnd v dom T 0 ∈ such that Tv = T ∗v. Then for all v dom T we have: 0 ∈

Tu,v0 = u,Tv0 = u,Tv = Tu,v . h iH h iH h iH h iH

As T is surjective, this imples v = v0 and then T = T ∗. Let us show the density of dom T in . Let h such that v,h = 0 for all V v dom T . There exists f such thatV h = Af∈, where V A (h ) isi the operator associated∈ with a. We have∈ then V ∈ L V

0= v,h = v, Af = a(v, f)= Tv,f . h iV h iV h iH As the vectors Tv cover the whole of as v runs through dom T , this imples f = 0 and h = 0. H

Note that one can indeed associate an operator T to any sesquilinear form a (non- symmetric, non-elliptic etc.) in literally the same way, but the properties of the operator are then unpredictable. An important point in the above consideration in the presence of a certain auxiliary Hilbert space . As a set, coincides with the domain of the form a. If a is symmetric andV positive, a(v,vV) α u 2 , it defines an alternative Hilbert structure ≥ k kV on . Namely, for u, v = dom a set u, v a := a(u, v), then , a satisfies all the propertiesV of the scalar∈ product. V If theh assumptionsi of Definitih·on·i 2.3 are satisfied, then, due to the boundedness and the -ellipticity, one can find C > 0 such that 2 1 2 V C u a(u, v) C− u for all u . This motivates the following definition: k kV ≤ ≤ k kV ∈ V Definition 2.7 (Closed forms). Let a be a symmetric sesquilinear form in a Hilbert space H with a dense domain dom a. We say that a is closed and positive and write a> 0 if

there exists α> 0 with a(u, u) α u 2 for all u dom a • ≥ k kH ∈ dom a equipped with the scalar product , is a Hilbert space. • h· ·ia We say that a is closed and semibounded from below if there exists c R such that the sesquilinear form a + c deﬁned on dom a by (a + c)(u, v)= a(u, v∈)+ c u, v is H closed positive (in this case we write a> c). h i − Proposition 2.8 (Operators deﬁned by closed forms).

Let a be a sesquilinear form in which is closed and positive, then the asso- • ciated linear operator T in isH self-adjoint and deﬁnes an isomorphism from dom T to . H H If a is closed and semibounded from below, then the associated operator T in • is self-adjoint and semibounded from below. H

11 Proof. In the ﬁrst case, one can take (dom a, , a) as , then a is bounded and -elliptic. The second case is an easy exercise. h· ·i V V Deﬁnition 2.9 (Closable form). Let us introduce another important notion. We say that a symmetric sesqulinear form a is closable, if there exists a closed form in (positive or just semibounded from below) which extends a. The closed sesqulinearH forma ˜ with the above property and with the minimal domain is called the closure of a.

The following proposition is rather obvious.

Proposition 2.10 (Domain of the closure of a form). If a c, c R, and a is a closable form, then doma ˜ is exactly the completion of dom≥a −with respect∈ to the norm π(x)= (a + c + 1)(x, x).

It is time to lookp at examples!

Example 2.11 (Non-closable form). Take = L2(R) and consider the form H a(u, v)= u(0)v(0) + uv dx R Z defined on dom a = L2(R) C0(R). This form is densely defined, symmetric and positive. Let us show that it∩ is not closable. By contradiction, assume that a is the closure of a. One should then have the following property: if (un) is a sequence of vectors from dom a which is π-Cauchy (see proposition 2.10 for the definitione of π), then it converges to some u dom a and a(u , u ) converges to a(u, u). But ∈ ⊂ H n n one can construct two sequences (un) and (vn) in dom a such that e e both converge to u in the L2-norm, • u (0) = 1 and v (0) = 0 for all n. • n n

Then both sequences are π-Cauchy, but the limits of a(un, un) and a(vn,vn) are diﬀerent. This shows that a cannot exist.

Let us give some “canonical”e examples of operators deﬁned by forms. Example 2.12 (Neumann boundary condition on the halﬂine). Take = L2(0, ). Consider the form H ∞

∞ 1 a(u, v)= u (x)v′(x)dx, dom a = H (0, ). ′ ∞ Z0 The form is semibounded below and closed (which is in fact just equivalent to the completeness of H1 in the respective Sobolev norm). Let us describe the associated operator T .

12 Let v dom T , then there exists f such that ∈ v ∈H ∞ ∞ u′(x)v′(x)dx = u(x)fv(x)dx Z0 Z0 1 for all u H . Taking here u Cc∞ we obtain just the deﬁnition of the derivative ∈ ∈ 2 in the sense of distributions: fv := (v′)′ = v′′. As fv L , the function v must 2 − − ∈ be in H (0, ) and Tv = v′′. ∞ − Now note that for v H2(0, ) and u H1(0, ) there holds ∈ ∞ ∈ ∞ x= ∞ ∞ ∞ u′(x)v′(x)dx = u(x)v′(x) u(x)v′′(x)dx. 0 x=0 − 0 Z Z

Hence in order to obtain the requested inequality a(u, v) = u,Tv the boundary h iH term must vanish, which gives the additional condition v′(0) = 0.

Therefore, the associated operator is TN := T acts as TN v = v′′ on the domain 2 − dom TN = v H (0, ) : v′(0) = 0 . It will be referred as the (positive) Laplacian with the Neumann{ ∈ boundary∞ condition.} Example 2.13 (Dirichlet boundary condition on the halﬂine). Take = L2(0, ). Consider the form which is a restriction of the previous one, H ∞

∞ 1 a (u, v)= u (x)v′(x)dx, dom a = H (0, ). 0 ′ 0 0 ∞ Z0 1 The form is still semibounded below and closed (as H0 is still complete with respect to the H1-norm), and the boundary term does not appear when integrating by parts, which means that the associated operator TD = T acts as TDv = v′′ on the domain 2 1 2 − dom TN = H (0, ) H (0, )= v H (0, ) : v(0) = 0 . It will be referred to as the (positive)∞ Laplacian∩ with∞ the{ Dirichlet∈ boundary∞ cond}ition. Remark 2.14. In the two previous examples we see an important feature: the fact that one form extends another one does not imply the same relation for the associated operators. Example 2.15 (Neumann/Dirichlet Laplacians). The two previous examples can be generalized to the multidimensional case. Let Ω be an open subset of Rd with suﬃciently regular boundary ∂Ω. In = L2(Ω) consider two sesqulinear forms: H a(u, v)= u vdx, dom a = H1(Ω), ZΩ 1 a0(u, v)= u vdx, dom a0 = H0 (Ω). ZΩ Both these forms are closed and semibounded from below, and one can easily show that the respective operators A and A act both as u ∆u. By a more careful 0 7→ − analysis and, for example, for a smooth ∂Ω, one can show that

dom A = H2(Ω) H1(Ω) = u H2(Ω) : u = 0 0 ∩ 0 { ∈ |∂Ω } 13 and ∂u dom A = u H2(Ω) : = 0 , { ∈ ∂n ∂Ω } where n denotes the normal vector on ∂Ω, and the restrictions to the boundary should be understood as the respective traces. If the boundary is not regular, the domains become more complicated, but in all the cases one calls A0 the Dirichlet Laplacian in Ω and A the Neumann Laplacian. Indeed, the whole construction only d makes sense in the boundary of Ω is non-empty. For example, A = A0 if Ω= R , as 1 Rd 1 Rd H ( )= H0 ( ).

2.2 Semibounded operators and Friedrichs extensions

Deﬁnition 2.16 (Semibounded operator). We say that a symemtric operator T in is semibounded from below if there exists a constant C R such that H ∈ u, T u C u, u for all u dom T, h i ≥ h i ∈ and in that case we write T C. ≥ Now assume that an operator T is semibounded from below and consider the indiced sesqulinear form t in , H t(u, v)= u,Tv , dom t = dom T. h i Proposition 2.17. The sesqulinear form t is semibounded from below and closable.

Proof. The semiboundedness of t from below follows directly from the analogous property for T . To show the closability we remark that without loss of generality one can assume T 1. By proposition 2.10, the domain of the completion of t must be the completion≥ of dom T with respect to the normV p(x) = t(x, x). More concretely, a vector u belongs to iﬀ there exists a sequence u dom T which is p- p n Cauchy such∈ that H u convergesV to u in . A natural candidate∈ for the norm of u n H is p(u) = lim p(un). Actually we just need to show that this limit is independent of the choice of the sequence un. Using the standard arguments we are reduced to prove the following: Assertion. If (u ) dom t be a p-Cauchy sequence converging to zero in , then n ⊂ H lim p(un) = 0.

To prove this assertion we observe ﬁrst that p(xn) is a non-negative Cauchy sequence, and is convergent to some limit α 0. Suppose by contradiction that α > 0. Now ≥ let us remark that t(un, um)= t(un, un)+ t(un, um un). Moreover, by the Cauchy- Schwartz inequality for p we have t(u , u u ) − p(u )p(u u ). Combining n m − n ≤ n m − n the two preceding expressions with the fact that un is p-Cauchy, we see that for any 2 ε > 0 there exists N > 0 such that t(un, um) α ε for all n,m > N. Take ε = α2/2 and take the associated N, then for n,m>N− ≤we have

α2 u , T u t(u , u ) . h n mi ≡ n m ≥ 2

14 On the other hand, the term on the left-hand side goes to 0 as n . So we obtain a contradiction, and the assertion is proved. → ∞

Example 2.18 (Schrödinger operators). A basic example for the Friedrichs extension is delivered by Schrödinger operators with semibounded potentials. Let 2 Rd 2 Rd W Lloc( ) and W 0. In = L ( ) consider the operator T acting as ∈ ≥ H d T u(x) = ∆u(x)+ W (x)u(x) on the domain dom T = C∞(R ). One has clearly − c T 0. The Friedrichs extension TF of T will be called the Schrödinger operator with≥ the potential W . Note that the sesqulinear form t associated with T is given by t(u, v)= u vdx + W uvdx. Rd ∇ ∇ Rd Z Z Denote by t the closure of the form t. One can easily show the inclusion dom t dom t H1 (Rd), where ⊂ ⊂ W e e H1 (Rd)= u H1(Rd) : √W u L2(Rd) . W ∈ ∈ 1 Rd Note that actually we have the equality dom t = HW ( ), see Theorem 8.2.1 in the book [4] for a rather technical proof, but the inclusion will be sufficient for our purposes. e

15 3 Spectrum and resolvent

3.1 Deﬁnitions

Actually most deﬁnitions of this chapter can be introduced in the Banach spaces, but we prefer to concentrate on the Hilbertian case.

Deﬁnition 3.1 (Resolvent set, spectrum, point spectrum). Let T be a linear operator in a Hilbert space . The resolvent set res T consists of the complex z H 1 for which the operator T z : dom T is bijective and the inverse (T z)− is bounded. The spectrum−spec T of T →is deﬁned H by spec T := C res T . The−point \ spectrum specp T is the set of the eigenvalues of T .

Note that very often the resolvent set and the spectrum are also denoted by ρ(T ) and σ(T ), respectively. The following two propositions explains why one usually deals with the study of spectra for closed operators only.

Proposition 3.2. If res T = , then T is a closed operator. 6 ∅ 1 Proof. Let z res H, then gr(T z)− is closed by the closed graph theorem, but ∈ − 1 then the graph of T z is also closed, as gr(T z) and gr(T z)− are isometric in . − − − H×H Proposition 3.3. For a closed operator T one has the following equivalence:

ker(T z)= 0 z res T iff − { } ∈ ran(T z)= . ( − H Proof. The direction follows from the definition. ⇒ Now let T be closed and z C with ker(T z) = 0 and ran(T z) = . The 1 ∈ − { } − H inverse (T z)− is then defined everywhere and has a closed graph (as the graph of T z is− closed), and is then bounded by the closed graph theorem. − Proposition 3.4 (Properties of the resolvent). The set res T is open and the set spec T is closed. The operator function

1 res T z R (z):=(T z)− ( ) ∋ 7→ T − ∈ L H called the resolvent of T is holomorph and satisﬁes the identities

R (z ) R (z )=(z z )R (z )R (z ), (3.1) T 1 − T 2 1 − 2 T 1 T 2 RT (z1)RT (z2)= RT (z2)RT (z1), (3.2) d R (z)= R (z)2 (3.3) dz T T for all z, z , z res T . 1 2 ∈ 16 Proof. Let z res H. We have the equality 0 ∈ T z =(T z ) 1 (z z )R (z ) . − − 0 − − 0 T 0

If z z0 < 1/ RT (z0) , then the operator on the right had sinde has a bounded inverse,| − which| meansk thatk z res T . Moreover, one has the series representation ∈ 1 R (z)= 1 (z z )R (z ) − R (z )= (z z )jR (z )j+1, (3.4) T − − 0 T 0 T 0 − 0 T 0 j=0 X which shows that RT is holomorph. The remaining properties can be proved is the same way.

3.2 Examples

Let us consider a series of examples showing several situations where an explicit calculation of the spectrum is possible. We emphasize that the point spectrum is not the same as the spectrum!

Example 3.5. Consider the multiplication operator Mf from Example 1.5. Recall that the essential range of a function f is deﬁned by

ess ran f = λ : µ x : f(x) λ < ǫ > 0 for all ǫ> 0 . | − | n o Indeed this notion makes sense in any measure space. For a continuous function f and the Lebesgue measure µ the essential range coincides with the closure of the usual range. Proposition 3.6 (Spectrum of multiplication operator). There holds

spec Mf = ess ran f, spec M = λ : µ x : f(x)= λ > 0 . p f { }

Proof. Let λ / ess ran f, then the operator M1/(f λ) is bounded, and easily checks − that this is the∈ inverse for M λ. f − On the other hand, let λ ess ran f. For any m N denote ∈ ∈ m S := x : f(x) λ < 2− m | − | and choose subset Sm Sem of strictly positive but ﬁnite measure. If φm is the ⊂ characteristic function of Sm, one has e 2 2 2 2m 2 (M λ)φ f(x) λ φ (x) dx 2− φ , f − m ≤ − m ≤ m ZSm

1 and the operator (M λ )− cannot be bounded. f − To prove the second assertion we remark that the condition λ spec M is equiv- ∈ p f alent to the existence of φ L2(Rd) such that f(x) λ φ(x) = 0 for a.e. x. ∈ − 17 This means that φ(x) = 0 for a.e. x with f(x) = λ, and specp Mf = if µ x : f(x) = λ = 0. On the other hand, if µ x 6 : f(x) = λ > 0, one∅ can { } { } choose a subset Σ x : f(x) = λ of a strictly positive but ﬁnite measure, then ⊂ { } the characteristic function φ of Σ is an eigenfunction of Mf corresponding to the eigenvalue λ. Example 3.7. It can be show that the spectrum is invariant under unitary trans- formations. Namely, the following proposition holds.

Proposition 3.8 (Spectrum and unitary equivalence). Let 1 and 2 be Hilbert spaces, and let U : be a unitary operator. Furthermore,H letHT be H1 → H2 j operators in such that such that dom T = U(dom T ) and that T = UT U ∗. (The Hj 2 1 2 1 operators T1 and T2 are then called unitarily equivalent.) Then spec T1 = spec T2 and specp T1 = specp T2. The proof is left as an exercise for the reader.

Example 3.9. Consider the operator T1 from Example 1.6 and use the notation of that example. If F is the Fourier transform, then we have shown already that the 2 operator T := F T F ∗ is exactly the operator of multiplication f(p) p f(p). By 1 7→ Propositions 3.6 and 3.8 there holds spec T1 = spec T = [0, + ). b ∞ Example 3.10 (Multiplication in ℓ2). Take = ℓ2(Z). Consider an aribtrary b function a : Z C and the associated operator TH: → dom T = (ξ ) ℓ2(Z): (a ξ ) ℓ2(Z) , (T ξ) = a ξ . n ∈ n n ∈ n n n Similarly to examples 1.5 and 3.6 one can show that T is a closed operator and that spec T := a : n Z . { n ∈ } Example 3.11 (Harmonic oscillator). Let = L2(R). Consider the operator T = d2/dx2 + x2 defined as the Friedrichs extensionH of the operator defined by the same− expression on (Rd). One can show that T has the simple eigenvalues λ = S n 2n 1, n N, and the associated eigenfunctions φn are given by φn(x)= cn(d/dx + n−1 ∈ 2 x) − φ1(x), φ1(x)= c1 exp( x /2), where cn are normalizing constants. It is known − 2 d that (φn) is an orthonormal basis in L (R ), and this means that the map U : 2 H → ℓ (N), (Uf)(n) = φ , f , is a unitary operator. Moreover, (UT U ∗ξ)(n) = λ ξ(n) h n i n for any ξ dom UT U ∗. Combining the results of Proposition 3.8 and Example 3.10 ∈ we can say that the spectrum of T coincides with the set of the eigenvalues λn. Actually one needs to show that the operator T obtained by the Friedrichs extension is really unitary equivalent to the multiplication in ℓ2, this technical part is left as an exercise. Example 3.12 (A finite-difference operator). Consider again the Hilbert space = ℓ2(Z) and the operator T in acting as (T u)(n)= u(n 1)+u(n+1). Clearly, TH ( ). To find its spectrum considerH the map − ∈ L H Φ : ℓ2(Z) L2(0, 1), (Φu)(x)= u(n)e2πinx, → n Z X∈ 18 where the sum on the right hand side should be understood as a series in L2. It is known that Φ is a unitary map. On the other hand, for any u ℓ2(Z) supported at ∈ a finite number of points we have

Φ(T u)(x)= (T u)(n)e2πinx n X = u(n 1)e2πinx + u(n + 1)e2πinx − n n X X 2πi(n+1)x 2πi(n 1)x = u(n)e + u(n)e − n n X X 2πix 2πinx 2πix 2πinx = e u(n)e + e− u(n)e n n X X = 2 cos(2πx)(Φu)(x).

Using the density argument we show that the operator ΦT Φ∗ is exactly the multil- ication by f(x) = 2 cos(2πx) in the space L2(0, 1), and its spectrum coincides with the segment [ 2, 2], i.e. with the essential range of f. So we have spec T = [ 2, 2]. − − Example 3.13 (Empty spectrum). Take = L2(0, 1) and consider the operator H T acting as T f = f ′ on the domain

dom T = f H1(0, 1) : f(0) = 0 . { ∈ } One can easily see that for any g L2(0, 1) and any z C the equation (T z)f = g has the unique solution explicitly∈ given by ∈ −

x z(x t) f(x)= e − g(t) dt, Z0 and the map g f is bounded in the norm of . So we obtained res T = C and spec T = . 7→ H ∅ Example 3.14 (Empty resolvent set). Let us modify the previous example. 2 Take = L (0, 1) and consider the operator T acting as T f = f ′ on the domain H 1 zx dom T = H (0, 1). Now for any z C we see that the function φz(x)= e satisﬁes (T z)φ = 0. Therefore, spec T ∈= spec T = C. − z p As we can see in the two last examples, for general operators one cannot say much on the location of the spectrum. In what follows we will study mostly self-adjoint operators, whose spectral theory is now understood much better than for the non- self-adjoint case.

3.3 Basic facts on spectra of self-adjoint operators

The following two propositions will be of importance during the whole course.

19 Proposition 3.15. Let T be a closed operator in a Hilbert space , then, for any z C, H ∈

ker(T ∗ z) = ran(T z)⊥, (3.5) − − ran(T z) = ker(T z)⊥. (3.6) ∗ − − Proof. Note that the second equality can be obtained from the ﬁrst one by taking the orthogonal complement in the both parts.

Let us prove the ﬁrst equality. As dom T is dense, the condition f ker(T ∗ z) is equivalent to ∈ − (T ∗ z)f,g = 0 g dom T, h − i ∀ ∈ which can be also rewritten as

T ∗f,g = z f,g g dom T. h i h i ∀ ∈

By the deﬁnition of T ∗, one has T ∗f,g = f,Tg and h i h i f,Tg z f,g f, (T z)g = 0 g dom T, h i − h i≡h − i ∀ ∈ i.e. f ran(T z). ⊥ − Proposition 3.16 (Spectrum of a self-adjoint operator is real). Let T be a self-adjoint operator in a Hilbert space , then spec T R, and for any z C R there holds H ⊂ ∈ \ 1 1 (T z)− . (3.7) − ≤ z |ℑ | Proof. Let z C R and u dom T . We have ∈ \ ∈ u, (T z)u = u, T u z u, u i z u, u . h − i h i − ℜ h i − ℑ h i As T is self-adjoint, the number u, T u is real. Therefore, h i z u 2 u, (T z)u (T z)u u , |ℑ |k k ≤ h − i ≤ − · k k which shows that (T z)u z u . (3.8) − ≥ |ℑ | · k k It follows from here that ran( T z) is closed, that ker(T z) = 0 and, by − −1 { } proposition 3.15, than ran(T z) = . Therefore, (T z)− ( ), and the estimate (3.7) follows from (3.8).− H − ∈ L H

The following proposition is of importance when studying bounded operators.

Proposition 3.17 (Spectrum of a continuous operator). Let T ( ), then spec T is a non-empty subset of z C : z T . ∈ L H ∈ | | ≤ k k

20 Proof. Let z C with z > T . Represent T z = z(1 T/z). As T/z < 1, the inverse to ∈T z is deﬁned| | k byk the series, − − − k k −

∞ 1 n n 1 (T z)− = T z− − . − n=0 X and z res T . This implies the sought inclusion. ∈ Let us show that the spectrum is non-empty. Assume that it is not the case. Then for any f,g the function C z F (z) := f, RT (z)g C is holomorph in C by proposition∈H 3.4. On the other hand,∋ 7→ it followsh from the abovei∈ series representation for the resolvent that for large z the norm of RT (z) tends to zero, which means that F is bounded. Therefore, F (z) is a constant function, and F ′(z)=0. On the 2 other hand, by proposition 3.4 we have F ′(z) = f, RT (z) g , and we must have R (z)2 = 0 for all z. This contradicts to the deﬁnition of the resolvent and shows T that the spectrum of T must be non-empty.

