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Home , Weak derivative

CHANGE: In order not to interrupt the flow of the exam you will pick all random papers at the beginning The most important Theorems of the entire course are the HVZ Theorem the RAGE Theorem, Zhislin’s Theorem, and exponential decay of eigenfunctions. The exam will begin as follows: you will choose one piece of paper out of four. The one of the four Theorems that you choose the Theorem with which the exam will begin. You will be asked about the statement and main ideas of the proof. Be prepared to explain the statement and its physical meaning within 1-2 minutes, and sketch the main ideas of the proof within 3-4 minutes. For that you should thing what are the most important things in the proof. Next you will pick randomly one of the following short Theorems/ examples to prove, and one of the following exercises of the exercise session. For one of the two papers you will have the alternative to pick another one and choose between the two (e.g. if you will you can fo the Theorem well but the exercise not so well, you may pick another paper of the exercises and you can choose one of the two) List of short Theorems/Examples 2.8, 3.4.1, Example 3.12, Example 3.15 (1) Theorems 4.1, 4.2, 5.3, 5.4, 5.5 (with exercise 17 as a hint) List of exercises: 4, 5, 7, 11, 17 (with hint), 21, 23, 31, 33, 35, 36, 42 (part 1 of it uses Theorem 9.8) Definitions and other Theorems with ideas of the proofs will be asked. You can see some hints about which theorems are more important. As mentioned in the class, if a Theorem is not marked as important it does not mean that it will not be asked at all, but we will not go so deeply into it. All definitions are important. The statement of the exercise 46 might be asked Brief Summary of Lectures The first lecture is going to be introductory and general. We are going to explain some motivations of Quantum mechanics, and we are also going to explain a little bit what it means Mathematical Methods in Quantum Mechanics. We are going to motivate the use of wavefunctions, why one uses self-adjoint operators and why the notion of spectrum is important. The last two notions play a very important part in Quantum mechanics. In the first few weeks we are going to deal with background needed to study them. During the Semester several open problems were mentioned. You might be asked to state one or two of them

1 An introduction to weak and Sobolev spaces

You will not be asked directly on this Chapter, but it is a language that is very much needed Let X be a vector space with a norm k.k. X is called a Banach space if every Cauchy sequence in (X, k.k) is convergent. If in addition X is a complex vector space it is called a complex Banach space. An important example We consider the space

2 n 2 n 2 n X = {φ ∈ L (R ): φ ∈ C (R ) and ∆φ ∈ L (R )}.

Then X equipped with the norm kφkL2(Rn) + k∆φkL2(Rn) is not a Banach space. This motivates as to define the Sobolev spaces. Some notation ∞ n n ∞ n n Cc (R ) := {φ : R → | φ ∈ C (R ), and supp φ is compact }, where supp φ := {x ∈ R : φ(x) 6= 0}. ∞ n An element of Cc (R ) is called a test . 1 n n 1 Lloc(R ) := {φ : R → C | φ ∈ L (K) for all compact sets K}.

1 1 n th Definition 1.1 (Weak derivatives). Suppose that u, w ∈ Lloc(R ). We say that w is the α -weak partial of u, and write ∂αu = v if Z Z α |α| ∞ n u∂ φdx = (−1) wφdx, ∀φ ∈ Cc (R ). Rn Rn 1 n Proposition 1.2 (Uniqueness of weak derivatives). If u ∈ Lloc(R ) has a weak derivative then it is unique.

Definition 1.3 (Sobolev spaces). Let k ∈ N and 1 ≤ p ≤ ∞. Then

k n 2 n α α 2 n H (R ) := {u ∈ L (R ) | ∂ u exists and ∂ u ∈ L (R ), for all multiindices α, with |α| ≤ k}.

k n We equip H (R ) with the norm k.kHk(Rn) defined by

1   2 X α 2 kukHk(Rn) :=  k∂ ukL2  . |α|≤k

k n Theorem 1.4. The H (R ) norm is equivalent to the norm Z  kuk := (1 + |ξ|2)k|uˆ(ξ)|2 dξ, (1) Rn 2 n where uˆ is the Fourier transformation of u. In particular, the H (R ) norm is equivalent to the   1 norm kuk := kuk2 + k∆uk2 2 . L2(Rn) L2(Rn)

Some elementary properties of Sobolev spaces

k n Theorem 1.5 (Completeness). For any k ∈ N the space (H (R ), k.kHk(Rn)) is a Banach space. k n Theorem 1.6 (Approximation by smooth functions). Let k ∈ N If u ∈ H (R ), then there exists ∞ n k n a sequence of functions um ∈ Cc (R ) such that um → u in H (R ).

2 Unbounded Operators, Spectrum and Resolvent

The most important Theorems of this Chapter are marked Let (X, kk) be a complex Banach space and D ⊂ X a linear subspace of X. Let A : D → X be a linear operator. If D = X then we say that A is densely defined. The range of A and the Kernel of A are respectively defined by

Ran(A) = {Ax : x ∈ D}, Kern(A) = {x ∈ D : Ax = 0}.