Proposition 3.18 (Location of spectrum). Let T = T ∗ ( ). Denote ∈ L H u, T u u, T u m = m(T ) = inf h i, M = M(T ) = sup h i, u=0 u, u u=0 u, u 6 h i 6 h i then

spec T [m, M], • ⊂ m, M spec T . • ∈ Proof. We proved already that spec T R. For λ (M, ) we have ⊂ ∈ ∞ u, (λ T )u (λ M) u 2, h − i ≥ − k k 1 and (T λ)− ( ) by the Lax-Milgram theorem. In the same way one shows that spec−T ( ∈ L,mH)= . ∩ −∞ ∅ Let us shows the inclusion M spec T (for m the proof is similar). Using the Cauchy-Schwarz inequality for the∈ semi-scalar product (u, v) u, (M T )v we obtain 7→ h − i 2 u, (M T )v u, (M T )u v, (M T )v . h − i ≤h − i·h − i Taking the supremum over all u with u 1 we arrive at ∈H k k ≤ (M T )v M T v, (M T )v . − ≤ k − k · − By assumption, one can construct a sequence (u ) with u = 1 such that n k nk un, T un M as n . By the above inequality we have then (M T )un 0, andh the operatori → M →T ∞cannot have bounded inverse. Thus M spec−T . → − ∈

Corollary 3.19. If T = T ∗ ( ) and spec T = 0 , then T = 0. ∈ L H { }

21 Proof. By proposition 3.18 we have m(T )= M(T ) = 0. This means that x, T x = 0 for all x , and the polar identity shows that x, T y = 0 for all x, y h . i ∈H h i ∈H Let us combine all of the above to show the following fundamental fact.

Theorem 3.20 (Non-emptiness of spectrum). The spectrum of a self-adjoint in a Hilbert space is non-empty.

Proof. Let T be a self-adjoint operator in a Hilbert space . By contradiction, 1 H assume that spec T = . Then, ﬁrst of all, T − ( ). Let λ C, λ = 0. One can easily show that the operator∅ ∈ L H ∈ 6

T 1 1 L := T − λ − λ − λ 1 1 belongs to (T ) and that (T − λ)Lλ = Id and Lλ(T − λ) = Id . Therefore, 1L − H − H 1 λ res T − . As λ was an arbitrary non-zero complex number, we have spec T − = ∈ 1 1 0 . On the other hand, T − is self-adjoint by proposition 1.21, and T − = 0 by { } Corollary 3.19, which contradicts to the deﬁnition of the inverse operator.

22 4 Spectral theory of compact operators

Recall that a linear operator T acting from a Hilbert space 1 to a Hilbert space 2 is called compact, if the image of the unit ball in is relativelyH compact in . WeH H1 H2 denote by ( 1, 2) the set of all such operators. If T is a compact operator, then the image ofK anyH H weakly convergent sequence is strongly convergent. We recall that any Hilbert space is locally compact in the weak topology, which means that any bounded sequence contains a weakly convergent subsequence. Recall also that any compact operator is continuous. If A is a continuous operator and B is a compact one, then the products AB and BA are compact. It is also known that a linear operator in a Hilbert space is compact if and only if it can be represented as the limit (with respect to the operator norm) of operators with ﬁnite-dimensional range. It is also known that the adjoint of a compact operator is compact. Below we consider some examples, but in these notes we prefer to discuss ﬁrst some basic questions of the spectral theory.

Theorem 4.1 (Fredholm alternative). Let T be a compact operator in a Hilbert space , then H (a) ker(1 T ) is of ﬁnite dimension, − (b) ran(1 T ) is closed and of ﬁnite codimension, − (c) ran(1 T )= if and only if ker(1 T )= 0 . − H − { }

Proof. We give the proof for the case T = T ∗ only. An interested reader may refer to Section VI.5 in [8] for the proof of the general case. To show (a) let us recall the Riesz theorem: if E is a normalized linear space such that the unit ball is relatively compact, then E is ﬁnite-dimensional. Let us apply this to E = ker(1 T ) with the same norm as in . For every u E we have − H ∈ T u = u. As the unit ball B in E is bounded, it is weakly compact. As T is compact, B = T (B) is a compact set, which means that E is ﬁnite-dimensional.

Let us prove (b). Show ﬁrst that ran(1 T ) is closed. Let (yn) ran(1 T ) such that y converges to y in the norm of −. To show that y ran(1⊂ T )− we choose n H ∈ − x ker(1 T )⊥ with y = (1 T )x . n ∈ − n − n We show ﬁrst that the sequence (xn) is bounded. Assume by contradiction that it is not the case, then one can choose a subsequence with norms growing to + . To keep ∞ simple notation we denote the subsequence again by xn and denote un := xn/ xn , then k k (1 T )x y u T u = − n = n . n − n x x k nk k nk As the norms of y are bounded, the vectors u T u converge to 0. As the sequence n n − n un is bounded, one can choose a subsequence unj which is weakly convergent, then the sequence T unj is convergent with respect to the norm to some u due to the compactness of T . On the other hand, as shown above, u T u ∈converge H to nj − nj

23 0, which means that u T u = 0 and u ker(1 T ). On the other hand, we have − ∈ − u ker(1 T )⊥, which means that u ker(1 T )⊥ too. This shows that u = 0, n ∈ − ∈ − but this contradicts to un = 1. This contradiction shows that (xn) is a bounded sequence. k k

As (xn) is bounded, one can ﬁnd a subsequence xnj which converges weakly to some x , and then T xnj converge in the norm to T x . Now we have xnj = ynj +T xnj , ∞ ∞ both∈H sequences on the right-hand side are convergent with respect to the norm, so xnj is also convergent to x in the norm. Finally we obtain x = y + T x , or ∞ ∞ ∞ y = (1 T )x , which means that y ran(1 T ). So we proved that ran(1 T ) is ∞ closed.− ∈ − −

For our particular case T = T ∗ we have ran(1 T ) = ker(1 T )⊥ by Proposition − − 3.15. Combining this with (a) we complete the proof of (b), and the item (c) is proved too.

In a sense, the Fredholm alternative show that the operators 1 T with compact T behave like operators in finite dimensional spaces. We know that− a linear operator in a finite-dimensional space is injective if and only if it is surjective, and we see a similar feature in the case under consideration. We remark that the Fredholm alternative also holds for compact operators in Banach spaces, but we are not discussing this direction. Theorem 4.2 (Spectrum of compact operator). Let be an infinite- dimensional Hilbert space and T ( ), then H ∈ K H (a) 0 spec T , ∈ (b) spec T 0 = spec T 0 , \ { } p \ { } (c) we are in one and only one of the following situations:

– spec T 0 = , \ { } ∅ – spec T 0 is a ﬁnite set, \ { } – spec T 0 is a sequence convergent to 0. \ { } (d) Each λ spec T 0 is isolated (i.e. has a neighbodhood containing no other values of∈ the spectrum),\ { } and dim ker(T λ) < . − ∞ 1 1 Proof. (a) Assume that 0 / spec T , then T − ( ), and the operator Id = T − T is compact. This is possible∈ only if is ﬁnite-dimensional.∈ L H H (b) If λ = 0 we have T λ = λ(1 T/λ), and by Fredholm alternative the condition λ spec6 T is equivalent− to ker(1− −T/λ) ker(T λ) = 0 . ∈ − ≡ − 6 { } (c) Here we actually need to prove the following assertion: if (λj) is a sequence distinct non-zero eigenvalues of T and if converging to some λ C, then λ = 0. For the proof, assume by contradiction that λ = 0. Let (e ) be∈ the normalized 6 j eigenvectors associated with the eigenvalues λj, Tej = λjej. Denote by En the linear subspace spanned by e ,...,e . Clearly, E E and E = E . For any n one 1 n n ⊂ n+1 n 6 n+1 24 can choose un En En⊥ 1 with un = 1. As T is compact and (un) is bounded, ∈ ∩ − k k one can extract a subsequence unk such that the sequence (T unk ) converges, and then the sequence T u /λ is also convergent. Let j > k 2. We can write nk nk ≥ 2 2 T un T u (T λn )un (T λ )u j nk = − j j − nk nk + u u (4.1) λ − λ λ − λ nj − nk nj nk nj nk

Note that for any k we have (T λnk )Enk Enk 1. On the right-hand side of (4.1) − ⊂ − one has unj Enj and all the other vectors are in the strictly smaller subspace Enj 1. ∈ − Therefore, unj is orthogonal to the other three vectors, which gives the estimate

2 T un T u j nk u 2 = 1. λ − λ ≥ k nj k nj nk

Therefore, (T unj /λnj ) cannot be a Cauchy sequence, which shows that λ = 0. The item (d) easily follows from (c) and from the part (a) of the Fredholm alternative.

Finally let us apply all of the above to show the main result of the spectrum of compact self-adjoint operators.

Theorem 4.3 (Spectrum of compact self-adjoint operator). Let T = T ∗ ( ), then has a hilbertian basis consisting of eigenfunctions of T . ∈ K H H

Proof. Let (λn)n 1 be the non-zero eigenvalues of T . As T is self-adjoint, they all ≥ are real. Set λ = 0, and for n 0 denote E := ker(T λ ). One can easily see 0 ≥ n − n that En Em for n = m. Denote by F the linear hull of all En. We are going to show that⊥ F is dense6 in . H Note that we have T (F ) F . Due to the self-adjointness of T we also have T (F ⊥) ⊂ ⊂ F ⊥. Denote by T the restriction of T to F ⊥. Clearly, T is compact, self-adjoint, and its spectrum equals 0 , so T = 0. But this means that F ⊥ ker T = ker T F { } ⊂ ⊂ and shows that eF ⊥ = 0 . Therefore F is dense in e. { } H Now taking an orthonormal basise in each En we obtain an orthonormale basis in the whole space . H

4.1 Integral and Hilbert-Schmidt operators

An important class of compact operators is delivered by integral operators. For simplicity we restrict our attention to the case = L2(Ω), where Ω Rd is an open set. An interested reader may generalize allH the considerations to⊂ the case more general measure spaces. Let K L1 (Ω Ω). Consider the operator T deﬁned by ∈ loc × K

TK u(x)= K(x, y)f(y) dy (4.2) ZΩ

25 on bounded functions with compact supports. One wonders under which conditions this can be extended to a bounded operator in . A standard result in this direction H is delivered by the following theorem. Theorem 4.4 (Schur test). Assume that

M1 = sup K(x, y) dy < , M2 = sup K(x, y) dx < , x Ω Ω ∞ y Ω Ω ∞ ∈ Z ∈ Z then the above expression (4.2) deﬁnes a continuous linear operator TK with the norm T √M M . k K k ≤ 1 2 Proof. We have 2 2 TK u(x) K(x, y) K(x, y) u(y) dy | | ≤ Ω | | | || | Z p p K(x, y) dy K(x, y) u(y) 2dy ≤ | | | || | ZΩ ZΩ M K(x, y) u(y) 2dy, ≤ 1 | || | ZΩ and

T u 2 M K(x, y) u(y) 2dydx k K k ≤ 1 | | ZΩ ZΩ M M u 2. ≤ 1 2k k To obtain a class of compact integral operators we introduce the following notion. We say that T ( ) is a Hilbert-Schmidt operator if for some orthonormal basis (e ) of the sum∈ L H n H T 2 = Te 2 k k2 k nk n X is ﬁnite. Proposition 4.5 (Hilbert-Schmidt norm). Let T be a Hilbert-Schmidt operator, then T (called the Hilbert-Schidt norm of T ) does not depend of the choice of the k k2 basis, and T T . Moreover, the adjoint operator T ∗ is also Hilbert-Schmidt k k ≤ k k2 with T ∗ = T . k k2 k k2

Proof. Take another orthonormal basis (fn). Using the Parseval identity we have

2 2 2 2 2 T = Te = f ,Te = T ∗f ,e = T ∗f , k k2 k nk h m ni h m ni k mk n n m m m m X X X X X X and the right-hand side does not depend on (en). It remains to show the inequality T T . If x and x := e , x , we have k k ≤ k k2 ∈H n h n i 2 2 T x 2 = x Te x Te x 2 Te 2 = T 2 x 2. k k n n ≤ | n|k nk ≤ | n| k nk k k2k k n n n n X X X X

26 Proposition 4.6. Any Hilbert-Schmidt operator is compact.

Proof. For any x we have ∈H

∞ T x = e , x Te . h n i n n=1 X For N 1 introduce the operators T by ≥ N N T x = e , x Te . N h n i n n=1 X One has N T T 2 T T 2 = Te 2 →∞ 0. k − N k ≤ k − N k2 k nk −−−→ n N+1 ≥X The operators TN have ﬁnite-dimensional range, hence T is compact.

The following proposition describes the class of integral operators which are Hilbert- Schmidt.

Proposition 4.7 (Integral Hilbert-Schmidt operators). Let = L2(Ω). An operator T in is Hilbert-Schmidt iﬀ there exists K L2(Ω Ω) suchH that T = T , H ∈ × K and in that case TK 2 = K L2(Ω Ω). k k k k × Proof. Let ﬁrst K L2(Ω Ω). Let us show that the associated operator T ∈ × K is Hilbert-Schmidt. Let (en) be an orthonormal basis in . Then the functions H 2 em,n(x, y) = em(x)en(y) form an orthonormal basis in L (Ω Ω). There holds H⊗H∼ ×

2 2 T e 2 = e , T e = e (x) K(x, y)e (y)dy dx k K nk h m K ni m n n m,n m,n ZΩ ZΩ X X X 2 2 = em,n,K = K L2(Ω Ω). h i k k × m,n X

Now let T be an arbitrary Hilbert-Schmidt operator in . We have, for any u and with u := e , u , H ∈H n h n i T u = e , u Te = e , u e ,Te e . h n i n h n ih m ni m n m,n X X Take K(x, y)= e (y) e ,Te e (x)= e ,Te e (x, y). n h m ni m h m ni m,n m,n m,n X X 2 One easily checks that K L (Ω Ω), that T = TK , and that the remaining properties hold as well. ∈ ×

27 One can easily see that the operator TK is self-adjoint iﬀ K(x, y)= K(y, x) for a.e. x, y. Hence together with the Hilbert-Schmidt condition this gives an important class of self-adjoint compact operators to which the previous considerations can be applied. Taking an orthonormal basis consisting of eigenfunctions we see that a compact self-adjoint operator T is a Hilbert-Schmidt one iﬀ the series

λ2 = T 2 n k k2 n X is convergent, where λn denote the non-zero eigenvalues of T taking with their multiplicities. Moreover, by Proposition 4.7, for T = TK one has the exact equality

2 2 λn = K L2(Ω Ω), k k × n X which may be used to estimate the eigenvalues using the integral kernel.

4.2 Operators with compact resolvent

Let us go back back and continue the discussion of operators deﬁned by forms, see Section 2.

1 Proposition 4.8. In the situation of Theorem 2.4 one has T − ( , ). ∈ L H V Proof. For any u dom T we have: ∈ 2 u T u u, T u = a(u, u) α u V Cα u V u , k kHk kH ≥ h iH | | ≥ k k ≥ k k k kH 1 1 i.e. T u Cα u V and T − u (Cα)− u . k kH ≥ k k k k V ≤ k kH This gives an important consequence: Corollary 4.9. In the situation of Theorem 2.4, assume that the embedding j : 1 V → is compact, then T − is a compact operator. H 1 1 Proof. Indeed we have T − = jL, where L is the operator T − viewed as an operator 1 from to . Hence T − is compact as a composition of a bounded operator and a compactH one.V

The above can be applied to a variety of cases. For example, take the Dirichlet Laplacian A0 deﬁned in example 2.15. If Ω is relatively compact, then the embedding 1 2 1 of = H0 (Ω) to = L (Ω) is compact. Therefore, the operator L = (A0 + 1)− is compact.V Moreover,H it is self-adjoint due to the previous considerations. By 2 Theorem 4.3 there exists an orthonormal basis (en) of L (Ω) such that Len = λnen, where λn is a real-valued sequenece converging to 0. By elementary operations, e dom A and A e = µ e with n ∈ 0 0 n n n 1 µn = 1. λn −

28 It is an easy exercise to show that the spectrum of A0 is exactly the union of all the µ and that µ + as n + . n n → ∞ → ∞ The values µn are called the Dirichlet eigenvalues of the domain Ω. It is an important domain of the modern analysis to study the relations between the geometric and topological properties of Ω and its Dirichlet eigenvalues. The preceding example can be easily generalized. More precisely, we say that an operator A with res A = has a compact resolvent if Rλ(A) is a compact operator for all λ res A. One can6 ∅ easily check that it is suﬃcient to verify this property at a single value∈ λ. Similar to the preceding constructions one can show:

Proposition 4.10 (Spectra of operators with compact resolvents). Let T be a self-adjoint operator with a compact resolvent in an inﬁnite-dimensional Hilbert space, then:

spec T = spec T , • p the eigenvalues of T form a sequence converging to . • ∞ The proof is completely the same as for the Dirichlet Laplacian if we manage to show that spec T = R. In principle this can be done in a rather directly, but we prefer to show it later6 using the spectral theorem, see Example 5.23 below.

4.3 Schr¨odinger operators with growing potentials

Let us discuss a particular class of operators with compact resolvents. Recall the following classical criterion of compactness in L2(Rd) (sometimes referred to as the Riesz-Kolmogorov-Tamarkin criterion):

Proposition 4.11. A subset A L2(Rd) is relatively compact in L2(Rd) if and only if the following three conditions⊂ are satisﬁed:

(a) A is bounded,

(b) there holds u(x) 2dx 0 as R x R | | → → ∞ Z| |≥ uniformly for u A, ∈ (c) For h Rd and v L2(Rd) denote v (x)= v(x+h). Assume that u u 0 ∈ ∈ h k h− k → as h 0 uniformly for u A. → ∈ An interested reader may refer to [5] for the proof and various generalizations. 2 Rd Now let W Lloc( ) and W 0. Consider the operator T = ∆+ W deﬁned ∈ ≥ Rd − as the Friedrichs extension starting from Cc∞( )and discussed in Example 2.18.

29 We know already that T is a self-adjoint and semibounded from below operator in = L2(Rd). We would like to identify a reasonable large class of potentials W for H which T has compact resolvent.

Theorem 4.12. For r 0 denote ≥ w(r) := inf W (x). x r | |≥

If limr w(r)=+ , then the associated Schr¨odinger operator T = ∆+ W has →∞ compact resolvent. ∞ −

Proof. As follows from Example 2.18, it is suﬃcient to show that the embedding 1 Rd 2 Rd 2 of = HW ( ) to L ( ) is compact, where is equipped with the norm u W = V2 V k k √ 2 u H2 + W u L . Let B be the unit ball in . We will show that B is relatively kcompactk k in L2(Rkd) using Proposition 4.11. V

The condition (a) holds due to the inequality u L2 u W . The condition (b) follows from k k ≤ k k

1 2 d 2 u(x + h) u(x) dx = u(x + th)dt dx Rd − Rd 0 dt Z 1 Z Z 1 2 2 = h u(x + th )dt dx h2 u(x + th) dtdx Rd · ∇ ≤ Rd ∇ Z Z0 Z Z0 1 2 2 2 2 2 2 h u (x + th) dxdt = h u L2 h u W . ≤ Rd ∇ k∇ k ≤ k k Z0 Z

The assumption of Theorem 4.12 is rather easy to check, but this condition in not optimal one. For example, it is known that the operator ∆+ W with W (x1, x2)= 2 2 − x1x2 has compact resolvent, while the condition cleraly fails. A rather simple necessary and suﬃcient condition is known in the one-dimensional case:

Proposition 4.13 (Molchanov criterium). The operator T = d2/dx2 + W has compact resolvent iﬀ − x+δ lim W (s)ds =+ x ∞ →∞ Zx for any δ > 0.

Necessary and suﬃcient conditions are also available for the multi-dimensional case, but their form is much more complicated. An advanced reader may refer to the paper [9] for the discussion of such questions.

30 5 Spectral theorem

Some points in this section are just sketched to avoid technicalities. The presentation is based on [4, Chapter 2] where more detailed proofs can be found. The aim of the present section is to deﬁne, for a given self-adjoint operator T , the operators f(T ), where f are suﬃciently general functions. To be provided with a certain motivation, let T be either a compact self-adjoint operator or a self-adjoint operator with compact resolvent in a Hilbert space . As shown in the previous section, there exists an orthonormal basis (e ) in andH real n H numbers λn such that, with

T x = λ e , x e for all x dom T, nh n i n ∈ n X and the domain dom T is characterized by

2 dom T = x : λ2 e , x < . ∈H n h n i ∞ n n X o

For f C (R) one can deﬁne an operator f(T ) ( ) by ∈ 0 ∈ L H f(T )x = f(λ ) e , x e . n h n i n n X This map f f(T ) enjoys a number of properties. For example, (fg)(T ) = 7→ f(T )g(T ), f(T ) = f(T )∗, spec f(T ) = f(spec T ) etc. The existence of such a construction allows one to write rather explicit expressions for solutions of some equations. For example, one can easily show that the initial value problem

x′(t)= iAx(t), x(0) = y dom T, x : R dom T, ∈ → itx has a solution that can be written as x(t) = ft(T )y with ft(x) = e . Informally speaking, for a large class of equations involving the operator T one may ﬁrst assume that T is a real constant and obtain a formula for the solution, and then one can give this formula an operator-valued meaning using the above map f f(T ). 7→ Moreover, if we introduce the map U : ℓ2(N) deﬁned by Ux =: (x ), x = H → n n en, x , then the operator UT U ∗ becomes a multiplication operator (xn) (λnxn), hcf. Examplei 3.10. 7→ At this point, all the preceding facts are proved for compact self-adjoint operators and self-adjoint operator with compact resolvent only. The aim of the present section is to develop a similar theory for general self-adjoint operators.

To avoid potential misunderstanding let us recall that C0(R) denotes the class of the continuous functions f : R C with lim x + f(x) = 0 equipped with the sup- | |→ ∞ norm. This should not be confused→ with the set C0(R) of the continuous functions on R.

31 5.1 Continuous functional calculus

2 We say that a function f : C C belongs to C∞(C) if the function of two real 2 → 2 variables R (x, y) f(x + iy) C belongs to C∞(R ). In the similar way one ∋ 7→C2 k C ∈ defines the classes Cc∞( ), C ( ) etc. In what follows we always use the notation z + z z z z =: x, z =: y for z C. Using x = and y = − , for f C1(C) one ℜ ℑ ∈ 2 2i ∈ defines the derivative ∂ 1 ∂ ∂ := + i ∂z 2 ∂x ∂y Clearly, ∂f/∂z = 0 if f is a holomorph function. Recall the Stokes formula written in this notation: if f C∞(C) and Ω C is a domain with a sufficiently regular boundary, then ∈ ⊂ ∂f 1 dxdy = f dz. ∂z 2i ZZΩ I∂Ω The following fact is actually known, but is presented in a slightly unusual form. C C Lemma 5.1 (Cauchy integral formula). Let f Cc∞( ), then for any w we have ∈ ∈ 1 ∂f 1 dxdy = f(w). π C ∂z w z ZZ − Proof. We note first that the singularity 1/z is integrable in two dimensions, and the integral is well-defined. Let Ω be a large ball containing the support of f and the point w. For small ε> 0 denote B := z C : z w ε , and set Ω := Ω B . ε { ∈ | − | ≤ } ε \ ε Using the Stokes formula we have:

1 ∂f 1 1 ∂f 1 dxdy = dxdy π C ∂z w z π ∂z w z ZZ − ZZΩ − 1 ∂f 1 1 1 = lim dxdy = lim f(z) dz ε 0 π ∂z w z ε 0 2πi w z → ZZΩε − → I∂Ωε − 1 1 1 1 = f(z) dz lim f(z) dz. 2πi ∂Ω w z − ε 0 2πi z w =ε w z I − → I| − | − The ﬁrst term on the right-hand side is zero, because f vanishes at the boundary of Ω. The second term can be calculated explicitly:

2π it 1 1 1 it iεe dt lim f(z) dz = lim f(w + εe ) it ε 0 2πi z w =ε w z ε 0 2πi 0 w (w + εe ) → I| − | − → Z − 1 2π = lim f(w + εeit)dt = f(w), − ε 0 2π − → Z0 which gives the result.

The main idea of the subsequent presentation is to deﬁne the operators f(T ), for a self-adjoint operator T , using an operator-valued generalization of the Cauchy integral formula.

32 Introduce first some notation. For z C we write ∈ z := 1+ z 2. h i | | For β < 0 denote by β the set of the smoothp functions f : R C satisfying the estimates S → (n) β n f (x) c x − ≤ nh i R for any n 0 and x , where the positive constant cn may depend on f. Set ≥ ∈ := β<0 β; one can show that is an alebra. Moreover, if f = P/Q, where P Aand Q are polynomialsS with deg PA < deg Q and Q(x) = 0 for x R, then f . For anyS n 1 one can introduce the norms on : 6 ∈ ∈ A ≥ A (r) r 1 f n := f (x) x − dx. k k R h i r=0n Z X One can easily see that the above norms on induce continuous embeddings R R A A → C0( ). Moreover, one can prove that Cc∞( ) is dense in with respect to any norm . A k · kn Now let f C∞(R). Pick n N and a smooth function τ : R R such that τ(s) = 1 for∈ s < 1 and τ(s) =∈ 0 for s > 2. For x, y R set σ(x,→ y) := τ(y/ x ). | | | | ∈ h i Define f C∞(C) by ∈ n (iy)r e f(z)= f (r)(x) σ(x, y). r! r=0 X e Clearly, for x R we have f(x) = f(x), so f is an extension of f. One can check the following identity:∈ e e ∂f 1 n (iy)r 1 (iy)n = f (r)(x) σ + iσ + f (n+1)(x) σ. (5.1) ∂z 2 r! x y 2 n! " r=0 e X Now let T be a self-adjoint operator in a Hilbert space . For f define an operator f(T ) in by H ∈ A H

1 ∂f 1 f(T ) := (T z)− dxdy. (5.2) π C ∂z − ZZ e This integral expression is called the Helffer-Sjöstrand formula. We need to show several points: that the integral is well-defined, that it does not depend in the choice of σ and n etc. This will be done is a series of lemmas. Note first that, as shown in Proposition 3.16, we have the norm estimate (T 1 n k − z)− 1/ z , and one can see from (5.1) that ∂f/∂z(x + iy) = O(y ) for any k ≤ |ℑ | fixed x, so the subintegral function in (5.2) is locally bounded. By additional technical efforts one can show that the integral is convergente and defines an continuous operator with f(T ) c f n+1 for some c > 0. Using this observation and the kR k ≤ k k density of Cc∞( ) in the most proofs will be provided for f Cc∞ and extended to and larger spacesA using the standard density arguments. ∈ A 33 2 Lemma 5.2. If F C∞(C) and F (z)= O(y ) as y 0, then ∈ c → 1 ∂F 1 A := (T z)− dxdy = 0. π C ∂z − ZZ Proof. By choosing a sufficiently large N > 0 one may assyme that the support of F is contained in Ω := z C : x < N, y < N . For small ε > 0 define Ω := z C : x

Proof. If f C∞(R), then obviously f C∞(C). One can ﬁnd a ﬁnite family ∈ c ∈ c of closed curves γr which do not meet the spectrum of T and enclose a domain U containing supp f. Using the Stokes formulae we have

1 ∂f 1 1 1 f(T )= e (T z)− dxdy = f(z)(T z)− dz. π ∂z − 2πi − U r γr ZZ e X I e All the terms in the sum are zero, because f vanishes on γr. Lemma 5.5. For f,g one has (fg)(T )= f(T )g(T ). ∈A e

Proof. By the density arguments is it suﬃcient to consider the case f,g C∞(R). ∈ c Let K and L be large balls containing the supports of f and g respectively. Using the notation w = u + iv, u, v R, one can write: ∈ e e 1 ∂f ∂g 1 1 f(T )g(T )= (T z)− (T w)− dxdydudv. π2 ∂z ∂w − − ZZZZK L × e e Using the resolvent identity

1 1 1 1 1 1 (T z)− (T w)− = (T w)− (T z)− − − w z − − w z − − − 34 we rewrite the preceding integral in the form

1 ∂g 1 ∂f 1 f(T )g(T )= (T w)− dxdy dudv π2 ∂w − ∂z w z ZZL ZZK − e e 1 ∂f 1 ∂g 1 (T z)− dudv dxdy. − π2 ∂z − ∂w w z ZZK ZZL − e e By Lemma 5.1 we have

∂f 1 ∂g 1 dxdy = πf(w), dudv = πg(z), ∂z w z ∂w w z − ZZK − ZZL − e e and we arrive at

1 ∂g 1 1 ∂f 1 f(T )g(T )= f(w) (T w)− dudv + g(z) (T z)− dxdy π ∂w − π ∂z − ZZL ZZK e e 1 e ∂(fg) 1 = (T z)− dxdy e π K L ∂z − ZZ ∪ ee 1 ∂fg 1 1 ∂(fg fg) 1 = (T z)− dxdy + − (T z)− dxdy π C ∂z − π C ∂z − ZZ ZZ f ee f 1 ∂(fg fg) 1 =(fg)(T )+ − (T z)− dxdy. π C ∂z − ZZ ee f By direct calculation one can see that (fg fg)(z)= O(y2) for small y, and Lemma 5.2 shows that the second integral is zero.− f e e 1 Lemma 5.6. Let w C R. Define a function rw by rw(z)=(z w)− . Then 1 ∈ \ − r (T )=(T w)− . w − Proof. We provide just the main line of the proof without technical details (they can be easily recovered). Use first the independence of n and σ. We take n = 1 and put σ(z)= τ(λy/ x ) where λ> 0 is sufficiently large, to have w / supp σ. Without loss of generalityh wei assume w> 0. For large m> 0 consider the∈ region ℑ x Ω := z C : x < m, h i

1 ∂rw 1 1 1 rw(T ) = lim (T z)− dxdy = lim rw(z)(T z)− dz. m π ∂z − m 2πi − →∞ ZZΩm →∞ I∂Ωm e By rather technical explicit estimates (which are omitted here)e one can show that

1 lim rw(z) rw(z) (T z)− dz = 0. m ∂Ωm − − →∞ I e 35 and we arrive at

1 1 1 rw(T )= lim (T z)− dz. 2πi m z w − →∞ I∂Ωm −

For suﬃciently large m one has the inclusion w Ωm. For any f,g the function 1 ∈ ∈H C z f, (T z)− g C is holomorph in Ωm, so applying the Cauchy formula, for∋ large7→m h we have− i∈

1 1 1 1 f, (T z)− g dz = f, (T w)− g , 2πi z w − − I∂Ωm −

1 which shows that r (T )=(T w)− . w − Lemma 5.7. For any f we have: ∈A

(a) f(T )= f(T )∗,

(b) f(T ) f . ≤ k k∞

Proof . The item (a) follows directly from the equalities

1 1 (T z)− ∗ =(T z)− , f(z)= f(z). − − To show (c), take an arbitrary c> f and deﬁnee g(s) :=e c c2 f(s) 2. One ∞ can show that g . There holdsk ffk 2cg + g2 = 0, and− using− the | preceding| ∈ A − p lemmas we obtain f(T )∗f(T ) cg(T ) cg(T )∗ + g(T )∗g(T ) = 0, and − − 2 f(T )∗f(T )+ c g(T ) ∗ c g(T ) = c . − − Let ψ . Using the preceding equality we have: ∈H 2 2 2 f(T )ψ f(T )ψ + c g(T ) ψ ≤ −

= ψ, f(T )∗f(T )ψ + ψ, c g(T ) ∗ c g(T ) ψ − − = cD2 ψ 2. E D E k k As c> f was arbitrary, this concludes the proof. k k∞ All the preceding lemmas put together lead us to the following fundamental result. Theorem 5.8 (Spectral theorem, continuous functional calculus). Let T be a self-adjoint operator in a Hilbert space . There exists a unique linear map H C (R) f f(T ) ( ) 0 ∋ 7→ ∈ L H with the following properties:

f f(T ) is an algebra homomorphism, • 7→ 36 f(T )= f(T )∗, • f(T ) f , • k k ≤ k k∞ 1 1 if w / R and r (s)=(s w)− , then r (T )=(T w)− , • ∈ w − w − if supp f does not meet spec T , then f(T ) = 0. •

Proof. Existence. If one replaces C0 by , everything is already proved. But R A R A is dense in C0( ) in the sup-norm, because Cc∞( ) , so we can use the density argument. ⊂A Uniqueness. If we have two such maps, they coincide on the functions f which are linear combinations of rw, w C R. But such functions are dense in C0 by the Stone-Weierstrass theorem, so∈ by the\ density argument both maps coincide on C0. Remark 5.9. One may wonder why to introduce the class of functions : one • A could just start by Cc∞ which is also dense in C0. The reason in that we have no intuition on how the operator f(T ) should look like if f Cc∞. On the 1 ∈ other hand, it is naturally expected that for rw(s)=(s w)− we should have 1 − r (T )=(T w)− , otherwise there are no reasons why we use the notation w − rw(T ). So it is important to have an explicit formula for a sufficiently large class of functions containing all such rw. The approach based on the Helffer-Sjöstrand formula, which is presented here, • is relatively new, and it allows one to consider bounded and unbounded self- adjoint operators simultaneously. The same results can be obtained by other methods, starting e.g. with polynomials instead of the resolvents, which is a more traditional approach, see, for example, Sections VII.1 and VIII.3 in the book [8].

5.2 Borelian functional calculus and L2 representation

Now we would like to extend the functional calculus to more general functions, not necessarily continuous and not necessarily vanishing at inﬁnity. Deﬁnition 5.10 (Invariant and cyclic subspaces). Let be a Hilbert space, L be a closed linear subspace of , and T be a self-adjoint linearH operator in . H H Let T be bounded. We say that L is an invariant subspace of T (or just T -invariant) if T (L) L. We say that L is a cyclic subspace of T with cyclic vector v if L coincides with the⊂ closed linear hull of all vectors p(T )v, where p are polynomials. Let T be general. We say that L is an invariant subspace of T (or just T -invariant) if (T z)1(L) L for all z / R. We say that L is a cyclic subspace of T with cyclic − ⊂ ∈ 1 vector v if L coincides with the closed linear space of all vectors (T z)− v with z / R. − ∈

Clearly, if L is T -invariant, then L⊥ is also T -invariant.

37 Proposition 5.11. Both deﬁnitions of an invariant/cyclic subspace are equivalent for bounded self-adjoint operators.

Proof. Let T = T ∗ ( ), and let a closed subspace L be T -invariant in the sense of the definition for∈ bounded L H operators. If z C and z > T , then z / spec T and ∈ | | k k ∈ 1 ∞ 1 T − n 1 n (T z)− = z 1 = z− − T . − − − z n=0 X If x L, then T nx L for any n. As the series on the right hand side converges in ∈ ∈ 1 the operator norm sense and as L is closed, (T z)− x belongs to L. 1 − Let us denote W = z res T : (T z)− (L) L . As just shown, W is non- empty. On the other hand,∈ W is closed− in res T ⊂in the relative topology: if x L, 1 1 ∈ zn W and zn converge to z W , then (T zn)− x L and (T zn)− x converge ∈ 1 ∈ − ∈ − to (T z)− x. On the other hand, W is open: if z0 W and z z0 is sufficiently small,− then ∈ | − | 1 n n 1 (T z)− = (z z ) (T z )− − , − − 0 − 0 n=0 X 1 see (3.4), and (T z)− L L. Therefore, W = res T , which shows that L is T -invariant in the− sense of the⊂ definition for general operators.

Now let T = T ∗ ( ), and assume that L is T -invariant in the sense of the ∈ L H 1 deﬁnition for general operators, i.e. (T z)− (L) L for any z / R. Pick any − ⊂ ∈ z / R and any f L. We can represent T f = g + h, where g L and h L⊥ ∈ ∈ 1 ∈ ∈ are uniquely deﬁned vectors. As L⊥ is T -invariant, (T z)− h L⊥. On the other hand − ⊂

1 1 (T z)− h =(T z)− (T f g) − − − 1 =(T z)− (T z)f + zf g − 1− − = f +(T z)− (zf g). − − 1 As zf g L, both vectors on the right-hand side are in L. Therefore, (T z)− h − ∈ 1 − ∈ L, and finally (T z)− h = 0 and h = 0, which shows that T f = g L. The equivalence of the− two definitions of an invariant subspace is proved. ∈ On the other hand, for both definitions, L is T -cyclic with cyclic vector v iff L is the smallest T -invariant subspace containing v. Therefore, both definitions of a cyclic subspace also coincide for bounded self-adjoint operators. Theorem 5.12 (L2 representation, cyclic case). Let T be a self-adjoint linear operator in and let S := spec T . Assume that is a cyclic subspace for T with H H a cyclic vector v, then there exists a bounded measure µ on S with µ(S) v 2 and a unitary map U : L2(S,dµ) with the following properties: ≤ k k H → a vector x is in dom T iff hUx L2(S,dµ), where h is the function on S • given by h(∈Hs)= s, ∈

1 for any ψ U(dom T ) there holds UT U − ψ = hψ. • ∈ 38 In other words, T is unitarily equivalent to the operator Mh of the multiplciation by h in L2(S,dµ).

Proof. Step 1. Consider the map φ : C (R) C deﬁned by φ(f) = v, f(T )v . 0 → Let us list the properties of this map:

φ is linear, • φ(f)= φ(f), • if f 0, then φ(f) 0. This follows from • ≥ ≥ 2 φ(f)= v, f(T )v = v, f(T ) f(T )v = f(T )v . p p p φ(f) f v 2. • ≤ k k∞ k k

By the Riesz representation theorem there exists a uniquely deﬁned Borel measure µ such that

φ(f)= fdµ for all f C0(R). R ∈ Z Moreover, for supp f S = we have f(T ) = 0 and φ(f) = 0, which means that supp µ S, and we can∩ write∅ the above as ⊂ v, f(T )v = fdµ for all f C (R). (5.3) ∈ 0 ZS

Step 2. Consider the map Θ : C (R) L2(S,dµ) deﬁned by Θf = f. We have 0 → Θf, Θg = fgdµ = φ(fg) h i ZS = v, f(T )∗g(T )v = f(T )v,g(T )v .

Denote := f(T )v : f C0(R) , then the preceding equality means that the mapM ∈ ⊂ H U : C (R) L2(S,dµ), U f(T )v = f, H⊃M→ 0 ⊂ is one-to-one and isometric. Moreover, is dense in , because v is a cyclic vector, 2 M H and C0(R) is dense in L (S,dµ). Therefore, U is uniquely extended to a unitary map from to L2(S,dµ), and we denote this extension by the same symbol. H Step 3. Let f, f C (R) and ψ := f (T )v, j = 1, 2. There holds j ∈ 0 j j

ψ1, f(T )ψ2 = f1(T )v, f(T )f2(T )v

= v, (f 1ff2)(T )v

= ff 1f2 dµ ZS = Uψ , M Uψ , h 1 f 2i

39 2 where Mf is the operator of the multiplication by f in L (S,dµ). In particular, for 1 2 any w / R and r (s)=(s w)− we obtain Ur (T )U ∗ξ = r ξ for all ξ L (S,dµ). ∈ w − w w ∈ The operator U maps the set ran rw(T ) dom T to the range of Mrw . In other words, U is a bijection from dom T to ≡

ran M = φ L2(S,dµ) : x xφ(x) L2(S,dµ) = dom M . rw ∈ 7→ ∈ h Therefore, if ξ L2(S,dµ ), then ψ := r ξ dom M , ∈ w ∈ h

T r (T )U ∗ξ =(T w)r (T )U ∗ξ + wr (T )U ∗ξ = U ∗ξ + wr (T )U ∗ξ w − w w w and, ﬁnally,

UT U ∗ψ = UT U ∗rwξ = UT rw(T )U ∗ξ = U U ∗ξ + wrw(T )U ∗ξ = ξ+ wrwξ = hψ. Theorem 5.13 (L2 representation). Let T be a self-adjoint operator in a Hilbert space with spec T =: S. Then there exists N N, a finite measure µ on S N and aH unitary operator U : L2(S N,dµ) with⊂ the following properties. × H → × Let h : S N R be given by h(s, n)= s. A vector x belongs to dom T • iff hUx ×L2(S→ N,dµ), ∈H ∈ × 1 for any ψ U(dom T ) there holds UT U − ψ = hψ. • ∈ Proof. Using the induction one can find N N and non-empty closed subspaces ⊂ with the following properties: Hn ⊂H

= n N n, • H ∈ H n each L is a cyclic subspace of T with cyclic vector v satisfying v 2− . • Hn n k nk ≤

The restriction Tn of T to n is a self-adjoint operator in n, and one can apply to all H H n these operators Theorem 5.12, which gives associated measures µn with µ(S) 4− , 2 ≤ and unitary maps Un : n L (S,dµn). Now one can deﬁne a measure µ on S N by µ Ω n = µ (Ω),H and→ a unitary map × × { } n U : L2(S N,dµ) L2(S,dµ ) H ≡ Hn → × ≡ n n N n N M∈ M∈ by U(ψn)=(Unψn), and one can easily check that all the properties are veriﬁed. Remark 5.14. The previous theorem shows that any self-adjoint operator is unitarily equivalent• to a multiplication operator in some L2 space, and this multiplication operator is sometimes called a spectral representation of T . Clearly, such a representation is not unique, for example, the decomposition of the Hilbert space in cyclic subspaces in not unique.

40 The cardinality of the set N is not invariant. The minimal cardinality among • all possible N is called the spectral multiplicity of T , and it generalizes the notion of the multiplicity for eigenvalues. Calculating a spectral multiplicity is a non-trivial problem.

Theorem 5.13 can be used to improve the result of Theorem 5.8. In the rest of the section we use the function h and the measure µ from Theorem 5.13 without further speciﬁcations. Introduce the set consisting of the Borel functions f : R C such that BT →

f ,T := essµ sup f h < . k k∞ | ◦ | ∞

T In what follows, we say that fn T converges to f T and write fn B f if the following two conditions hold: ∈ B ∈ B −→

there exists c> 0 such that fn ,T c for all n and µ-a.e. • k k∞ ≤ f (x) f(x) for µ-a.e. x. • n → Definition 5.15 (Strong convergence). Wa say that a sequence A ( ) n ∈ L H converges strongly to A ( ) and write A = s lim An if Ax = lim Anx for any x . ∈ L H − ∈H Theorem 5.16 (Borel functional calculus). (a) Let T be a self-adjoint operator in a Hilbert space . There exists a map T f f(T ) ( ) extending the map from TheoremH 5.8 and satisfyingB the∋ same7→ properties∈ except L H that one can improve the estimate f(T ) f by f(T ) f ,T . k k ≤ k k∞ k k ≤ k k∞ T (b) This extension is unique if we assume that the condition fn B f implies f(T ) = s lim f (T ). −→ − n Proof. Consider the map U from Theorem 5.8. Then it is sufficient to define f(T ) := U ∗Mf hU, then one routinely check that all the properties hold, and (a) is ◦ proved. To prove (b) we remark first that the map just defined satisfies the requested con- 2 T dition: If x L (S,dµ) and fn B f, then fn(h)x converges to f(h)x in the norm of L2(S∈ N,dµ) by the dominated−→ convergence. But this means exactly that f(T ) = s lim×f (T ). − n On the other hand, C0(R) is obviously dense in T with respect to the T convergence, which proves the unicity of the extension.B B

We have a series of important corollaries, whose proof is an elementary modiﬁcation of the constructions given for the multiplication operator in Example 3.6.