A is called bounded if kAk := sup{kAxk : x ∈ D, kxk = 1} < ∞. Otherwise A is called unbounded. The operator A is called closed if the graph of A

ΓA = {(x, y): x ∈ D, y = Ax} is closed in the Graph Norm kφkA := kφk + kAφk.

2 Remark 2.1. From the definition one can see that A is closed if and only the following property holds: If (xn)n∈N is a sequence in D(A) with xn → x and Axn → y for some x, y ∈ X then x ∈ D and y = Ax. The resolvent set of A is defined as −1 ρ(A) = {z ∈ C :(z − A): D(A) → X is a bijection and (z − A) is bounded}. −1 As the resolvent of A one defines the mapping RA : ρ(A) → L(X),RA(z) := (z − A) . The spectrum of A is defined as σ(A) = C \ ρ(A). Lemma 2.2. (1) If A is not a closed operator then σ(A) = C. (2) If A is a closed operator and D = X then A is bounded. Let L(X) denote the set of bounded operators from X to itsself. Lemma 2.3. IMPORTANT Let K ∈ L(X). If kKk := sup{kKxk : kxk = 1} < 1, then the operator I+K is invertiable and ∞ X (I + K)−1 = (−K)n. k=0 Theorem 2.4. IMPORTANT Let X be a complex Banach space and A : D → X a linear operator. Then ρ(A) is open, σ(A) is closed and RA is analytic in ρ(A). More precisely if z0 ∈ A then −1 −1 B(z0, kRA(z0)k ) ⊂ ρ(A) and for all z ∈ B(z0, kRA(z0)k ) we have ∞ X n n+1 n RA(z) = (−1) RA(z0) (z − z0) . n=0 1 In particular kR (z0)k ≥ . A dist(z0,σ(A)) n Theorem 2.5. Let V : R → C be continuous. Consider the multiplication operator TV : D ⊂ 2 n 2 n 2 2 L (R ) → L (R ) defined on D := {f ∈ L : V f ∈ L } where (V f)(x) = V (x)f(x) by TV f := V f. Then σ(TV ) = Ran(V ). In particular, if V is real-valued, then σ(TV ) ⊂ R. Remark 2.6. Note that in a similar way the multiplication operator can be defined for a measurable −1 function V and in that case σ(TV ) = essran(V ) = {y ∈ C : µ(V (B(y, ))) > 0, ∀ > 0}. Remark 2.7. From now on we will mostly denote the multiplication operator mostly by V instead of TV . 2 n 2 n 2 n Theorem 2.8. IMPORTANT For the Laplacian operator −∆ : H (R ) ⊂ L (R ) → L (R ) we have that σ(−∆) = [0, ∞).

Definition 2.9. Let X be a complex Banach space and A : D1 ⊂ X → X,B : D2 ⊂ X → X linear operators. We set that B is an extension of A if and write A ⊂ B if D1 ⊂ D2 and Ax = Bx for all x ∈ D1. Definition 2.10. Let X be a complex Banach space and A : D ⊂ X → X a linear operator. We set that A is closable if it has a closed extension, namely an extension which is a closed operator. Theorem 2.11. Let X be a complex Banach space and A : D ⊂ X → X a linear operator. Then the following statements are equivalent: (a) A is closable (b) ΓA (closure of the graph of A in the graph norm) is the graph of a linear operator. (c) From xn → 0 and Axn → y it always follows that y = 0.

3 3 Symmetric and self-adjoint operators

Theorems 3.9 and after that in this Chapter are important Let H be a complex and A : D(A) ⊂ H → H densely defined. Then the adjoint operator A∗ : D(A∗) ⊂ H → H is defined as follows: If for a given φ ∈ H there exists a unique φ∗ such that hφ, Aψi = hφ∗, ψi for all ψ ∈ D(A), then φ ∈ D(A∗) and A∗φ = φ∗. The operator A∗ is linear. Lemma 3.1. D(A∗) = {φ ∈ H : ψ → hφ, Aψi is continuous in D(A)} Remark 3.2. If A ⊂ B then B∗ ⊂ A∗. Remark 3.3. A∗ is defined only because A is densely defined. Similarly A∗∗ can be defined only if A∗ is densely defined which is not always the case. Theorem 3.4. Let H be a complex Hilbert space and A : D(A) ⊂ H → H densely defined. Then 1. A∗ is a closed operator IMPORTANT because it follows that self-adjoint operators are closed 2. If A ist closable then (A)∗ = A∗. 3. A is closable if and only if A∗ is densely defined and in this case A = A∗∗. Theorem 3.5. Let A : D(A) ⊂ H ⊂ H be densely defined. Then Kern(A∗) = Ran(A)⊥. In particular Ran(A) ⊂ D(A∗) and Kern(A∗) = {0} ⇔ Ran(A) = H. Definition 3.6 (Symmetry and self-adjointness). A densely defined operator A : D(A) ⊂ H → H is called symmetric if A ⊂ A∗ and self-adjoint if A = A∗. Remark 3.7. A densely defined operator is symmetric if and only if hAφ, ψi = hφ, Aψi for all φ, ψ ∈ D(A). A is self-adjoint if and only if A is symmetric and D(A) = D(A∗). Remark 3.8. From the definition of self-adjointness and Theorem 3.4 it follows that a self-adjoint operator is always closed.