Corollary 5.17. spec T = ess ran h, • µ for any f one has spec f(T ) = ess ran f h, • ∈ BT µ ◦ 41 in particular, f(T ) = ess sup f h . • k k µ | ◦ | Example 5.18. One can also define the operators ϕ(T ) with unbounded functions ϕ by ϕ(T )= U ∗Mϕ hU. These operators are in general unbounded, but they are self- ◦ adjoint for real-valued ϕ; this follows from the self-adjointness of the multiplication operators Mϕ h. ◦ Example 5.19. The usual Fourier transform is a classical example of a spectral representation. For example, Take = L2(R) and T = id/dx with the natural domain dom T = H1(R). If is theH Fourier transform, then− T is exactly the operator of multiplication x F xf(x), and spec T = R. F F 7→ In particular, for bounded Borel functions f : R C one can define the operators → f(T ) by f(T )h = ∗M , where M is the operator of multiplication by f, i.e. in F f F f general one obtains a pseudodifferential operator. Let us look at some particular examples. Consider the shift operator A in which is defined by Af(x)= f(x + 1). It is a bounded operator, and for any u H(R) we ip iT ∈ S have A ∗u(p) = e u(p). This means that A = e , and this gives the relation spec AF=Fz : z = 1 .F { | | } An another example one may look at the operator B defined by

x+1 Bf(x)= f(t)dt. x 1 Z − Using the Fourier transform one can show that B = ϕ(T ), where ϕ(x) = 2 sin x/x with spec B = ϕ(R). Example 5.20. For practical computations one does not need to have the canonical representation from Theorem 5.13 to construct the Borel functional calculus. It is 2 suﬃcient to represent T = U ∗Mf U, where U : L (X,dµ) and Mf is the multiplcation operator by some function f. Then forH →any Borel function ϕ one can put ϕ(T )= U ∗Mϕ f U. ◦ For example, for the free Laplacian T in = L2(Rd) the above is realized with X = Rd and U being the Fourier transform,H and with f(p) = p2. This means that the operators ϕ(T ) act by

1 2 ipx ϕ(T )f(x)= d/2 ϕ(p )f(p)e dx. (2π) Rd Z For example, b 1 √ 2 ipx ∆ + 1f(x)= d/2 1+ p f(p)e dx − (2π) Rd Z p and one can show that dom √ ∆+1= H2(Rd). b − Example 5.21. Another classical example is provided by the Fourier series. Take 2 d d c2 m = ℓ (Z ) and let a function t : Z R satisfy t(m) c1e− | | with some cH,c > 0. Deﬁne T by → ≤ 1 2 T u(m)= t(m n)u(n ). − n Zd X∈ 42 One can easily see that T is bounded and self-adjoint. If one introduces the unitary map Φ : L2(Td), T := R/Z, H → Φu(x)= e2πimxu(m), mx := m x + + m x , 1 1 ··· d d m Zd X∈ then T = Φ∗Mτ Φ with τ(x)= t(m)e2πimx. m Zd X∈

Example 5.22. A less obvious example is given by the Neumann Laplacian TN on the half-line deﬁned in Example 2.12. 2 R 2 R 2 R Let T be the free Laplacian in L ( ). Denote by := Lp( ) the subspace of L ( ) consisting of the even functions. Clearly, is an invariantG subspace for T (the second derivate of an even function is also an evenG function), and the restriction of T to is a self-adjoint operator; denote this restriction by A. Moreover, is an invariantG subspace of the Fourier transform (the Fourier image of an even functionG is also an F 2 1/2 even function). Introduce now the a map Φ : L (R+) by Φf(x) = 2− f x . One checks easily that Φ is unitary and that dom A =→ Φ(dom G T ). | | N So we have TN = Φ∗AΦ and A = ∗Mh , where Mh is the multiplication by the 2 F F function h(p)= p in . Finally, M = ΦM Φ∗, where M is the multiplication by G h h h h is . f f G At the end of the day we have Tf= U ∗M U with U = Φ∗ Φ, and U is unitary N h F being a composition of three unitary operators. By direct calculation, for f L2(R ) L1(R ) one has ∈ + ∩ +

2 ∞ Uf(p)= cos(px)f(x) dx. π r Z0 This transform U is sometimes called the cos-Fourier transform. Roughly speaking, U is just the Fourier transform restricted to the even functions together with some identiﬁcations. An interested reader may adapt the preceding constructions to the Dirichlet Lapla- cian TD on the Half-line, see Example 2.13. Example 5.23 (Operators with compact resolvent). Let us ﬁll the gap which is missing in Subsection 4.2. Namely let us show that if a self-adjoint T has a compact resolvent, then spec T = R. 6 1 Assume that spec T = R and consider the function g given by g(x)=(x i)− . 1 − Then g(T )=(T i)− is a compact operator, and its spectrum with at most one accumulation point.− On the other hand, using Corollary 5.17 and the continuity of g one has the equality spec g(T ) = g(spec T ) = g(R), and this set has no isolated points.

43 6 Some applications of spectral theorem

In this chapter we discuss some direct applications of the spectral theorem to the estimates of the spectra of self-adjoint operators. We still use without special noti- ﬁcation the measure µ and the function h from Theorem 5.13. Theorem 6.1 (Distance to spectrum). Let T be a self-adjoint operator in a Hilbert space , and 0 = x dom T , then for any λ C one has the estimate H 6 ∈ ∈ (T λ)x dist(λ, spec T ) − . ≤ x k k Proof. If λ spec T , then the left-hand side is zero, and the inequality is valid. ∈ 1 Assume now that λ / spec T . By Corollary 5.17, one has, with ρ(x)=(x λ)− , ∈ − 1 1 (T λ)− = ess sup ρ h = , k − k µ | ◦ | dist(λ, spec T ) which gives

1 1 x = (T λ)− (T λ)x (T λ)x . k k k − − k ≤ dist(λ, spec T ) k − k Remark 6.2. The previous theorem is one of the basic tools for the constructing approximations of the spectrum of the self-adjoint operators. It is important to understand that the resolvent estimate obtained in Theorem 6.1 uses in an essential way the self-adjointness of the operator T . For non-self-adjoint operators the estimate fails even in the ﬁnite-dimensional case. For example, take = C2 and H 0 1 T = , 0 0 then spec T = 0 , and for z = 0 we have { } 6 1 1 z 1 (T z)− = . − −z2 0 z 1 2 For the vectors e1 = (1, 0) and e2 = (0, 1) one has e1, (T z)− e2 = z− , which h − 2i − shows that the norm of the resolvent near z = 0 is of order z− . In the inﬁnite 1 n dimensional-case one can construct examples with (T z)− dist(z, spec T )− for any power n. k − k ∼

6.1 Spectral projections

Deﬁnition 6.3 (Spectral projection). Let T be a self-adjoint operator in a Hilbert space and Ω R be a Borel subset. The spectral projection of T on H ⊂ Ω is the operator ET (Ω) := χΩ(T ), where χΩ is the characteristic function of Ω. This exchange between the index and the argument is due to the fact that the spectral projections are usually considered as functions of subsets Ω (with a ﬁxed operator T ).

44 The following proposition summarizes the most important properties of the spectral projections. Proposition 6.4. For any self-adjoint operator T acting a in a Hilbert space there holds:

1. for any Borel subset Ω R the associated spectral projection ET (Ω) is an orthogonal projection commuting⊂ with T . In particular, E (Ω) dom T dom T . T ⊂ 2. E (a, b) = 0 if and only if spec T (a, b)= . T ∩ ∅

3. for any λ R there holds ran ET ( λ ) = ker(T λ), and f ker(T λ) iﬀ f = E ( λ∈)f. { } − ∈ − T { } 4. spec T = λ R : E (λ ε,λ + ε) = 0 for all ε> 0 . { ∈ T − 6 } 2 Proof. To prove (1) we remark that χΩ = χΩ and χΩ = χΩ, which gives ET (Ω)ET (Ω) = ET (Ω) and ET (Ω) = ET (Ω)∗ and shows that ET (Ω) is an orthogonal projection. To prove the commuting with T we restrict ourselves by considering T realized as a multiplication operator from Theorem 5.8. Let x dom T , 2 2 ∈ then hx L (S, N,µ) and, subsequently, h χΩ h x L , which means that χ x dom∈ T . The× commuting follows now from· h ◦χ · h∈x = χ h h x. Ω ∈ · Ω ◦ · Ω ◦ · · To prove (2) we note that the condition ET (a, b) = 0 is, by deﬁnition, equivalent to χ h = 0 µ-e.a., which in turn means that (a, b) ess ran h = , and it (a,b) ◦ ∩ µ ∅ remains to recall that essµ ran h = spec T , see Corollary 5.17. The items (3) and (4) are left as elementary exercises.

As an important corollary of the assertion (4) one has the following description of the spectra of self-adjoint operators, whose proof is another simple exercise. Corollary 6.5. Let T be self-adjoint, then λ spec T if and only if there exists a sequence x dom T with x 1 such that ∈(T λ)x converge to 0. n ∈ k nk ≥ − n One can see from Proposition 6.4 that the spectral projections contains a lot of useful information about the spectrum. Therefore, it is a good idea to understand how to calculate them at least for simple sets Ω. Proposition 6.6 (Spectral projection to a point). For any λ R there holds ∈ 1 ET ( λ )= i s lim ε(T λ iε)− . { } − ε− 0+ − − → Proof. For ε> 0 consider the function iε f (x) := . ε −x λ iε − − One has the following properties:

f 1, • | ε| ≤ 45 f (λ) = 1, • ε if x = λ, then f (x) tends to 0 as ε tends to 0. • 6 ε T This means that fε B χ λ . By Theorem 5.16, ET ( λ ) = s limε 0+ fε(T ), and −→ { } 1 { } − → it remains to note that f (T )=(T λ iε)− by Theorem 5.8. ε − − Proposition 6.7 (Stone formula). For a < b one has: 1 1 b ET (a, b) + ET [a, b] = s lim R(λ + iε) dλ. 2 π ε− 0+ ℑ → Za Proof. For ε> 0 consider the function 1 b 1 f (x)= dλ. ε π ℑx λ iε Za − − By direct computation we have 1 b ε 1 b λ a λ f (x)= dλ = arctan − arctan − . ε π (x λ)2 + ε2 π ε − ε Za − Therefore, f 1, and | ε| ≤ 0, x / [a, b], ∈ 1 lim fε(x)= 1, x (a, b), = χ(a,b)(x)+ χ[a,b](x) , ε 0+ 1 ∈ 2 →  , x a, b , 2 ∈ { }  and the rest follows as in the previous proposition. Finally the following formula can be useful for the computation of spectral projections on isolated components of the spectrum. Proposition 6.8 (Spectral projection on isolated part of spectrum). Let Γ C be a smooth closed curve oriented in the anti-clockwise sense which does not meet⊂ spec T , and let Ω be the intersection of the interior of Γ with R, then

1 1 E (Ω) = (T z)− dz. T 2πi − IΓ Proof. If x is an intersection point of Γ with R, then, by assumption x / essµ ran h. On the other hand, for x R Γ there holds, using the Cauchy formula,∈ ∈ \

1 1 1, x is inside Γ, (x z)− dz = 2πi − 0, x is outside Γ. IΓ ( Therefore, µ-a.e. one has

1 1 (h z)− dz = χ h, 2πi − Ω ◦ IΓ and one can replace h by T using Theorem 5.16.

46 As a ﬁnal remark we mention that the map Ω ET (Ω) can be viewed an operator- valued measure, and one can integrate reasonable7→ scalar function (bounded Borel ones or even unbounded) with respect to this measure using e.g. the Lebesgue integral sums. Then one obtains the integral representations,

T = λdET (λ), f(T )= f(λ)dET (λ), R R Z Z and the associated integral sums can be viewed as certain approximations of the respective operators.

6.2 Generalized eigenfunctions

Let T be a self-adjoint operator in a Hilbert space . Let be another Hilbert H H+ space which is continuously embedded in , and let be the dual of +. The − triple is usually referred toH as a GelfandH triple or riggingH of . To H+ ⊂H⊂H+ H keep simple notation we denote the action of f on g + as f,g . ∈H− ∈H h i Deﬁnition 6.9 (Generalized eigenfunction). We say that a vector ψ is − a generalized eigenfunction of T with the generalized eigenvalue λ R if ∈ψ, H(T λ)ϕ = 0 for all ϕ dom T with T ϕ . ∈ − ∈ ∩H+ ∈H+ One has the following fundamental result.

Theorem 6.10 (Existence of expansion in generalized eigenfunctions). One can ﬁnd a rigging such that:

the set := ϕ dom T + : T ϕ + is a core of T , e.g. T = T , • D { ∈ ∩H ∈H } |D there exists a measure space (M,µ) and a map Φ : M with the following − • properties: →H

2 – the map + h h L (M,dµ) deﬁned by h(m) := Φ(m),h extends to a unitaryH operator∋ 7→ from∈ to L2(M,dµ), h i H – there exists a measurableb function a : M R bsuch that Φ(m) is a generalized eigenfunction of T with the generalized→ eigenvalue a(m) for µ-a.e. m M. ∈ We are not giving any proof here, an interested reader may refer to a good concise discussion in Supplement S1.2 of the book [2] or to the detailed study in the book [1].

Example 6.11 (Generalized eigenfunctions of Laplacian). Let = L2(Rn) 2 n 2 n H and T = ∆. One can take + = H (R ), then = H− (R ). One can easily − show that− for any p R3 theH function ψ, ψ(x)= eipxH is a generalized eigenfunction ∈ of T with the generalized eigenvalue p2. The associated map h h is the usual Fourier transform, and (M,dµ) is just Rn with the Lebesgue measure.7→ b 47 Theorem 6.10 just gives a special form of a unitary transform U from Theorem 5.13. Informally speaking, the theorem says that calculating the spectrum is in a sense equivalent to solving the eigenvalue problem Tψ = λψ but in a certain larger space . On the main difficulties in applying such an approach is that in concrete − examplesH the spaces are described in a rather implicit way, and it difficult to ± decide if a given vector/distributionH belongs to this space or not. Some particular cases are indeed well-studied. For example, one has the following nice description of the spectrum for Schrödinger operators, which we state without proof (see e.g. [3], Chapter 2):

2 n 2 Theorem 6.12 (Schnol theorem). Let = L (R ), V Lloc, V 0, T = ∆+ V (we take the operator deﬁned byH the Fridrichs extension).∈ De≥note by Σ the− set of the real numbers λ for which there exists a non-zero solution u to the diﬀerential equation ( ∆+ V )u = λu with the subexponential growth, i.e. such that − a x n for any a> 0 there exists C > 0 such that u(x) Ce | | for all x R . Then the spectrum of T coincides with the closure of Σ. ≤ ∈

Note that there are various versions of the above result for diﬀerential operators on manifolds and other related spaces, then the subexponentional growth condition should be replaced by a suitable relation comparing the growth of generalized eigenfunctions with the growth of the volume of balls at inﬁnity. Example 6.13. One can look again at the operator T = d2/dx2 in = L2(R). − H for any λ R the equation u′′ = λu has two linearly independent solutions. For λ< 0 all non-zero∈ solutions are− exponentially growing for x + or for x , and such values λ do not belong to the spectrum. For λ→= 0∞ one has either→ −∞ a constant or a linear function, and for λ> 0 the both solutions are bounded, which gives again spec T = [0, + ). ∞

6.3 Tensor products

Let Aj be self-adjoint operators in Hilbert spaces j, j = 1,...,n. With any m1 mn N H m1 mn monomial λ1 ...λ2 , mj , one can associate the operator A1 An in := · deﬁned by∈ ⊗ · H H1 ⊗···⊗Hn (Am1 Amn )(ψ ψ )= Am1 ψ Amn ψ , ψ dom Amj , 1 ⊗···⊗ n 1 ⊗···⊗ n 1 1 ⊗···⊗ n n j ∈ j 0 and then extended by linearity; here the zero powers Aj equal the identity operators in the respective spaces. Remark 6.14. For an operator A in a Hilbert space the domain dom An is usually deﬁned in a recursive way: H

0 n n 1 dom A = and dom A = x dom A : Ax dom A − for n N. H { ∈ ∈ } ∈ n n As an exercise one can show that for a self-adjoint A one has dom A = ran RA(z) with any z res A and that dom An is dense in for any n. ∈ H 48 Using the above construction one can associate with any polynomial P of λ1,...,λn of degree N a linear operator P (A ,...,A ) in deﬁned on the set consisting of 1 n H H the linear combinations of the vectors of the form ψ ψ with ψ dom AN . 1 ⊗···⊗ n j ∈ j Theorem 6.15 (Spectrum of tensor product). Denote by B the closure of the above operator P (A1,...,An), then B is self-adjoint, and

spec B = P (λ ,...,λ ) : λ spec A . 1 n j ∈ j Sketch of the proof. The complete proof involves a number of technicalities, see e.g. Section III.10 in [8], but the main idea is rather simple. By the spectral theorem, it is suﬃcient to consider the case when Aj is the multiplication by a certain function f in := L2(M ,dµ ). Then j Hj j j = L2(M,dµ), M = M M , µ = µ µ , H 1 ×···× n 1 ⊗···⊗ n and P (A ,...,A ) acts in as the multiplication by p, p(x ,...,x ) = 1 n H 1 n P f1(x1),...,fn(xn) , and its domain includes at least all the linear combinations of the functions ψ ψ where ψ are L2 with compact supports. It is a routine 1 n j to show that the closure⊗···⊗ of this operator is just the usual multiplication operator by p, which gives the sought relation.

Example 6.16 (Laplacian in rectangle). A typical example of the above construction is given by the Laplacians in rectangles. Namely, let a,b > 0 and Ω=(0,a) (0, b) R2, = L2(Ω), and T be the Dirichlet Laplacian in Ω. One can show that×T can be⊂ obtainedH using the above procedure using the representation

T = L 1 + 1 L , a ⊗ ⊗ b where by L we denote the Dirichlet Laplacian in := L2(0,a), i.e. a Ha 2 1 L f = f ′′, dom L = H (0,a) H (0,a). a − a ∩ 0

It is known (from the exercises) that the spectrum of La consists of the simple eigenvalues (πn/a)2, n N, with the eigenfunctions x sin(πnx/a), and this means that the spectrum∈ of T consists of the eigenvalues 7→ πm 2 πn 2 λ (a, b)= + , m,n N, m,n a b ∈ and the associated eigenfunctions are the products of the respective eigenfunctions for La and Lb. The multiplicity of each eigenvalue λ is exactly the number of pairs (m, n) N2 for which λ = λ . ∈ m,n Note that the closure of the set λ can be omitted as this is a discrete set. { m,n} The same constructions hold for the Neumann Laplacians, one obtains the same formula for the eigenvalues but now with m, n N 0 . ∈ ∪ { }

49 7 Perturbations

7.1 Kato-Rellich theorem

We have seen since the beginning of the course that one needs to pay a great attention to the domains when dealing with unbounded operators. The aim of the present subsection is to describe some classes of operators in which such problems can be avoided.

Deﬁnition 7.1 (Essentially self-adjoint operator). We say that a linear operator T is essentially self-adjoint on a subspace dom T if the closure of the restriction of T to is a self-adjoint operator. IfD the ⊂ closure of T is self-adjoint, then we simply sayD that T is essentially self-adjoint.

Proposition 7.2. An essentially self-adjoint operator has a unique self-adjoint extension.

Proof. Let T be an essentially self-adjoint operator, and let S be its self-adjoint extension. As S is closed, the inclusion T S implies T S. On the other hand, ⊂ ⊂ S = S∗ (T )∗ = T (as T is self-adjoint). This shows that S = T . ⊂ Theorem 7.3 (Self-adjointness criterion). Let T be a closed symmetric operator in a Hilbert space , then the following three assertions are equivalent: H 1. T is self-adjoint,

2. ker(T ∗ + i) = ker(T ∗ i)= 0 , − { } 3. ran(T + i) = ran(T i)= . − H Proof. The implication 1 2 is obvious: a self-adjoint operator cannot have non- real eigenvalues. ⇒

To show the implication 2 3 recall ﬁrst that ker(T i) = ran(T i)⊥. Therefore, ⇒ ± ∓ it is suﬃcient to show that the subspaces ran(T i) are closed. For any f dom T we have: ± ∈

2 (T i)f = (T i)f, (T i)f = TfTf + f, f i Tf,f f,Tf ± ± ± h i h i ± h i−h 2 i 2 = T f + f . k k k k Let fn ran(T i) such that fn converge to some f . Find ϕn dom T with f = (T∈ i)ϕ ±, then due to the preceding equality∈ (ϕ H) and (T ϕ ∈) are Cauchy n ± n n n sequences. As T is closed, ϕn converge to some ϕ dom T and T ϕn converge to T ϕ, and then f =(T i)ϕ converge to (T i)ϕ =∈f and f ran(T i). n ± n ± ∈ ± It remains to the prove the implication 3 1. Let ϕ dom T ∗ and let ψ dom T ⇒ ∈ ∈ such that (T i)ψ = (T ∗ i)ϕ. As T T ∗, we have (T ∗ i)(ψ ϕ)=0. On − − ⊂ − − the other hand, due to ran(T + i)= we have ker(T ∗ i) = 0, which means that ϕ = ψ dom T . H − ∈ 50 Note that during the proof we obtained the following simple fact: Proposition 7.4. Let T be a symmetric operator, then ran(T i) = ran(T i). ± ± This leads as to the following assertion: Corollary 7.5 (Essential self-adjointness criterion). Let T be a symmetric operator in a Hilbert space , then the following three assertions are equivalent: H 1. T is self-adjoint,

2. ker(T ∗ + i) = ker(T ∗ i)= 0 , − { } 3. ran(T + i) and ran(T i) are dense in . − H Remark 7.6. The above theorem can be modiﬁed is various ways. For example, it still holds if one replaces T i by T iλ with any λ R 0 . For semibounded operators we haven another± version: ± ∈ \ { } Theorem 7.7 (Self-adjointness criterion for semibounded operators). Let T be a closed symmetric operator in a Hilbert space and T 0 and let a > 0, then the following three assertions are equivalent. H ≥

1. T is self-adjoint,

2. ker(T ∗ + a) = 0, 3. ran(T + a)= . H This is left as an exercise. The analogues of Proposition 7.4 and Corollary 7.5 hold as well.

Now we would like to apply the above assertions to the study of some perturbations of self-adjoint operators. Deﬁnition 7.8 (Relative boundedness). Let A be a self-adjoint operator in a Hilbert space and B be a symmetric operator with dom A dom B. Assume that there existHa,b > 0 such that ⊂

Bf a Af + b f for all f dom A, k k ≤ k k k k ∈ then B is called relatively bounded with respect to A or, for short, A-bounded. The inﬁmum of all possible values a is called the relative bound of B with respect to A. If the relative bound is equal to 0, then B is called inﬁnitesimally small with respect to A. Theorem 7.9 (Kato-Rellich). krel Let A be a self-adjoint operator in and let B be a symmetric operator in which is A-bounded with a relative bound 0 such that ∈ Bu a Au + b u , for all u dom A. (7.1) k k ≤ k k k k ∈ Step 1. As seen many times, for any λ> 0 one has

2 2 (A + B iλ)u = (A + B)u + λ2 u 2. ± k k Therefore, for all u dom A one can estimate ∈ 2 (A + B iλ)u (A + B)u + λ u Au Bu + λ u ± ≥ k k ≥ k k − k k k k = (1 a) Au +(λ b) u . (7.2) − k k − k k Let us pick some λ > b. Step 2. Let us show that A+B with the domain equal to dom A is a closed operator. Let (un) dom A and fn := (A+B)un such that both un and fn converge in . By (7.2), Au⊂is a Cauchy sequence. As A is closed, u converge to som u domHA and n n ∈ Aun converge to Au. By (7.1), Bun is a Cauchy sequence and is hence convergent to some v . Let us show that Bun converge exactly to un (actually this would follow from∈ the H closedness of B, but we did not assume that B is closed!). Take any h dom A, then v,h = lim Bu ,h = lim u ,Bh = u,Bh = Bu,h . So ∈ h i h n i h n i h i h i ﬁnally (A + B)un converge to (A + B)u. This shows that A + B is closed. Step 3. Let us show that the operators A + B iλ : dom A are bijective at least for large λ. As previously, we have (A iλ±)u 2 = Au →2 + Hλ2 u 2. Then k ± k k k k k b b Bu a Au + b u a (A iλ)u + (A iλ)u = a + (A iλ)u . k k ≤ k k k k ≤ ± λ ± λ ± | | | | As a (0, 1), we can choose λ suﬃciently large to have a + b/ λ < 1. This means ∈ 1 | | that for such λ we have B(A iλ)− < 1. Now we can represent ±

1 A + B iλ = 1+ B(A iλ)− (A iλ). ± ± ± As A is self-adjoint, the operators A iλ : dom A are bijections, and 1 + 1 ± → H B(A iλ)− is a bijection from to itself. Therefore, A + B iλ are bijective, in particular,± ran(A + B iλ) =H . By Theorem 7.3 and Remark± 7.6, A + b is self-adjoint. ± H For the part concerning the essential self-adjointness one needs just need to show the relation A + B = A + B, which is an elementary exercise.

7.2 Essential self-adjointness of Schr¨odinger operators

The Kato-Rellich theorem is one of the tools used to simplify the consideration of the Schr¨odinger operators.

52 p d d Theorem 7.10. Let V L (R )+ L∞(R ) be real-valued with p = 2 if n 3 and p>d/2 if d> 3, then the∈ operator T = ∆+ V with the domain dom T =≤H2(Rd) 2 Rd − Rd is a self-adjoint operator in L ( ), and it is essentially self-adjoint on Cc∞( ).

Proof. We give the proof only for the dimension d 3. For all f (Rd) and λ> 0 we have the representation ≤ ∈ S

1 ipx f(x)= d/2 e f(p) dp (2π) Rd Z 1 1 b 2 = d/2 2 (p + λ)f(p)dp (2π) Rd p + λ Z 1 1 (p2 + λb)f(p) ≤ (2π)d/2 p2 + λ ·

1 1 2 p f(p) b+ λ f = aλ ∆f + bλ f ≤ (2π)d/2 p2 + λ · k k k k k k k k with b b 1 1 λ 1 a = , b = . λ (2π)d/2 p2 + λ λ (2π)d/2 p2 + λ

By density, for all f H2(Rd) and all λ> 0 we have ∈

f aλ ∆f + bλ f . k k∞ ≤ k k k k 2 d d By assumption we can represent V = V1 + V2 with V1 L (R ) and V2 L∞(R ). Using the preceding estimate we arrive at ∈ ∈

2 d V f V1f + V2f V1 2 f + V2 f aλ ∆f + bλ f , f H (R ), k k ≤ k k k k ≤ k k k k∞ k kk k ≤ k k k k ∈ with aλ = V1 2aλ and bλ = V1 2bλ + V2 . As aλ can bee made arbitrary small ∞ e by a suitablek choicek of λ, we seek thatk thek multiplicationk operator V is inﬁnitesimally smalle with respect to thee free Laplacian, and the result follows from the Kato-Rellich theorem. 2 1 The above proof does not work for d 3 as the function p (p + λ)− does not belong to L2(Rd) anymore. The respective≥ parts of argument7→ should be replaced by suitable Sobolev embedding theorems.

Example 7.11 (Coulomb potential). Consider the three-dimensional case and the potential V (x) = α/ x , α R. For any bounded open set Ω containing the 2| | 3 ∈ 3 2 3 origin, one has χΩV L (R ) and (1 χΩ)V L∞(R ), and ﬁnally V L (R )+ 3 ∈ − ∈ ∈ L∞(R ). This means that the operator T = ∆+α/ x is self-adjoint on the domain H2(Rd). − | |

Let us mention some other conditions guaranteeing the essential self-adjointness of the Schr¨odinger operators for other types of potentials.

53 Theorem 7.12. Let = L2(Rd) and let V C0(Rd) be real-valued such that for some c R one has theH inequality ∈ ∈ u, ( ∆+ V )u c u 2 h − i ≥ k k Rd for all u Cc∞( ). Then the operator T = ∆+ V is essentially self-adjoint on Rd ∈ − Cc∞( ).

Proof. By adding a constant to the potential V one can assume that T 1. In other words, using the integration by parts, ≥

2 2 2 u(x) dx + V (x) u(x) dx u(x) dx (7.3) Rd ∇ Rd ≥ Rd Z Z Z

Rd 1 Rd for all u Cc∞( ), and this extends by density at least to all u Hcomp( ). By Theorem∈ 7.7 it is suﬃcient to show that the range of T is dense. ∈ 2 Rd Rd Let f L ( ) such that f, ( ∆+ V )u = 0 for all u Cc∞( ). Note that T preserve∈ the real-valuedness, and− we can suppose without loss∈ of generality that f is d real-valued. We have at least ( ∆+ V )f = 0 in the sense of ′(R ), and ∆f = V f. − 2 Rd D As V is locally bounded, the function V f is in Lloc( ), and the elliptic regularity 2 Rd gives f Hloc( ). ∈ d Let us pick a real-valued function ϕ C∞(R ) such that ϕ(x) = 1 for x 1, ∈ c | | ≤ that ϕ(x) = 0 for x 2 and that 0 ϕ 1, and introduce functions ϕn by ϕ (x)= ϕ(x/n). For| any| ≥ u H1 (Rd) we≤ have,≤ by a standard computation: n ∈ loc

(ϕnf) (ϕnu)dx + V ϕnfϕnudx Rd ∇ ∇ Rd Z Z d 2 ∂u ∂u ∂ϕn 2 = ϕn fudx + f f ϕn dx + f, T ϕnu . (7.4) Rd ∇ Rd ∂x − ∂x ∂x h i Z j=1 Z j j j X

2 Rd As ϕnu Cc∞( ), the last term vanishes. Taking now u = f and using (7.3) we arrive at∈ 2 2 2 2 2 2 ϕn f dx ϕnf dx ϕnf dx, Rd ∇ ≥ Rd ≥ Z Z ZΩ where Ω is any ball. As n tends to inﬁnity, the left-hand side goes to 0. On the other side, the restriction of ϕnf to Ω coincides with f for suﬃciently large n, and this means that f vanishes in Ω. As Ω is arbitrary, f = 0.

Another condition, which complements the preceding theorems, will be given without proof.

d Theorem 7.13. Let V Lloc(R ) be non-negative, then the operator ∆+ V is ∈ Rd − essentially self-adjoint on Cc∞( ).

54 7.3 Discrete and essential spectra

Up to know we just distinguished between the whole spectrum and the point spectrum, i.e. the set of the eigenvalues. Let us introduce another classification of spectra, which is useful when studying various perturbations. Definition 7.14 (Discrete spectrum, essential spectrum). Let T be a self- adjoint operator in a Hilbert space . We define its discrete spectrum specdisc T by H

spec T := λ spec T : ε> 0 with dim ran E (λ ε,λ + ε) < . disc ∈ ∃ T − ∞ n o The set spec T := spec T spec T is called the essential spectrumof T . ess \ disc The following proposition gives an alternative description of the discrete spectrum.

Proposition 7.15. A real λ belongs to specdisc T iﬀ and only it is an isolated eigenvalue of T of ﬁnite multiplicity.

Proof. Let λ specdisc T , then there exists ε0 > 0 such that the operators ET (λ ε,λ+ε) do not∈ depend on ε if ε (0,ε ). On the the other hand, this limit operator− ∈ 0 is non-zero, as λ spec T . This means ET λ = s limε 0+ ET (λ ε,λ + ε) = → 0, and λ spec∈T by Proposition 6.4(3).{ At} the− same time, E (λ− ε ,λ) =6 ∈ p T − 0 E (λ,λ + ε ) = 0, and Proposition 6.4(2) show that λ is an isolated point of the T 0 spectrum. Now let λ be an isolated eigenvalue of ﬁnite multiplicity. Then there exists ε0 > 0 such that ET (λ ε0,λ) = ET (λ,λ+ε0) = 0, and dim ran ET ( λ ) = dim ker(T λ) < . Therefore,− { } − ∞ dim ran E (λ ε ,λ + ε ) = dimran E (λ ε ,λ) T − 0 0 T − 0 + dimran E (λ,λ + ε ) + dimran E ( λ ) < . T 0 T { } ∞ Therefore, we arrive at the following direct description of the essential spectrum

Proposition 7.16. A value λ spec T belongs to specess T iﬀ at least one of the following three conditions holds:∈ λ / spec T , • ∈ p λ is an accumulation point of spec T , • p dim ker(T λ)= . • − ∞ Furthermore, the essential spectrum is a closed set.

Proof. The ﬁrst part just describes the points of the spectrum which are not isolated eigenvalues of ﬁnite multiplicity.

For the second part we note that specess T is obtained from the closed set spec T by removing some isolated points. As the removing an isolated point does not change the property to be closed, specess T is also closed.

55 Let us list some examples.

Proposition 7.17 (Essential spectrum for compact operators). Let T be a compact self-adjoint operator in an infinite-dimensional space , then specess T = 0 . H { } Proof. By Theorem 4.2, for any ε > 0 the set spec T ( ε,ε) consists of a finite number of eigenvalues of finite multiplicity, hence we have:\ − spec T ( ε,ε) = ess \ − ∅ and dim ran ET R ( ε,ε) < . On the other hand, dim = dimran ET R ( ε,ε) + dimran \E −( ε,ε) ,∞ and dimran E ( ε,ε) mustH be infinite for any\ − T − T − ε> 0, which means that 0 spec T . ∈ ess Proposition 7.18 (Essential spectrum of operators with compact resolvents). The essential spectrum of a self-adjoint operator is empty if and only if the operator has a compact resolvent.

Proof is left as an exercise. Sometimes one uses the following terminology:

Deﬁnition 7.19 (Purely discrete spectrum). We say that an self-adjoint operator T has a purely discrete spectrum in some interval (a, b) if spec T (a, b)= . ess ∩ ∅ If one has simply specess T = , then we say simply that the spectrum of T is purely discrete. As follows from the∅ previous proposition, this exactly means that T has compact resolvent.

Example 7.20. As seen several times, the free Laplacian in L2(Rd) has the spectrum [0, + ). This set has no isolated points, so this operator has no discrete spectrum. ∞ The main diﬀerence between the discrete and the essential spectra comes from their behavior with respect to perturbations. This will be discussed in the following sections.

7.4 Weyl criterion and relatively compact perturbations

Let T be a self-adjoint operator in a Hilbert space . H The following proposition is an exercise.

Proposition 7.21. Let λ be an isolated eigenvalue of T , then there exists c > 0 such that (T λ)u c u for all u ker(T λ). k − k ≥ k k ⊥ − The following theorem gives a description of the essential spectrum using approxi- mating sequences.

Theorem 7.22 (Weyl criterion). The condition λ specess T is equivalent to the existence of a sequence (u ) dom T satisfying the following∈ three properties: n ⊂ 1. u 1, k nk ≥ 56 2. un converge weakly to 0, 3. (T λ)u converge to 0 in the norm of . − n H Such a sequence will be called a singular Weyl sequence for λ. Moreover, as will be shown in the proof, one can replace the conditions (1) and (2) just by:

1’. un form an orthonormal sequence. Proof. Denote by W (T ) the set of all real numbers λ for which one can ﬁnd a singular Weyl sequence.

Show ﬁrst the inclusion W (T ) specess T . Let λ W (T ) and let (un) be an associated singular Weyl sequence,⊂ then we have at least∈ λ spec T . Assume by ∈ contradiction that λ specdisc T and denote by Π the orthogonal projection to ker(T λ) in . As∈ Π is a ﬁnite-rank operator, it is compact, and the sequence Πu converge− toH 0. Therefore, the norms of the vectors w := (1 Π)u satisfy n n − n wn 1/2 for large n. On the other hand, the vectors (T λ)wn = (1 Π)(T λ)un convergek k ≥ to 0, which contradicts to Proposition 7.21. − − −

Conversely, if λ specess T , then dimran ET (λ ε,λ + ε) = for all ε > 0. In particular, one∈ can ﬁnd a strictly decreasing to− 0 sequence (ε∞) with E (I n T n I ) = 0, where I := (λ ε ,λ + ε ). Now we can choose u with u =\ 1 n+1 6 n − n n n k nk and ET (In In+1)un = un. These vectors form an orthonormal sequence and, in particular, converge\ weakly to 0. On the other hand, (T λ)u = (T λ)E (I I )u ε u = ε , k − nk k − T n \ n+1 nk ≤ nk nk n which shows that the vectors (T λ)un converge to 0. Therefore, (un) is a singular Weyl sequence, and spec T W−(T ). ess ⊂ The following theorem provides a starting point to the study of perturbations of self-adjoint operators. Theorem 7.23 (Stability of essential spectrum). Let A and B be self-adjoint operators such that for some z res A res B the diﬀerence of their resolvents 1 1 ∈ ∩ K(z):=(A z)− (B z)− is a compact operator, then spec A = spec B. − − − ess ess Proof. One can easily see, using the resolvent identities (Proposition 3.4), that K(z) is compact for all z res A res B. ∈ ∩ Let λ specess A and let (un) be an associated singular Weyl sequence. Without loss of∈ generality assume that u = 1 for all n. We have k nk 1 1 1 1 lim (A z)− u = lim (A z)− (A λ)u = 0. (7.5) − − λ z n z λ − − n − − On the other hand, as K(z) is compact, the sequence K(z)un converges to 0 with respect to the norm, and

1 1 1 1 lim (B λ)(B z)− u = lim (B z)− u z λ − − n − − λ z n − − 1 1 = lim (A z)− u lim K(z)u = 0. − − λ z n − n − 57 1 Now denote v := (B z)− u . The preceding equality shows that (B λ)v converge n − n − n to 0, and one can easily show that vn converge weakly to 0. It follows again from 1 (7.5) and from the compactness of K(z) that lim vn = λ z − . Therefore, (vn) is a singular Weyl sequence for B and λ, and λ k speck B| .− So| we have shown the ∈ inclusion specess A specess B. As the participation of A and B is symmetric, we have also spec A ⊂ spec B. ess ⊃ ess Let us describe a class of perturbations which can be studied using the preceding theorem. Deﬁnition 7.24 (Relatively compact operators). Let A be a self-adjoint operator in a Hilbert space , and let B a closable linear operator in with dom A dom B. We say thatHB is compact with respect to A (or simply A-compactH ) ⊂ 1 if B(A z)− is compact for at least one z res A. (It follows from the resolvent identitites− that this holds then for all z res∈A. ∈ Proposition 7.25. Let B be A-compact, then B inﬁnitesimally small with respect to A.

Proof. We show ﬁrst that

1 lim B(A iλ)− = 0 (7.6) λ + → ∞ −

Assume that (7.6) is false. Then one can ﬁnd a constant α> 0, non-zero vectors un 1 and a positive sequence λn with lim λn such that B(A iλ)− un > α un for all 1 2 −2 2 k 2 k 2 n. Set vn := (A iλ)− vn. Using un = (A iλn)vn = Avn + λn vn we obtain − k k − k k k k

Bv 2 > α2 Av 2 + α2λ2 v 2. k nk k nk nk nk Without loss of generality one may assume the normalization Bvn = 1, then the sequence Av is bounded and v converge to 0. Let z res Ak, thenk (A z)v is n n ∈ − n also bounded, one can extract a weakly convergent subsequence (A z)vnk . Due 1 − to the compactness, the vectors B(A z)− (A z)v = Bv converge to some − · − nk nk w with w = 1. On the other hand, as shown above, vnk converge to 0, and the∈ closability H k ofk B shows that w = 0. This contradiction shows that (7.6) is true. 1 Now, for any a > 0 one can ﬁnd λ > 0 such that B(A iλ)− u a u for all 1 k − 1 k ≤ k k u . Denoting v := (A iλ)− u and noting that (A iλ)− is a bijection between ∈Hand dom A we see that− − H Bv a (A iλ)v a Av + aλ v k k ≤ k − k ≤ k k k k for all v dom A. As a> 0 is arbitrary, we get the result. ∈ So a combination of the preceding assertions leads us to the following observation: Theorem 7.26 (Relatively compact perturbations). Let A be a self-adjoint operator in a Hilbert space and let B be symmetric and A-compact, then the operator A + B with dom(A +HB) = dom A is self-adjoint, and the essential spectra of A and A + B coincide.

58 Proof. The self-adjointness of A + B follows from the Kato-Rellich theorem, and it remains to show that the diﬀerence of the resolvents of A + B and A is compact. This follows directly from the obvious identity

1 1 1 1 (A z)− (A + B z)− =(A + B z)− B(A z)− , − − − − − which holds at least for all z / R. ∈ As an exercise one can show the following assertion, which can be useful in some situations.

Proposition 7.27. Let A be self-adjoint, B be symmetric and A-bounded with a relative bound < 1, and C be A-compact, then C is also (A + B) compact.

7.5 Essential spectra for Schr¨odinger operators

Deﬁnition 7.28 (Kato class potential). We say that a measurable function V : Rd R belongs to the Kato class if for any ε > 0 one can ﬁnd real-valued p →d d Vε L (R ) and V ,ε L∞(R ) such that Vε + V ,ε = V and V ,ε < ε. Here ∞ ∞ ∞ ∞ p =∈ 2 for d 3 and p>d/∈ 2 for d 4. k k ≤ ≥ Theorem 7.29. If V is a Kato class potential in Rd, then V is compact with respect to the free Laplacian T = ∆ in L2(Rd), and the essential spectrum of ∆+ V is equal to [0, ). − − ∞ Proof. We give the proof for d 3 only. Let denote the Fourier transform, then ≤ F for any f L2(Rd) and z res T we have ∈ ∈ 1 2 1 (T z)− f (p)=(p z)− f(p). F − − F 1 2 This means that (T z)− f = gz ⋆ f, where gz is the L function with gz(p) = 2 1 − F (p z)− , and ⋆ stands for the convolution product. In other words, −

1 (T z)− f = gz(x y)f(y)dy. − Rd − Z 1 Let ε> 0 and let Vε and V ,ε be as in Deﬁnition 7.28. The operator Vε(T z)− is ∞ an integral one with the integral kernel K(x, y)= V (x)g (x y), i.e. − ε z −

1 Vε(T z)− f(x)= K(x, y)f(y) dy. − Rd Z One has

2 2 2 K(x, y) dxdy = Vε(x) dx gz(y) dy Rd Rd Rd Rd Z Z Z Z 2 2 = Vε gz < , k 2k ∞

59 1 which means that Vε(T z)− is a Hilbert-Schmidt operator and, therefore, is compact, see Subsection 4.1.− At the same time we have the estimate

1 1 V ,ε(T z)− ε (T z)− . k ∞ − k ≤ k − k 1 Therefore, the operator V (T z)− is compact as it can be represented as the norm − 1 limit of the compact operators V (T z)− as ε tends to 0. ε − Example 7.30 (Coulomb potential). The previous theorem easily applies e.g. to the operators ∆+ α/ x . It is suﬃcient to represent − | | 1 χ (x) 1 χ (x) = R + − R , x x x | | | | | | where χR is the characteristic function of the ball of radius R > 0 and centered at the origin with a suﬃciently large R. So the essential spectrum of ∆+ α/ x is − | | always the same as for the free Laplacian, i.e. [0, + ). ∞ Another typical application of the Weyl criterion can be illustrated as follows.

d Theorem 7.31. Let V L∞(R ) be real-valued. Assume that there exists α R such that the set Ω := ∈x Rd : V (x) < α has a ﬁnite Lebesgue measure, then∈ ∆+ V has a purely discrete∈ spectrum in ( , α). − −∞

Proof. Let χΩ be the characteristic function of Ω. Denote U := (V α)χΩ and W := V U. Then U Lp(Rd) for any p 1 (as U is bounded and− supported by a set− of ﬁnite measure),∈ in particular, U≥is of Kato class. At the same time, d W L∞(R ) and W α. By Proposition 7.27, U is ( ∆+ W )-compact, ∈ ≥ − spec ( ∆+ V ) ( , α) = spec ( ∆+ W + U) ( , α) ess − ∩ −∞ ess − ∩ −∞ = spec ( ∆+ W ) ( , α). ess − ∩ −∞ On the other hand, spec ( ∆+ W ) spec( ∆+ W ) [α, + ). ess − ⊂ − ⊂ ∞ Note that we have no troubles with the domains, as all the operators ∆+ V , ( ∆+ W )+ U and ∆+ W are deﬁned on the same domain H2(Rd) due− to the − − boundedness of the potentials.

Remark 7.32. In the physics literature, the situation of Theorem 7.31 is sometimes referred to as a potential well below α. The same result holds without assumptions on V outside Ω (i.e. for unbounded potentials), but the proof would then require a slightly diﬀerent machinery.

7.6 Perturbations of discrete spectra

In this section we discuss rather brieﬂy the behavior of the discrete spectrum of self-adjoint operators under small perturbations. Let A be a self-adjoint operator in a Hilbert space , and let B , ε R, be a family of symmetric operators with H ε ∈ 60 1 dom A dom Bε such that for some z0 res A the operators Bε(A z0)− are bounded⊂ and continuously depend on ε and∈ that −

1 lim Bε(A z0)− = 0 (7.7) ε 0 → − (and then the same holds for any z res A). It follows from the Kato-Rellich 0 ∈ theorem that for suﬃciently small ε the operators Aε := A + Bε are self-adjoint on the domain dom Aε = dom A. A canonical example is Bε = εB, where B is symmetric and A-bounded.

Proposition 7.33. Let K res A be compact, then K res Aε for sufficiently small ε, and ⊂ ⊂ 1 1 lim sup (Aε z)− (A z)− = 0. (7.8) ε 0 z K − − − → ∈ 1 Proof. Denote Cε := Bε(A z0)− with some z0 res A. There holds, for any z res A, − ∈ ∈ A z = A + B z = A + C (A z ) z ε − ε − ε − 0 − 1 = 1+ C (A z )(A z)− (A z) ε − 0 − − (7.9) h i 1 = 1+ C +(z z )C (A z)− (A z) ε − 0 ε − − h i1 Due to the spectral theorem, for any z K we have (A z)− c with c := ∈ − ≤1 1/ dist(K, spec A), and one find d> 0 such that 1+(z z )(A z)− d for all − 0 − ≤ z K. Due to the assumption (7.7), one can find ε > 0 such that for ε ( ε ,ε ) ∈ 0 ∈ − 0 0 we have then

1 dε := sup Dε < 1,Dε := Cε +(z z0)Cε(A z)− , z K k k − − ∈ and it follows from (7.9) that, for the same ε, the operator Aε z has a bounded inverse for all z K. Moreover, Eq. (7.9) shows that for ε ( −ε ,ε ) one has ∈ ∈ − 0 0 ∞ 1 1 1 1 1 1 j (A z)− (A z)− =(A z)− 1+D )− (A z)− =(A z)− ( D ) , ε − − − − ε − − − − ε j=1 X and for z K one obtains ∈ ∞ 1 1 j cdε (A z)− (A z)− c d = . ε − − − ≤ ε 1 d j=1 ε X −

Noting that limε 0 dε = 0 we obtain (7.8). →

Theorem 7.34 (Perturbations of discrete eigenvalues). Let λ specdisc A and m := dim ker(A λ). Then for any δ > 0 such that [λ δ, λ + δ] ∈ spec A = λ − − ∩ { } there exists εδ > 0 such that dim ran E (λ δ, λ + δ) = m Aε − for all ε ( ε ,ε ). ∈ − δ δ 61 Proof. Let Γ be the complex circle of radius δ centered at λ. By Proposition 7.33, for suﬃciently small ε we have Γ res A . Proposition 6.8 gives the representation ⊂ ε

1 1 P := E (λ δ, λ + δ) = (A z)− dz. ε Aε − 2πi ε − IΓ Using the constructions similar to those in the proof of proposition 7.33 one can show that Pε depends continuously on ε if ε is close to 0. At the same time, Eq. (7.8) shows that P0 = limε 0 Pε. Denote now Qε := 1 Pε and consider the operators → −

Sε := QεQ0 + PεP0, Tε := Q0Qε + P0Pε.

It is clear that these two operators continuously depend on ε near ε = 0. As S0 = T0 = 1, both Sε and Tε are invertible for small ε. On the other hand, Sε maps ran EA (λ δ, λ + δ) ker(A λ) to ran EAε (λ δ, λ + δ) , and Tε maps ran E (λ δ, λ−+δ) to ker(≡A λ). As− both S and T are− bijective for small ε, the Aε ε ε subspaces ran− E (λ δ, λ + δ−) and ker(A λ) have the same dimension m. Aε − − Informally speaking, the preceding theorem shows that under small perturbations an isolated eigenvalue of multiplicity m splits into exactly m eigenvalues (counting multiplicities), but no additional parts of the spectrum can appear. If the perturbation is analytic with respect to ε, then one can show that the suitably numbered perturbed eigenvalues are also analytic functions of ε. We refer to the fundamental monograph [10] providing a detailed study of related questions.

62 8 Variational principle for eigenvalues

8.1 Max-min and min-max principles

Throughout the subsection we denote by T a self-adjoint operator in an inﬁnite- dimensional Hilbert space , and we assume that T is semibounded from below. If spec T = , we denote ΣH := + , otherwise we put Σ := inf spec T . ess ∅ ∞ ess Theorem 8.1 (Max-min principle). For n N introduce the following numbers: ∈ ϕ, T ϕ µn = µn(T ) = sup inf h i, ψ1,...,ψn−1 ϕ dom T, ϕ=0 ϕ, ϕ ∈H ϕ ψ∈j ,j=1,...,n6 1 ⊥ − h i then we are in one and only one of the following situations:

(a) µn is the nth eigenvalue of T (when ordering all the eigenvalues in the non- decreasing order with their multiplicities), and T has a purely discrete spectrum in ( ,µ ). −∞ n (b) µ = Σ, and µ = µ for all j n. n j n ≥ Proof. Step 1. Let us prove ﬁrst two preliminary assertions:

dim ran E ( ,a) < n for a<µ , (8.1) T −∞ n dim ran E ( ,a) n for a>µ . (8.2) T −∞ ≥ n

Proof of (8.1). Assume that the assertion is false, then dim ran ET ( ,a) n for some a<µ , and one can find an n dimensional subspace V ran−∞E ( ≥ ,a) . n T As T is semibounded, V dom T .− By dimension considerations,⊂ for any−∞ vectors ⊂ ψ1,...,ψn 1 there exists a non-zero vector ϕ V orthogonal to all ψj, j = 1,...,n − ∈ − 1, and the inclusion ϕ ran ET ( ,a) implies ϕ, T ϕ a ϕ, ϕ . Therefore, for any choice of ψ the infimum∈ in the−∞ definition of µ his not greateri ≤ h thani a, which gives j n the inequality µ a, which contradicts to the assumption. Eq. (8.1) is proved. n ≤ Proof of (8.2). Again, assume by contradiction that the assertion is false, then for some a>µn we have dim ran ET ( ,a) n 1. Let ψ1,...,ψn 1 be some vectors − spanning ran E ( ,a) . Due to−∞ the equality≤ −E ( ,a) + E [a, + ) = Id, T T T for every ϕ dom−∞T with ϕ ψ , j = 1,...,n 1, one−∞ has ϕ = E [a, +∞) ϕ and j T ϕ, T ϕ a∈ϕ, ϕ , which shows⊥ that µ a. This− contradiction proves (8.2).∞ h i ≥ h i n ≥ Step 2. Let us prove that µn < + for any n (note that the equality µn > follows from the semiboundedness∞ of T ). Assume that µ = + , then, by (8.1),−∞ n ∞ one has dim ran ET ( ,a) < n for any a R, and dim n, which contradicts to the assumption. −∞ ∈ H ≤ Now we have two possibilities: either dim ran ET ( ,µn + ε) = for all ε > 0 or dim ran E ( ,µ + ε) < for some ε> 0.−∞ Let us consider them∞ separately. T −∞ n ∞ Step 3. Assume that dim ran ET ( ,µn + ε) = for all ε > 0. We are going to show that the case (b) of the−∞ theorem is realized.∞ Due to (8.1), one has 63 dim ran ET (µn ε,µn + ε) = for all ε > 0, and µn specess T . On the other hand, again by (8.1),− spec T ∞( ,µ ε)= for all∈ε> 0, which proves that ess ∩ −∞ n − ∅ µn = Σ. It remains to show that µn+1 = µn. Assume that µn+1 >µn, then, by (8.1), for any ε<µn+1 µn we have dim ran ET ( ,µn + ε) n + 1, which contradicts to the assumption.− Therefore, µ = µ . −∞ ≤ n+1 n Step 4. Assume now that dim ran ET ( ,µn + ε) < for some ε > 0. It follows directly that the spectrum of T is−∞ purely discrete in∞ ( ,µ + ε). More- −∞ n over, one can find ε1 > 0 such that ET ( ,µn] = ET ( ,µn + ε1) . As dim ran E ( ,µ +ε ) n by (8.1), we have−∞ dim ran E ( −∞,µ ] n, which T n 1 T n means that T−∞has at least n≥eigenvalues λ λ (counting−∞ with multiplicities)≥ 1 ≤···≤ n in ( ,µn]. If λn < µn, then dimran ET ,λn] n, which contradicts to (8.1).−∞ This proves the equality µ = λ . − ∞ ≥ n n Remark 8.2. If T is defined by a closed sesquilinear form t, then one may replace the above definition of the numbers µn by t(ϕ, ϕ) µn = sup inf . ψ1,...,ψn−1 ϕ dom t, ϕ=0 ϕ, ϕ ∈H ϕ ψ∈j ,j=1,...,n6 1 ⊥ − h i This follows from the fact that dom T is dense in dom t is the topology defined by t, see Theorem 2.6 and the subsequent discussion. Another elementary observation is given in the following corollary. Corollary 8.3. If there exists ϕ dom t with t(ϕ, ϕ) < Σ ϕ 2, then T has at least one of the eigenvalue in ( , Σ)∈. k k −∞

Indeed, in this case one has µ1 < Σ, which means that µ1 is an eigenvalue.

By similar considerations one can obtain another variational formula for the eigenvalues, one may refer to Section 4.5 in [4] for its proof: Theorem 8.4 (Min-max principle). All the assertions of Theorem 8.1 hold with

ϕ, T ϕ t(ϕ, ϕ) µn := inf sup h i = inf sup . L dom T L dom t ⊂ ϕ L ϕ, ϕ ⊂ ϕ L ϕ, ϕ dim L=n ϕ∈=0 h i dim L=n ϕ∈=0 h i 6 6 The max-min and min-max principles are powerful tools for the analysis of the behavior of the eigenvalues with respect to various parameters. As a basic example we mention the following situation, which will be applied later to some speciﬁc operators: Corollary 8.5. Let a and b be two closed symmetric sesqulinear forms, both semibounded from below, in a Hilbert space with the following two properties: H dom b dom a, • ⊂ a(u, u) b(u, u) for all u dom a. • ≤ ∈ 64 Let A and B be the associated with a and b self-adjoint operators in , and assume that they both are compact. If λ (A) and λ (B), j N, denote theirH eigenvalues j j ∈ taken with their multiplicities and enumerated in the non-decreasing order, then λ (A) λ (B) for all j N. j ≤ j ∈

8.2 Negative eigenvalues of Schr¨odinger operators

As seen above in Proposition 7.27, if V is a Kato class potential in Rd, then the associated Schr¨odinger operator T = ∆+ V acting in = L2(Rd) has the same − H essential spectrum as the free Laplacian, i.e. specess T = [0, + ) and Σ=0. In the present section we would like to discuss the question on the∞ existence of negative eigenvalues. We have rather a simple suﬃcient condition for the one- and two-dimensional cases.

d 1 d Theorem 8.6. Let d 1, 2 and V L∞(R ) L (R ) be real-valued such that ∈ { } ∈ ∩

V0 := V (x)dx < 0, Rd Z then the associated Schr¨odinger operator T = ∆+ V has at least one negative eigenvalue. −

Proof. We assumed the boundedness of the potential just to avoid additional tech- 2 Rd nical issues concerning the domains. It is clear that V L ( ), and specess T = [0, + ) in virtue of Theorem 7.29. By Corollary 8.3 it is∈ now suﬃcient to show that one can∞ ﬁnd a non-zero ϕ H1(Rd) with ∈ 2 2 τ(ϕ) := ϕ(x) dx + V (x) ϕ(x) dx < 0. Rd ∇ Rd Z Z

Consider ﬁrst the case d = 1. For ε > 0 and consider the function ϕε given by ε x /2 1 ϕ (x) := e− | | . Clearly, ϕ H (R) for any ε > 0, and the direct computation ε ε ∈ shows that

2 ε 2 ϕε′ (x) dx = and lim V (x) ϕε(x) dx = V0 < 0. R 2 ε 0+ Rd Z → Z

Therefore, for suﬃciently small ε one obtains τ(ϕε) < 0. x ε/2 Now let d = 2. Take ε> 0 and consider ϕε(x) deﬁned by ϕε(x)= e−| | . We have

65 and, as previously, 2 lim V (x) ϕε(x) dx = V0 < 0, ε 0+ Rd → Z and for suﬃciently small ε we have again τ(ϕ ε) < 0.

We see already in the above proof that ﬁnding suitable test functions for proving the existence of eigenvalues may become very tricky and depending on various parameters. One may easily check that the analog of ϕε for d = 1 does not work for d = 2 and vice versa. It is a remarkable fact that the analog of Theorem 8.6 does not hold for the higher dimensions. This diﬀerence is explained by the following classical inequality.

d Proposition 8.7 (Hardy inequality). Let d 3 and u C∞(R ), then ≥ ∈ c 2 2 2 (d 2) u(x) u(x) dx − 2 dx. Rd ∇ ≥ 4 Rd x Z Z | | Proof. For any γ R one has ∈ xu(x) 2 u(x)+ γ 2 dx dx 0, Rd ∇ x ≥ Z | | which may be rewritten in the form

u(x) u(x) 2 x x u(x) 2 + x u(x) 2 dx = u(x) 2 dx Rd · ∇ x · ∇ x Rd ∇ · x Z | | | | Z | | 2 2 x u(x) = u(x) div 2 dx = (d 2) 2 dx. − Rd x − − Rd x Z Z | | | | Inserting this equality into (8.3) gives

2 2 u(x) u(x) dx γ (d 2) γ 2 dx, Rd ∇ ≥ − − Rd x Z Z | | and in order to optimize the coeﬃcient before the integral on the right-hand side we take γ =(d 2)/2. −

66 Note that the integral in the right-hand side of the Hardy inequality is not deﬁned 1 2 for d 2, because the function x x − does not belong to L anymore. ≤ 7→ | | loc Proposition 8.8. Let d 3 and let V : Rd R be bounded with a compact support. ≥ → For λ R consider the Schr¨odinger operators Tλ := ∆+ λV , then there exists λ > 0∈such that spec T = [0, + ) for all λ ( λ , +− ). 0 λ ∞ ∈ − 0 ∞

Proof. Due to the compactness of supp V one can ﬁnd λ0 > 0 in such a way that

(d 2)2 λ V (x) − for all x Rd. 0 ≤ 4 x 2 ∈ | | d Using the Hardy inequality, for any u C∞(R ) and any λ ( λ , + ) we have ∈ c ∈ − 0 ∞

2 2 u, Tλu = u(x) dx + λ V (x) u(x) dx h i Rd ∇ Rd Z Z 2 2 u(x) dx λ 0 V (x) u(x) dx ≥ Rd ∇ − Rd · Z Z 2 2 2 ( d 2) u(x) u(x) dx − 2 dx 0. ≥ Rd ∇ − 4 Rd x ≥ Z Z | | Rd As Tλ is essentially self-adjoint on Cc∞( ), see Theorem 7.10, this inequality extends to all u dom Tλ, and we obtain Tλ 0, and this means that spec Tλ [0, + ). On the other∈ hand, spec T = [0, + )≥ as λV is of Kato class (see Theorem⊂ 7.29).∞ ess λ ∞

67 9 Laplacian eigenvalues for bounded domains

9.1 Dirichlet and Neumann eigenvalues

In this section we discuss some application of the general spectral theory to the eigenvalues of the Dirichlet and Neumann Laplacians in bounded domains. Let us recall the setting. Let Ω Rd be a bounded open set with a regular boundary (for example, piecewise smooth∈ and lipschitzian); all the domains appearing in this section will be supposed to have a regular boundary without further speciﬁcations. Then the embedding of H1(Ω) into := L2(Ω) is a compact operator. By deﬁnition, the Dirichlet Laplacian T = ∆H and the Neumann Laplacian T = ∆ are D − D N − N the self-adjoint operators in associated with the sesqulinear forms tD and tN respectively, H

t (u, v)= u(x) v(x) dx, dom t = H1(Ω), D ∇ · ∇ D 0 ZΩ t (u, v)= u(x) v(x) dx, dom t = H1(Ω). N ∇ · ∇ N ZΩ

We know that both TD and TN have compact resolvent, and their spectra are purely discrete (see Section 4.2). Denote by λD/N = λD/N (Ω), j N, the eigenvalues of j j ∈ TD/N repeated according to their multiplicities and enumerated in the non-decreasing order. The eigenvalues are clearly non-negative, and they are usually referred to as the Dirichlet/Neumann eigenvalues of the domain Ω (the presence of the Laplacian is assumed implicitly). Let us discuss some basic properties of these eigenvalues.

N Proposition 9.1. (a) λ1 = 0. If Ω is connected, then ker TN is spanned by the constant function u(x) = 1.

D (b) λ1 > 0.

Proof. (a) Note that u = 1 is clearly an eigenfunction of TN with the eigenvalue 0. As all the eigenvalues are non-negative, λN = 0. Now let u ker T , then 1 ∈ N 2 0= u, T u = t (u, u)= u(x) dx, h N i N ∇ ZΩ which shows that u = 0. Therefore, v is constant on each maximal connected component of Ω. ∇ D D (b) We have at least λ1 0. Assume that λ1 = 0 and let v be an associated eigenfunction. We have as≥ above v = 0, so v must be constant on each maximal connected component of Ω. But∇ the restriction of v to the boundary of Ω must vanish, which gives v = 0.

A direct application of Corollary 8.5 gives

Proposition 9.2. For any j N one has λN (Ω) λD(Ω). ∈ j ≤ j

68 Another important aspect is the dependence of the eigenvalues on the domain.

Proposition 9.3 (Monotonicity with respect to domain, Dirichlet case). Let Ω Ω, then λD(Ω) λD(Ω) for all n N. ⊂ n ≤ n ∈ 1 Proof. Wee observe ﬁrste that if f H0 (Ω), then its extension f to Ω by zero belongs 1 ∈ to H0 (Ω). This allows one to write the following chain of equalities and inequalities: e e e f 2 D k∇ kL2(Ω) λn (Ω) = sup inf 2 2 f H1(Ω),f=0 f ψ1,...,ψn−1 L (Ω) ∈ 0 6 L2(Ω)e ∈ f,ψj =0 k k h iL2(Ω) e e e e e 2 2 f 2 f 2 sup inf k∇ kL (Ω) = sup inf k∇ kL (Ω) 1 2 1 2 ≤ − 2 f H0 (Ω),f=0 f − 2 f H0 (Ω),f=0 f 2 ψ1,...,ψn 1 L (Ω) ∈ 6 L2(Ω)e ψ1,...,ψn 1 L (Ω) ∈ 6 L (Ω) ∈ f,ψj =0 k ke ∈ f,ψj L2(Ω)=0 k k h iL2(Ω) h i e e e 2 e e e f 2 k∇ kL (Ω) D = sup inf 2 = λn (Ω). 2 f H1(Ω),f=0 f ψ1,...,ψn−1 L (Ω) ∈ 0 6 L2(Ω) ∈ f,ψj 2 =0 k k h iL (Ω)

Note that there is no easy generalization of this result to the Neumann case. The reason can be understood at a certain abstract level. As can be seen from the proof, 1 1 for Ω Ω there exists an obvious embedding τ : H0 (Ω) H0 (Ω) (extension by zero) such⊂ that τu = u for all u H1(Ω). If one replaces→ the spaces H1 by k k k k ∈ 0 0 H1, thene the existence of a bounded embedding and the estimatese for its norm in terms of the two domains become non-trivial. We mention at least one important case where a kind of the monotonicity can be proved.

Proposition 9.4 (Neumann eigenvalues of composed domains). Let Ω Rd ⊂ be a bounded open domain with a regular boundary, and let Ω1 and Ω2 be non- intersecting open subsets of Ω with regular boundaries such that Ω= Ω1 Ω2, then λN (Ω Ω ) λN (Ω) for any j N. ∪ n 1 ∪ 2 ≤ j ∈ 1 1 Proof. Under the assumptions made, any function f H (Ω) belongs to H (Ω1 Ω ), while the spaces L2(Ω) and L2(Ω Ω ) coincide,∈ and we have ∪ 2 1 ∪ 2 2 f 2 N L (Ω1 Ω2) λ (Ω1 Ω2) = sup inf k∇ k ∪ n 1 2 ∪ ψ ,...,ψ − L2(Ω Ω ) f H (Ω1 Ω2),f=0 f 2 1 n 1 1 2 ∈ ∪ 6 L (Ω1 Ω2) ∈ ∪ f,ψj 2 ∪ =0 k k ∪ h iL (Ω1 Ω2) 2 f 2 L (Ω1 Ω2) sup inf k∇ k ∪ 1 2 ≤ ψ ,...,ψ − L2(Ω Ω ) f H (Ω),f=0 f 2 1 n 1 1 2 ∈ 6 L (Ω1 Ω2) ∈ ∪ f,ψj 2 ∪ =0 k k ∪ h iL (Ω1 Ω2) 2 f 2 k∇ kL (Ω) N = sup inf 2 = λn (Ω). 2 f H1(Ω),f=0 f ψ1,...,ψn−1 L (Ω) ∈ 6 L2(Ω) ∈ f,ψj 2 =0 k k h iL (Ω)

69 Remark 9.5. Under the assumptions of proposition 9.4 for any n N we have λD(Ω) λD(Ω Ω ), which follows from the inclusion Ω Ω Ω. Therefore,∈ for n ≤ n 1 ∪ 2 1 ∪ 2 ⊂ any n N one has the chain ∈ λN (Ω Ω ) λN (Ω) λD(Ω) λD(Ω Ω ), 1 ∪ 2 ≤ ≤ ≤ n 1 ∪ 2 and this is the key argument of the so-called Dirichlet-Neumann bracketing which is used e.g. for estimating the asymptotic behavior of the eigenvalues (see below).

Let us mention another useful fact:

Proposition 9.6 (Dependence on domain, Neumann case). Let Ω Ω. As- 1 1 ⊂ sume that there exists a bounded linear map τ : H (Ω) H (Ω) with τu Ω = u. N 2 N N → | Then λj (Ω)+1 τ λj (Ω) + 1 for all j . e ≤ k k ∈ e Proof. Lete us use the min-max principle (Theorem 8.4). Denote C := τ . We have k k

u 2 + u 2 u 2 N k∇ kL2(Ω) k kL2(Ω) k kH1(Ω) λj (Ω) + 1 = inf sup 2 = inf sup 2 L H1(Ω) u L u L H1(Ω) u L u e L2(Ω) e L2(Ω)e dim⊂ L=j u∈=0 k k dim⊂ L=j u∈=0 k k 6 6 e e 2 e 2 u e τu e k kH1(Ω) k kH1(Ω) inf sup 2 = inf sup 2 ≤ L H1(Ω) u L H1(Ω) u L τu ⊂ u τ(L) L2(Ω)e ⊂ L2(Ω)e dim L=j ∈u=0 k k dim L=j u∈=0 k k 6 6 2 e u 1 e C2 inf sup k kH (Ω) = C2 λN (Ω) + 1 . 1 2 j ≤ L H (Ω) u L τu L2(Ω) dim⊂ L=j u∈=0 k k 6 We complete the discussion by proving the continuity of the Dirichlet eigenvalues with respect to domain.

Proposition 9.7 (Continuity with respect to domain, Dirichlet). If Ωj N D D N⊂ Ωj+1 for all j , and Ω= ∞ Ωj, then λn (Ω) = limj λn (Ωj) for any n . ∈ j=1 →∞ ∈

Proof. Let us pick n N, andS let f1,...fn be the mutually orthogonal normalized ∈ D D eigenfunctions associated with the eigenvalues λ1 (Ω),...,λn (Ω). If U denotes the 2 subspace spanned by f1,...,fn, then for any f U one has the estimate f D 2 ∈ k∇ k ≤ λn (Ω) f . k k 1 Now take an arbitrary ε> 0. Using the density of Cc∞(Ω) in H0 (Ω) one can approx- imate every fj by uj Cc∞(Ω) in such a way that u1,...,un will be linearly inde- ∈ 2 D 2 pendent and that u L2(Ω) λn (Ω)+ε u L2(Ω) for all u from the n-dimensional subspace V spannedk∇ byk u ,...,u≤ . Let K k Ωk be a compact subset containing the 1 n ⊂ supports of all uj and, as a consequence, the supports of all functions from V . One can ﬁnd M N such that K Ωm for all m M, and then for all m M we 1∈ ⊂ ≥ 2 ≥ have V H0 (Ωm). Now let ψ1,...,ψn 1 be arbitrary functions from L (Ωm). As V ⊂ −

70 is n-dimensional, there exists a non-zero v V which is orthogonal to all ψj. This means that ∈

2 2 2 u 2 v 2 v 2 inf k∇ kL (Ωm) k∇ kL (Ωm) = k∇ kL (Ω) λD(Ω) + ε. 1 2 2 2 n u H (Ωm),u=0 u 2 ≤ v 2 v 2 ≤ ∈ 0 6 L (Ωm) L (Ωm) L (Ω) u ψ1,...ψn−1 k k k k k k ⊥ D D Due to the arbitrariness of ψj we have λn (Ωm) λn (Ω)+ ε for all m M. On the other hand, λD(Ω) λD(Ω ) by monotonicity.≤ ≥ n ≤ n m

9.2 Weyl asymptotics

In this subsection we will discuss some aspects of the asymptotic behavior of the Laplacian eigenvalues. We introduce the Dirichlet/Neumann counting functions ND/N (λ, Ω) by

N (λ, Ω) = the number of j N for which λD/N (Ω) ( ,λ]. D/N ∈ j ∈ −∞

Clearly, ND/N (λ, Ω) is ﬁnite for any λ, and it has a jump at each eigenvalue; the jump is equal to the multiplicity. We emphasize the following obvious properties:

N (λ, Ω) N (λ, Ω) (9.1) D ≤ N N D/N (λ, Ω Ω )= N D/N (λ, Ω )+ N D/N (λ, Ω ) for Ω Ω= . (9.2) 1 ∪ 2 1 2 1 ∩ ∅ N (λ, Ω) N (Ω) for Ω Ω. (9.3) D ≤ D ⊂ We are going to discuss the following rathere general resulte on the behavior of the counting functions N and N as λ + : D N → ∞ Theorem 9.8 (Weyl asymptotics). For ⋆ D,N we have ∈ { } N (λ, Ω) ω lim ⋆ = d vol(Ω), λ + d/2 d → ∞ λ (2π)

d where ωd denotes the volume of the unit ball in R .

To keep simple notation we proceed with the proof for the case d = 2 only. Due to ω2 = π we are reduced to prove N (λ, Ω) area(Ω) lim ⋆ = , ⋆ D,N . (9.4) λ + → ∞ λ 4π ∈ { } The proof consists of several steps.

Lemma 9.9. The Weyl asymptotics is valid for rectangles.

71 Proof. Let Ω = (0,a) (0, b), a,b > 0. As shown in Example 6.16, the Neumann eigenvalues of Ω are the× numbers πm 2 πn 2 λ(m, n) := + a b with m, n N0 := 0 N, and the Dirichlet spectrum consists of the eigenvalues λ(m, n) with∈ m, n {N}∪. Denote ∈ x2 y2 λ D(λ) := (x, y) R2 : + , x 0, y 0 , ∈ a2 b2 ≤ π2 ≥ ≥ n o then ND(λ, Ω)=#D(λ) N N and NN (λ, Ω)=#D(λ) N0 N0 , where # denotes the cardinality. ∩ × ∩ × First, counting the points (n, 0) and (0, n) with n N0 inside D(λ) we obtain the majoration ∈ a + b N (λ) N (λ) + 2 √λ, λ> 0. N − D ≤ π At the same time, D(λ) contains the union of the unit cubes [m 1,m] [n 1, n] with (m, n) D(λ) N N . As there are exactly N (λ, Ω) such− cubes,× we− have ∈ ∩ × D λab N (λ, Ω) area D(λ)= . D ≤ 4π We also observe that D(λ) is contained in the union of the unit cubes [m,m + 1] [n, n + 1] with (m, n) D(λ) N N . As the number of such cubes is exactly× ∈ ∩ 0 × 0 NN (λ, Ω), this gives λab N (λ, Ω) area D(λ)= . N ≥ 4π Putting all together we arrive at λab a + b λab a + b N (λ, Ω) N D(λ, Ω) + + 2 √λ + + 2 √λ, 4π ≤ N ≤ π ≤ 4π π and it remains to recall that area(Ω) = ab.

Deﬁnition 9.10 (Domains composed from rectangles). We say that a domain Ω with a regular boundary is composed from rectangles if there exists a ﬁnite family of non-intersecting open rectangles Ωj, j = 1,...,k, with

Ω= Ωj. j=1 [ Lemma 9.11. The Weyl asymptotics holds for domains composed from rectangles.

Proof. Let Ω be a domain composed from rectangles, ant let Ωj, j = 1,...,k, be the rectangles as in Deﬁnition 9.10. Using Remark 9.5 and the equality (9.2) we

72 obtain the chain N (λ, Ω )+ + N (λ, Ω ) N (λ, Ω Ω ) N (λ, Ω) N 1 ··· N k = N 1 ∪···∪ k N λ λ ≤ λ N (λ, Ω) N (λ, Ω Ω ) N (λ, Ω )+ + N (λ, Ω ) D D 1 ∪···∪ k = D 1 ··· D k , ≤ λ ≤ λ λ and the result is obtained by applying Lemma 9.9 to the quotients ND/N (λ, Ωj)/λ and by noting that area(Ω) = area(Ω )+ + area(Ω ). 1 ··· k Proof of the Weyl asymptotics, Dirichlet case. Let us show that the Weyl asymptotics holds for the Dirichlet counting function. Let Ω be a domain with a regular boundary. It is a standard result of the analysis that for any ε> 0 one can ﬁnd two domains Ωε and Ωε such that: both Ω and Ω are composed from rectangles, • ε ε e Ω Ω Ω , • ε ⊂ ⊂ ε e area(Ω Ω ) <ε. • ε \ eε Using (9.1)e and the monotonicity of the Dirichlet eigenvalues with respect to domain we have: N (λ, Ω ) N (λ, Ω) N (λ, Ω ) N (λ, Ω ) D ε D D ε N ε . λ ≤ λ ≤ λ ≤ λ By Lemma 9.11, we can ﬁnd λε > 0 such that e e area(Ω ) ε N (λ, Ω) area(Ω )+ ε ε − D ε for λ>λ . 4π ≤ λ ≤ 4π ε e At the same time, area(Ω ) area(Ω) ε and area(Ω ) area(Ω) + ε, so for λ>λ ε ≥ − ε ≤ ε we have area(Ω) 2ε N (λ, Ω) area(Ω) + 2ε − D e , 4π ≤ λ ≤ 4π which gives the sought result.

The proof for the Neumann case is much more involved due to the absence of monotonicity with respect to domain. We provide just a sketch to show to overcome this diﬃculty and to illustrate some relations with other areas of the analysis.

Sketch of the proof for the Neumann case. First of all, due to the minoration N (λ, Ω) N (λ, Ω) we have at least D ≤ N N (λ, Ω) area(Ω) lim inf N . λ + → ∞ λ ≥ 4π To obtain the majoration one can proceed first as for the Dirichlet case. For ε> 0 choose Ωε and Ωε as in the proof for Dirichlet case. We have at least the inequalities N (λ, Ω) N (λ, Ω ) N (λ, Ω Ω ) e N N ε + N \ ε . (9.5) λ ≤ λ λ 73 Note that U := Ωε Ωε is a domain composed from rectangles. So we can find open non-intersecting rectangles\ U ,...,U with U = U U , and we have 1 k 1 ∪···∪ k e N (λ, Ω Ω ) N (λ, U Ω) + N (λ, U Ω). N \ ε ≤ N 1 ∩ N k ∩ The main technical ingredient, which is not presented here, is to show that for a 1 suitable choice of Ωε, Ωε and Uj there exist a bounded extension maps τj : H (Uj 1 ∩ Ω) H (Uj) with τju Uj Ω = u whose norms admit a uniform bound τj C for → | ∩ k k ≤ sufficiently small ε. Thee existence of such maps can be proved in a rather direct way by cutting the domain into small pieces of controllable geometry, see e.g. VI.4 in [11]. Now it follows from Proposition 9.6 that one can find a > 0 such§ that for sufficiently large λ one has the estimates NN (λ, Uj Ω) NN (aλ,Uj) for all j = 1,...,k and sufficiently small ε> 0. It follows that ∩ ≤

N (λ, Ω Ω ) N (aλ,U )+ + N (aλ,U ) N \ ε N 1 ··· N k , λ ≤ λ and for large λ we have

N (λ, Ω Ω ) area(U )+ + area(U ) area(Ω Ω) 2aε N \ ε 2a 1 ··· k = 2a ε \ . λ ≤ 4π λ ≤ 4π e Substituting this into (9.5) we arrive at

N (λ, Ω) area(Ω ) 2aε area(Ω) + 2aε lim sup N ε + . λ + λ ≤ 4π 4π ≤ 4π → ∞ As ε> 0 is arbitrary, we have the result.

The Weyl asymptotics is one of the basic results on the relations between the Dirich- let/Neumann eigenvalues and the geometric properties of the domain. It states, in particular, that the spectrum of the domain contains the information on its dimension and its volume. There are various reﬁnements involving lower order terms with respect to λ, and the respective coeﬃcients contains some information on the topology of the domain, on its boundary etc.

9.3 Simplicity of the lowest eigenvalue

In the present section Ω is a connected bounded domain with a regular boundary. Our aim is to complete Proposition 9.1 with the following fundamental fact.

Theorem 9.12 (Smallest eigenvalue of the Dirichlet laplacian). The small- D est Dirichlet eigenvalue λ1 (Ω) is simple, and the associated eigenfunction does not vanish in Ω.

For the proof we need the following rather standard and technical proposition whose proof we omit here (it may be found e.g. in Section 7.2 of [6]).

74 1 + Lemma 9.13. Let u H0 (Ω) be real-valued. Denote u := max(u, 0) and u− := ∈ 1 + max( u, 0), then u± H (Ω) and u u− 0. − ∈ 0 ∇ · ∇ ≤ Lemma 9.14. The Dirichlet Laplacian ∆D is positivity preserving in the following 2 − 1 sense: if f L (Ω) and f 0, then ( ∆ + 1)− f 0. ∈ ≥ − D ≥ 2 1 Proof. We need to show the following: Let u H (Ω) H0 (Ω) such that ∆u+u = f, where f L2(Ω) and f 0, then u 0. ∈ ∩ − ∈ ≥ ≥ As the Laplacian maps real-valued functions to real-valued functions, the function 2 1 u is real-valued: if u = u1 + iu2 with real-valued u1, u2 H (Ω) H0 (Ω), then ∆u + u = ( ∆+1)u + i( ∆+1)u = f, which implies∈ ( ∆+1)∩ u = 0, and − − 1 − 2 − 2 u2 = 0 due to the inclusion spec( ∆D) [0, + ). + − ⊂ ∞ + Now we can deﬁne u := max(u, 0) and u− := max( u, 0), then u± 0, u = u u− + − ≥ − and u , u− = 0. So we have, using Lemma 9.13, h i 0 u−, f = u−, ∆u + u−, u = u−, u + u−, u ≤h i h − i +h i h∇ ∇ i h + i = u−, u u−, u− + u−, u u−, u− h∇ ∇ i − h∇ ∇ i h i−h i u−, u− u−, u− 0, ≤ −h∇ ∇ i−h i ≤ which gives u− = 0.

Proof of theorem 9.12. Let u be an eigenfunction of ∆D for the eigenvalue D − λ = λ1 (Ω); this eigenvalue is strictly positive by Proposition 9.1. Without loss of generality we can assume that u is real-valued (otherwise instead of u we can consider its real or imaginary part). We also observe that u C∞(Ω) due to the elliptic regularity of the Laplacian. ∈ 1 Consider the operator B := ( ∆D + 1)− . We know that B is compact and that − 1 B 0. Moreover, µ := (λ + 1)− B is its maximal eigenvalue, and u is an associated≥ eigenfunction. ≡ k k + Introduce again u := max(u, 0) and u− := max( u, 0). Due to the representations + + − u = u u− and u = u + u− we have − | | + + + + u, Bu = u−, Bu− + u , Bu u−, Bu u , Bu− , h i h i h + +i−h +i−h + i u , B u = u−, Bu− + u , Bu + u−, Bu + u , Bu− . h| | | |i h i h i h i h i Due to Lemma 9.14 we have Bu± 0, which implies u±, Bu∓ and, ﬁnally, ≥ h i ≥ µ u 2 = u, Bu u , B u . On the other hand, for any v L2(Ω) we have k k h i ≤ h| | | |i ∈ 2 v,Bv B v 2 = µ v 2. As u = u , we have u , B u = µ u , and hthis showsi ≤ k thatk · ku kis alsok ank eigenfunctionk k | for| B associatedh| | with| |i the eigenvalue| | µ, | | and it is automatically an eigenfunction of ∆ for the eigenvalue λ. So we have − D u± ker( ∆D λ) too. ∈ − − + Let v denote either u or u−. We have v 0 and ∆v = λv 0. Therefore, v is a ≥ − ≤ subharmonic function. Let BR(y) denote the ball of radius R centered at y. By the maximum principle, for any y Ω and any R> 0 such that B (y) Ω we have ∈ R ⊂ 1 v(y) v(x) dx. ≥ vol B (y) R ZBR(y) 75 If u(y) = 0 for at least one y Ω, then v 0 due to the fact that Ω is connected. + ∈ ≡ Therefore, either v u−, u is strictly positive in Ω or v 0. On the other hand, + ∈ { } ≡ + if both u and u− are strictly positive, one has the contradiction with u , u− = 0. Hence, one of them must be identically zero, which shows that the eigenfunctionh i u does not vanish in Ω.

It remain to show the equality dim ker( ∆D λ) = 1. Assume by contradiction dim ker( ∆ λ) 2, then one can− ﬁnd− two non-zero real-valued functions − D − ≥ u1, u2 ker( ∆D λ) which are orthogonal. On the other hand, the previous consideration∈ − shows− that each of these functions is either strictly positive or strictly negative in Ω, and their scalar product cannot be zero. This contradiction shows that dim ker( ∆ λ) = 1. − D − Note the simplicity of the lowest eigenvalue is a rather general property which holds for the operators having the so-called positivity improving property (an operator A in a suitable measure space is called positivity improving if for f 0 and f = 0 one has Af > 0 a.e.); actually above we have proved that this property≥ holds6 for the resolvent of the Dirichlet Laplacian. This property holds also for other classes of operators, for example, for Schr¨odinger operators with reasonable potentials, but one needs more advanced methods to prove it.

76 10 Self-adjoint extensions

In the present chapter we would like to discuss the existence and the description of self-adjoint extensions of symmetric operators. We have shown previously that semibounded symmetric operators always have self-adjoint extensions (for example, the Friedrichs one), but, in general, a symmetric operator can have several self- adjoint extensions (for example, the Dirichlet and the Neumann Laplacians can be represented as extensions of the same symmetric operator).

10.1 Description of self-adjoint extensions

In this section, T denotes a closed symmetric operator in a Hilbert space . H Proposition 10.1.

The number dim ker(T ∗ λ) is the same for all λ with λ> 0. • − ℑ

The number dim ker(T ∗ λ) is the same for all λ with λ< 0. • − ℑ Proof. By symmetry it is sufficient to prove the first assertion. Let λ = 0. Recall the estimate (T λ)u λ u which holds for all u dom Tℑ. This6 implies − ≥ |ℑ | · k k ∈ that ran(T λ) is closed, and one has the identity ker(T λ) = ran(T λ) . ∗ ⊥ − − − Now let us pick any λ with λ > 0 and any η C. Let u dom T ∗ with u = 1 ℑ ∈ ∈ k k such that T ∗u = (λ + η)u. Assume that u v for all v ker(T ∗ λ). This means ⊥ ∈ − that u ker(T ∗ λ)⊥ = ran(T λ), so we can find ϕ dom T with u =(T λ)ϕ. ∈ − − 1 ∈ − By the previous estimates we have ϕ λ − , and we arrive at the following chain of estimates: k k ≤ |ℑ |

0= ϕ, 0 = ϕ, T ∗ (λ + η) u = (T λ)ϕ, u η ϕ, u h i − − − h i D E η = u, u η ϕ, u 1 η ϕ 1 | | . h i − h i ≥ − | | · k k ≥ − λ |ℑ | These inequalities are impossible if η < λ . Therefore, we have proved the following: If λ> 0 and η < λ , then| | |ℑ | ℑ | | |ℑ |

ker T ∗ (λ + η) ker(T ∗ λ)⊥ = 0 , − ∩ − { } which means that dim ker T ∗ (λ + η) dim ker(T ∗ λ). By the same argument, − ≤ − if η < λ /2, then dim ker T ∗ λ dim ker T ∗ (λ + η) . As dim ker(T ∗ λ) is| locally| |ℑ constant,| it is constant− in the≤ upper half-plane.− − It is also constant in the lower half-plane, but the value may be different. Definition 10.2 (Deficiency indices, deficiency subspaces). The numbers n n (T ) := dim ker(T ∗ λ) with λ > 0 are called the deficiency indices ± ≡ ± − ±ℑ of T , and the subspaces (T ) := ker(T ∗ λ) are called its deficiency sub- Nλ ≡ Nλ − spaces. For brevity we will denote := i. N± N±

77 Remark 10.3. As already seen above (Corollary 7.5), T is self-adjoint iff its deficiency indices are zero. It is important to emphasize that the deficiency indices (both or one of them) can be infinite.

Example 10.4. It is easy to construct an example of a closed symmetric operator having distinct deﬁciency indices. Take = L2(0, ) and let T = id/dx with 1 H ∞ the domain dom T = H0 (0, ), then the adjoint operator is T ∗ = id/dx with 1 ∞ dom T ∗ = H (0, ). ∞ x 1 Let f ker(T ∗ i), then f ′ = f, and f(x)= f(0)e . As f must belong to H (0, ), ∈ − ∞ we obtain f 0, and this gives n+(T ) = 0. On the other hand, ker(T ∗+i) is spanned ≡ x by the function f(x)= e− , which means that n (T ) = 1. − The deﬁciency subspaces can be used for the description of the domain of the adjoint operator.

Proposition 10.5. There holds dom T ∗ = dom T +˙ ++˙ . Here and below the N N− symbol +˙ means that the sum is direct (but not necessarily orthogonal).

Proof. Recall that dom T ∗ becomes a Hilbert space if considered with the scalar product u, v T := u, v + T ∗u, T ∗v . It is also clear that dom T and are ± T -closedh (i.e.i closedh in thei topologyh deﬁnedi by this scalar product) subspacesN of dom T ∗. Moreover, one can easily show that these three subspaces are mutually T -orthogonal. For example, if u dom T and v , then ∈ ∈N+

u, v = u, v + T ∗u, T ∗v = u, v + Tu,iv h iT h i h i h i h i = i i u, v + Tu,v = i (T + i)u, v − h i h i = i u, (T ∗ i)v = i u, 0 = 0. − h i The two remaining cases are considered in a similar way. Therefore, the sum dom T + + + is direct, and it remains to show that this sum coincides with dom T ∗. N N− Let ψ dom T ∗ such that ψ is T -orthogonal to all the three subspaces. In particular, ∈ for any ϕ dom T we have ∈

ψ, ϕ = ψ, ϕ + T ∗ψ, T ∗ϕ = 0. h iT h i h i

As T ∗ϕ = T ϕ, we have ψ, ϕ = T ∗ψ, T ϕ . By the deﬁnition of the adjoint h i −h i operator this means that T ∗ψ dom T ∗ and that T ∗T ∗ψ = ψ. On the other hand, ∈ −

0=(T ∗T ∗ + 1)ψ =(T ∗ + i)(T ∗ i)ψ, − which means that (T ∗ i)ψ . For any ϕ one has − ∈N− ∈N−

i ϕ, (T ∗ i)ψ = i ϕ, iψ + iϕ, T ∗ψ = ϕ, ψ + T ∗ϕ, T ∗ψ = ϕ, ψ = 0, − h − i h− i h i h i h iT which means that (T ∗ i)ψ = 0 and ψ +. On the other hand, ψ was chosen orthogonal to , which− shows that ψ =∈ 0. N N+

78 Our next aim is develop a certain machinery that allows one to describe all self- adjoint extensions of a symmetric operator and to analyze its spectral properties. Let us introduce the following notion: Deﬁnition 10.6 (Boundary triple). Let S be a closed symmetric operators in a Hilbert space . Let be another Hilbert space and Γ, Γ′ : dom S be linear H G → H maps with the following three properties:

1. for all f,g dom S one has the identity f,Sg Sf,g = Γf, Γ′g ∈ h iH −h iH h iG − Γ′f, Γg , h iG 2. the map dom S f (Γ, Γ′)f := (Γf, Γ′f) is surjective, ∋ 7→ ∈G×G 3. ker(Γ, Γ′) is dense in , H then ( , Γ, Γ′) is called a boundary triple for S. G The following proposition shows a link between boundary triples and symmetric operators. Proposition 10.7. Let T be a closed symmetric operators with equal deﬁciency indices n+ = n = n, then its adjoint T ∗ has a boundary triple ( , Γ, Γ′) with − G dim = n. G Scheme of the proof. The complete proof can be found e.g. in Section 1.2 of [12], here we just give the main idea.

Denote by P the projections from dom T ∗ to corresponding to the expansion ± N± in Proposition 10.5, and let U : + be an arbitrary unitary operator (its − existence is guaranteed by the factN that→ N the deﬁciency indices are equal). By the direct computation one can show that the triple ( , Γ, Γ′) with = +,Γ= iUP G G N − − iP+ and Γ′ = P+ + UP satisﬁes all the requested properties. −

In the above deﬁnition we do not ask for any boundedness of the maps Γ and Γ′, but it follows automatically:

Proposition 10.8. Let ( , Γ, Γ′) be a boundary triple for S. If dom S is considered G as the Hilbert space with the scalar product u, v S = u, v + Su,Sv , then Γ, Γ′ : dom S are bounded. h i h i h i → G

Proof. We show that Γ and Γ′ are closed, then their boundedness follows from the closed graph theorem. Let g dom S such that g converge to g and Sg converge n ∈ n n to Sg. Assume that Γg and Γ′g converge to some u and v , respectively. n n ∈ G ∈ G We need to show that Γg = u and Γ′g = v. For any f dom S we have: ∈

Γf, Γ′g Γ′f, Γg = f,Sg Sf,g h i−h i h i−h i = lim f,Sgn Sf,gn = lim Γf, Γ′gn Γ′f, Γgn n n →∞ h i−h i →∞ h i−h i h i h = Γf,v i Γ′f, u , h i−h i

79 which can be rewritten as Γf, Γ′g v = Γ′f, Γg v . Chosing f dom S with h − i h − i ∈ Γf =Γ′g v and Γ′f = 0 we obtain Γ′g = v. For Γf =0 and Γ′f =Γg we obtain − u Γg = u. Definition 10.9 (Linear relations). Let be a Hilbert space. G Any linear subspace Λ will be called a linear relation in . • ⊂G×G G A linear relation is called closed, is it is a closed subspace of . • G × G If Λ is a linear relation in , then the adjoint linear relation Λ∗ is defined by • G Λ∗ := JΛ⊥, where J : is defined by J(x, y)=( y, x), and the orthotognal complementG×G→G×G is taken in . − G × G A linear relation Λ is called symmetric if Λ Λ∗ and is called self-adjoint if • ⊂ Λ=Λ∗. Remark 10.10. Clearly, the notions of a closed/adjoint/symmetric/self-adjoint linear relation generalize those for the linear operators if we identify each linear operator with its graph. This means, in particular, that a linear operator is closed/symmetric/self-adjoint iff its graph is a linear relation with the same property. Nevertheless, the notion of a linear relation is much larger. For example, the “vertical subspace” 0 is a self-adjoint linear relation in while it cannot be represented as the graph{ } × of G a linear operator. G Proposition 10.11. Let S be a closed symmetric operator in a Hilbert space , H and let ( , Γ, Γ′f) be an associated boundary triple, then: G 1. If Λ is a closed linear relation in , then the restriction S to dom S := f G Λ Λ { ∈ dom S : (Γf, Γ′f) Λ is a closed operator, ∈ } 2. Any closed restriction of S is of the above form SΛ,

3. There holds SΛ∗ =(SΛ)∗.

Proof. The ﬁrst assertion follows immediately from Proposition 10.8. For the second assertion, if S is any closed restriction of S, then S = S with Λ := (Γf, Γ′f) : Λ { f dom S , and the closedness of Λ follows again from Proposition 10.8. The third assertion∈ is} checkede directly. e e Corollary 10.12. Let T be a closed symmetric operator, then T has self-adjoint extensions iﬀ n+(T )= n (T ). −

Proof. Let n+(T ) = n (T ). By Proposition 10.7 one can construct a boundary − triple ( , Γ, Γ′) for T ∗. If Λ is a self-adjoint relation in , then (T ∗)Λ is a self-adjoint operatorG by Proposition 10.11, and it is an extension ofG T .

Now assume that T has a self-adjoint extension A. For z1, z2 / spec A consider 1 ∈ the operator U(z1, z2)=(A z2)(A z1)− . This is a bijective operator, and one can check directly that U(z −, z ) −= . Taking z = i and z = i we obtain 1 2 Nz2 Nz1 1 2 − n+ = n . − 80 Corollary 10.13. If T is a closed symmetric operator and ( , Γ, Γ′) is a bound- G ary triple for T ∗, then there is a on-to-one correspondence between the self-adjoint extensions of T and the self-adjoint linear relations in . G Remark 10.14. In view of the preceding assertion one may wonder how to describe all self-adjoint linear relations. First of all, the graph of any self-adjoint linear operator is a self-adjoint linear relation, and it will be sufficient for all the examples below. Nevertheless, one can prove the following general result (it will not be used below, but it is a good exercise to prove it): Proposition 10.15. Let be a Hilbert space. For A, B ( ) consider the linear relation ΛA,B := (x, y) G : Ax = By , then ∈ L G { ∈G×G } ΛA,B is self-adjoint iff A and B satisfy the following two conditions: • A B AB∗ = BA∗, ker = 0. BA− any self-adjoint linear relation is of the above form. • Example 10.16 (Schrödinger operators on half-line). As a first example one can consider the case = L2(0, ) and take the operator T = d2/dx2 +V defined 2 H ∞ 2 − on dom T = H0 (0, ); here V L∞(0, )+ L (0, ) is a real-valued potential. Note that T commutes∞ with the∈ complex∞ conjugation,∞ which means automatically that the deficiency indices of T are equal.

One can easily check that the adjoint T ∗ is given by the same differential expression 2 on the domain dom T ∗ = H (0, ). Using the integration by parts one can check the identity ∞ f, T ∗g T ∗f,g = f(0)g′(0) f (0)g(0) h i−h i − ′ valid for all f,g dom T ∗. Checking the remaining properties in Definition 10.6 we ∈ show that as a boundary triple for T ∗ one can take (C, Γ, Γ′) with Γf = f(0) and Γ′f(0) = f ′(0). As follows from the previous considerations, all the self-adjoint extensions of T 2 are the restrictions of T ∗ to the functions f H (0, ) satisfying the boundary ∈ ∞ conditions af(0) = bf ′(0), where a and b are complex numbers with ab = ba and (a, b) = (0, 0). One can easily check that such boundary conditions can be rewritten 6 equivalently as cos ϕf(0) + sin ϕf ′(0) = 0 with some ϕ [0,π). ∈ Example 10.17 (Schrödinger operators on an interval). Consider the case 2 2 2 2 = L (0, 1), T = d /dx + V defined on dom T = H0 (0, ), here again V H2 − ∞ ∈ L (0, 1) is a real-valued potential. One shows that T ∗ is given by the same differential 2 expression and that dom T ∗ = H (0, 1). The integration by parts shows that as a boundary triple ( , Γ, Γ′) for T ∗ one can take G 2 f(0) f ′(0) = C , Γf = , Γ′f = . G f(1) f ′(1) − A description of all possible self-adjoint extensions of T is indeed much more involved compared to the previous case.

81 Remark 10.18 (Laplacian in a bounded domain). Let Ω Rd (d 2) be a domain with a regular boundary. One is faced with numerous diﬃculties⊂ if≥ one tries to describe all self-adjoint extensions of the operator T = ∆ acting in = L2(Ω) 2 − H with the domain H0 (Ω). By analogy to the preceding examples one could try to use the integration by parts. It is well known that for f,g C2(Ω) one has the Green identity ∈ ∂g ∂f f, ∆g ∆f,g = f dℓ g dℓ, h − i−h− i ∂n − ∂n I∂Ω I∂Ω which looks very close to what is expected for the boundary triples and one could 2 have the intention to take = L ( ), Γf = f ∂Ω, Γ′f = ∂f/∂n ∂Ω as a boundary triple. Nevertheless, one hasG the followingG principal| obstacle: if| one calculates the adjoint T ∗, then one can see that the boundary traces of the functions from dom T ∗ and their normal derivatives do not belong to L2(∂Ω). Of course, the existence of a boundary triple is guaranteed by Proposition 10.7, but its explicit constructions involves rather sophisticated constructions with pseudodiﬀerential operators. De- scribing all the self-adjoint extensions of T is a form suitable for the subsequent analysis is still a non-trivial problem.

10.2 Spectral analysis

Now we would like to understand how to study the spectral properties of various self-adjoint extensions. In this section, T denotes a closed symmetric operator in a Hilbert space with equal deﬁciency indices, ( , Γ, Γ′) is a boundary triple for H 0 G T ∗. Furthermore, we denote by H the restriction of T ∗ to ker Γ; it is a self-adjoint 0 0 0 1 operator by Proposition 10.11. By R (z) we denote its resolvent, R (z)=(H z)− . − 0 0 Example 10.19. In Example 10.16, the operator H acts as H = f ′′ + V f on dom H0 = f H2(0, ) : f(0) = 0 , i.e. corresponds to the Dirichlet− boundary conditions.{ Similarly,∈ in∞ Example 10.17 the respective operator H0 acts as H0 = 0 2 f ′′ + V f on dom H = f H (0, ) : f(0) = f(1) = 0 . − { ∈ ∞

Our considerations start with a certain special-type decomposition of dom T ∗.

0 0 Proposition 10.20. For any z res H one has dom T ∗ = dom H +˙ . ∈ Nz 0 1 Proof. Let f dom T ∗. Denote f0 := (H z)− (T ∗ z)f. One has clearly f dom H0. At∈ the same time, − − 0 ∈ 0 1 (T ∗ z)(f f0)=(T ∗ z)f (T ∗ z)(H z)− (T ∗ z)f − − 0 − −0 − 1 − − =(T ∗ z)f (H z)(H z)− (T ∗ z)f =(T ∗ z)f (T ∗ z)f = 0. − − − − − − − − 0 Therefore, dom T ∗ = dom H + z, and we need to show that this decomposition N0 0 is direct. Assume that f dom H and g z with f + g = 0, then (H z)f = ∈ ∈ N 0 − (T ∗ z)f = (T ∗ z)g = 0. Due to z res H one has f = 0, and then g = 0 too.− − − ∈

82 Definition 10.21 (Boundary operators). Let z res H0. Define the operators ∈ 1 − γ(z) := Γ z : z |N G→N ⊂H (which is well-defined due to Proposition 10.20) and M(z):=Γ′γ(z) : . The maps z γ(z) and z M(z) will be called the γ-field and the WeylG → function G respectively.7→ 7→ Remark 10.22 (Weyl functions for Schrödinger operators). In other words, for ξ , z res H0 the vector f := γ(z)ξ is the unique solution to the abstract ∈ G ∈ boundary value problem (T ∗ z)f =0, Γf = ξ. − 2 In Example 10.16, define by x c(x, z) the unique H -solution to y′′ + V y = zy with y(0) = 1, then γ(z)ξ(x) =7→ξc(x, z), and the associated Weyl− function is just the multiplication by a constant, M(z) := c′(0; z). This function is known as the so- called Weyl-Titchmarsch m-function and is defined at least for z / R. For example, for V = 0, i.e. for the free Laplacian, one has c(x, z) = ei√zx ∈and M(z) = i√z, where the continuous branch of the square root is chosen by the condition √z > 0 for z < 0. ℑ In Example 10.17, for z / R denote by x ϕ(x, z) and x ψ(x, z) the unique ∈ 7→ 7→ solutions to y′′ + V y = zy with ϕ(0, z) = ψ(1, z) = 1 and ϕ(1, z) = ψ(0, z) = 0, then − ξ γ(z) 1 (x)= ξ ϕ(x, z)+ ξ ψ(x, z), ξ 1 2 2 and ξ ξ ϕ (0, z)+ ξ ψ (0, z) ϕ (0, z) ψ (0, z) ξ M(z) 1 = 1 ′ 2 ′ = ′ ′ 1 . ξ ξ ϕ′(1, z) ξ ψ′(1, z) ϕ′(1, z) ψ′(1, z) ξ 2 − 1 − 2 − − 2 For V = 0 one has, using the same branch of the square root as above, sin √z(1 x) sin √zx ϕ(x, z)= − , ψ(x, z)= , sin √z sin √z which gives √z cos √z 1 M(z)= − . sin √z 1 cos √z − The following proposition summarizes the main properties of γ and M: Proposition 10.23. The γ-field and the Weyl function enjoy the following properties:

1. γ(z) ( , ) for any z res H0, ∈ L G H ∈ 0 0 2. for any z1, z2 res H one has γ(z1) γ(z2)=(z1 z2)R (z1)γ(z2). In particular, γ : res∈ H0 ( , ) is holomorph.− − → L G H d 3. γ(z)= R0(z)γ(z), dz

83 0 4. for any z1, z2 res H one has M(z1) M(z2)∗ = (z1 z2)γ(z2)∗γ(z1). In particular, M :∈ res H0 ( ) is holomorph.− − → L G

5. M ∗(z)= M(z). d 6. M(z)= γ(z) γ(z). dz ∗ Proof. To show the assertion 1 we recall that Γ : is bounded in with respect Nz → G to the norm deﬁned by T ∗. By on this norm is equivalent to the usual norm of Nz . Hence the graph of Γ z is a closed set, so the graph of γ(z) z is closedH too, and the boundedness⊂N × follows G from the closed graph theorem.⊂G×N To prove assertion 2 consider the bijective operator

0 1 0 U(z , z )=(H z )(H z )− 1+(z z )R (z ). 1 2 − 2 0 − 1 ≡ 1 − 2 1 0 Let ξ . Denote f := γ(z2)ξ and g := U(z1, z2)f. We have g = f+(z1 z2)R (z1)f, and Γ∈R G0(z )f = 0 due to the inclusion R0(z )f dom H0 = ker Γ.− Therefore, 1 1 ∈ Γg =Γf = ξ. On the other hand,

0 (T ∗ z1)g =(T ∗ z1)f +(z1 z2)(T ∗ z1)R (z1)f − − − − 0 0 =(T ∗ z )f +(z z )(H z )R (z )f − 1 1 − 2 − 1 1 =(T ∗ z )f +(z z )f =(T ∗ z )f = 0. − 1 1 − 2 − 2

So we have Γf = ξ and g z1 , which means that g = γ(z1)ξ, and ﬁnally we obtain ∈N 0 γ(z1)= U(z1, z2)γ(z2), which is exactly the requested equality. As the resolvent R is holomorph, the map γ is holomorph too. Assertion 3 follows trivially. To shows the assertion 4, take arbitrary φ,ψ and denote f := γ(z )ψ and ∈ G 2 g := γ(z1)φ. We have:

f, T ∗g T ∗f,g (z z ) f,g = f, (T ∗ z )g (T ∗ z )f,g = 0. (10.1) h i−h i − 1 − 2 h i − 1 − − 2

On the other hand, f,g = γ(z )ψ,γ( z )φ = ψ,γ (z ) ∗γ(z )φ , and h i 2 1 2 1

f, T ∗g T ∗f,g = Γf, Γ′g Γ′f, Γg h i−h i h i−h i = Γγ(z )ψ, Γ′γ(z )φ Γ′γ(z )ψ, Γγ(z )φ 2 1 − 2 1 = ψ, M(z )φ M(z )ψ,φ = ψ, M( z ) M(z )∗ φ . 1 − 2 1 − 2 D E Substituting these two equalities into (10.1) we obtain the result. The assertion 5 and 6 follow respectively by taking z1 = z2 and by taking the limit as z1 tends to z2. Remark 10.24. Recall that with any A ( ) one can associate the self-adjoint operators ∈ L G A + A∗ A A∗ A := , A = − , ℜ 2 ℑ 2i

84 and then A = A + i A. As follows from the assertion 4 of Proposition 10.23, for any z / R oneℜ has theℑ equality ∈ M(z) ℑ = γ(z)∗γ(z). z ℑ 0 Note that for any z res H one can ﬁnd a constant az > 0 such that γ(z)ξ a ξ for all ξ ;∈ this follows from the fact that γ(z) : has a bounded≥ zk k ∈ G G→Nz inverse which is Γ. Therefore, for any z with z > 0 one can ﬁnd a constant b > 0 ℑ z with M(z) bz. A holomorph map M : z : z > 0 ( ) with this property is usually called≥ an operator-valued Herglotz{ ℑor Nevalninna} → L G function. Such maps appear in various domains besides the spectral analysis, for example, in the measure theory or in the harmonic analysis.

Our aim now is to show how the Weyl functions can be useful for the spectral analysis.

Deﬁnition 10.25 (Simple operators). A closed symmetric operator T in a Hilbert space is said to be non-simple or reducible, if there exists a non-trivial closed subspaceH with the following properties: T ( dom T ) and K⊂H K ∩ ⊂ K T dom T is a self-adjoint operator in . An operator which is not reducible will be K∩ called| simple or irreducible. K

Proposition 10.26. The closed linear span of the vectors γ(z)ξ, z / R, ξ , coincides with if and only if T is simple. ∈ ∈ G H Proof. Denote by the orthogonal complement to the span in question, and let f dom T . OneK has then f,γ(z)ξ = 0 for all z / R and ξ , and ∈ ∩ K ∈ ∈ G

Tf,γ(z)ξ = f, T ∗γ (z)ξ = z f,γ(z)ξ = 0, i.e. T f . Obviously, the restriction T of T to dom T is a closed symmetric ∈ K K∩ operator. Moreover, e K ran(T i)⊥ = ran(T i)⊥ = ker(T ∗ i) = 0 , ± ± ∩ K ∓ ∩ K { } which means thateT is self-adjoint in . K The other implication is a simple exercise. e We can now formulate the main result concerning the relation between the spectral properties of H0 and the Weyl function M.

Theorem 10.27 (Weyl function and spectrum). Let T be simple and λ R, then λ res H0 if and only if M has an analytic extension to λ. ∈ ∈

Proof. First, if λ res H∗0, then M is holomorph near λ in virtue of Proposition 10.23(4). ∈

85 Now let λ R and assume that M has an analytic extension to a certain neighbor- hood of∈λ. Take µ,ν,z C R with z = ν, ν = µ, z = µ. Applying several times O ∈ \ 6 6 6 the identities of Proposition 10.23 we arrive at

0 1 γ(z) γ(ν) γ(µ)∗(H z)− γ(ν)= γ(µ)∗ − − z ν 1 − = γ(µ)∗γ(z) γ(µ)∗γ(ν) z ν − −1 hM(z) M(µ) M(ν)i M(µ) = − − z ν z µ − ν µ − − − Mh(z) M(ν) iM(µ) = + . (z ν)(z µ) − (z ν)(ν µ) (ν µ)(z µ) − − − − − − (10.2) 0 Choose any interval (a, b) containing λ and with a,b / specp H , and denote by E(a, b) to spectral projection⊂O of H0 to (a, b). Combining∈ the above representation with the Stone formula (Proposition 6.7), we obtain γ(µ)ξ,E(a, b)γ(ν)η = 0 for any µ,ν C R and any ξ,η . By Proposition 10.26, the vectors γ(µ)ξ with ∈ \ ∈ G µ / R and ξ span the whole space , which means that E(a, b) = 0, and (a,∈ b) spec H0∈= G by Proposition 6.4. H ∩ ∅ Up to know we only considered the spectral properties of H0. Let us show how the above constructions cen be applied to the analysis of other self-adjoint extensions of T . Let L = L∗ ( ). Consider the self-adjoint extension H of T given by ∈ L G L

dom H = f dom T ∗ : Γ′f = LΓf . L { ∈ }

We would like to have a description of the spectra of HL similar to the one of Theorem 10.27. For this purpose we use the non-uniqueness of the boundary triple.

Proposition 10.28. Denote Γ := LΓ Γ′ and Γ′ := Γ, then ( , Γ , Γ′ ) is a L − L G L L boundary triple for T , and the associates Weyl function ML is given for z / C is 1 ∈ given by M (z)= L M(z) − . L − Proof. We need to check the three conditions of Deﬁnition 10.6. The ﬁrst condition holds due to

Γ f, Γ′ g Γ′ f, Γ g = LΓf Γ′f, Γg Γf,LΓg Γ′g h L L i−h L L i h − i−h − i = Γf, Γ′g Γ′f, Γg + LΓf, Γg Γf,LΓg , h i−h i h i−h i and the term in parenthesis vanishes due to L = L∗.

Let us check the surjectivity condition. Let ξ, ξ′ . As ( , Γ, Γ′) is a boundary ∈ G G triple, there is f dom T ∗ with Γf = ξ′ and Γ′f = Lξ′ ξ. We have then Γ f = ξ ∈ − L and ΓL′ = ξ′, so the property holds.

Finally, the third property holds due to ker(Γ, Γ′) = ker(ΓL, ΓL′ ).

86 Now let us calculate the associated γ-ﬁeld γ and the Weyl function M . Let ξ L L ∈ G and z / R. By deﬁnition f := γ (z)ξ is the unique vector f ker(T ∗ z) with ∈ L ∈ − Γ f = ξ or, equivalently, with LΓf Γ′f = ξ. On the other hand, we have L −

f = γ(z)Γf and Γ′f =Γ′γ(z)Γf = M(z)Γf, which gives L M(z) Γf = ξ. As seen above (Remark 10.24), we have 0 res M(z) and − M(z) 0 for z > 0, and it follows easily that the operator∈ ℑ ±ℑ ≥ ±ℑ 1 L M(z) has a bounded inverse. This gives Γf = L M(z) − ξ and, ﬁnally, − 1 1 − f = γ(z) L M(z) − ξ, i.e. γL(z)= γ(z) L M(z) − . For the Weyl function we − 1− 1 have M (z)=Γ′ γ (z)=Γγ(z) L M(z) − = L M(z) − . L L L − − With respect to the new boundary triple, the operator HL becomes the restriction of T ∗ to ker ΓL, and the combination of Theorem 10.27 and Proposition 10.28 leads to the following result:

Corollary 10.29. Let T be simple. A real value λ belongs to res HL iff the map 1 C R z L M(z) − ( ) extends analytically to λ. \ ∋ 7→ − ∈ L G Calculating the Weyl function can be viewed as an abstract approach to the boundary measurement: one has no access to the interior of an object, but one is able to measure the behavior of some special solutions at the object’s boundary and to study their dependence on a certain controllable parameter (z in our case). Combin- ing the constructions of Remark (10.22) with Proposition 10.27 and Corollary 10.29 one can recover the spectral properties of the Schrödinger operators on the half-line and the intervals with various boundary conditions by considering the behavior of the respective solutions at the boundary points. Note that a similar approach exists in various settings, in which the boundary triple machinery is difficult to apply. For example, if Ω Rd is a bounded domain with a sufficiently regular boundary, one can define its Dirichlet-to-Neumann⊂ map D(z) as follows: if ϕ C∞(∂Ω), then ψ = D(z)ϕ iff there exists a function u defined in Ω ∈ such that ∆u = zu, u ∂Ω = ϕ and ∂u/∂n ∂Ω = ψ. By applying various techniques one can show− that D(z)| can be viewed as an| operator between some Sobolev spaces at the boundary, and it is a very active domain of the analysis to study the relations between the Dirichlet-to-Neumann map, the spectra of the associated Laplacians with various boundary conditions and the geometry of the domain.

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