We define C+ = {z ∈ C : Im(z) > 0}, C− = {z ∈ C : Im(z) < 0}. Theorem 3.9. IMPORTANT Let A : D(A) ⊂ H → H be symmetric. 2 2 2 2 1. For all λ, µ ∈ R and all φ ∈ D(A) we have k(A − λ − iµ)φk = k(A − λ)φk + µ kφk . 2. (i) If Ran(A − z+) = H for one z+ ∈ C+ then C+ ⊂ ρ(A). (ii) If Ran(A − z−) = H for one z− ∈ C− then C− ⊂ ρ(A). Theorem 3.10. IMPORTANT Let A be a symmetric operator. Then the following statements are equivalent: 1. A = A∗ 2. σ(A) ⊂ R 3. Ran(A + z±) = H for a z+ ∈ C+ and a z− ∈ C−. ∗ 4. A is closed and Kern(A + z±) = {0} for a z+ ∈ C+ and a z− ∈ C−. Remark 3.11. From σ(A) ⊂ R alone self-adjointness does not necessarily follow. 2 n 2 n 2 n Example 3.12. IMPORTANT The operator A : H (R ) ⊂ L (R ) → L (R ) Aφ = −∆φ is self-adjoint. Theorem 3.13 (Kato-Rellich). Let A be self-adjoint and B symmetric with D(A) ⊂ D(B). If

kBφk ≤ akAφk + bkφk, ∀φ ∈ D(A), where a, b ∈ R and 0 < a < 1 then is A + B : D(A) ⊂ H → H self-adjoint.

4 Theorem 3.14 (Hardy’s inequality). Ex 17 might be asked but not the proof of this, as it is too 1 3 long. Statement very important. Let ψ ∈ H (R ). Then we have that Z |ψ(x)|2 Z dx = 4 |∇ψ(x)|2dx. |x|2

2 3 2 3 2 3 Example 3.15 (Important). 1. The operator A : H (R ) ⊂ L (R ) → L (R )(Aφ)(x) = (−∆ − 1 |x| )φ(x) is self-adjoint. 3 N 3N 2 3N 2 3N 2. We use the notation x := (x1, ..., xN ) ∈ (R ) = R . The operator B : H (R ) ⊂ L (R ) → 2 3N L (R ) N X  N  X 1 (Bφ)(x) = −∆xj − φ(x) + φ(x) (2) |xj| |xi − xj| j=1 1≤i

A symmetric operator is called essentially self-adjoint if A is self-adjoint. The proofs of the next two Theorems will rather not be asked as they are similar to the proofs of two previous ones

Theorem 3.16. Let A be a symmetric operator. Then the following statements are equivalent: 1. A is essentially self-adjoint. 2. Ran(A + z±) = H for a z+ ∈ C+ and a z− ∈ C−. ∗ 3. Kern(A + z±) = {0} for a z+ ∈ C+ and a z− ∈ C−. Theorem 3.17 (Kato-Rellich general form). Let A be essentially self-adjoint and B symmetric with D(A) ⊂ D(B). If kBφk ≤ akAφk + bkφk, ∀φ ∈ D(A), (3) where a, b ∈ R and 0 < a < 1 then is A + B : D(A) ⊂ H → H is essentially self-adjoint and D(A + B) = D(A).

4 The Schr¨odingerequation and Existence of dynamics

Only Chapters 4 and 11 have information about dynamics and you will certainly asked about dynamics Let H : D(H) ⊂ H → H be densely defined. We consider the initial value problem ( i d φ = Hφ dt t t (4) φ0 = u

Let I be a nontrivial interval with 0 ∈ I. A solution (4) is a differentiable function φ : I → H, with the following properties 1. φt ∈ D(H), ∀t ∈ I d φt+h−φt 2. i dt φt = i limh→0 h = −iHφt. 3. φ0 = u Theorem 4.1. IMPORTANT Let H : D(H) ⊂ H → H be densely defined. If (4) has for all u ∈ D a solution with constant norm then H is symmetric. If H is symmetric then the problem has for every u ∈ D(H) at most one solution locally in time.

5 Theorem 4.2. IMPORTANT Let H : D(H) ⊂ H → H be symmetric. If (4) has for all u ∈ D a unique solution which is global in time then H is essentially self-adjoint. Theorem 4.3 (Existence of Solution bounded case). IMPORTANT Let H : H → H be bounded and −itH −itH P∞ (−itH)n self-adjoint. Then for all u ∈ H (4) has the unique solution e u, where e = n=0 n! . −iHt The solution is global in time. The map U : R → L(H), U(t) = e satisfies the following properties 1) U(t) is unitary and U(t + s) = U(t)U(s) for all s, t ∈ R 2) limt→0 U(t)ψ → ψ for all ψ ∈ H.

Definition 4.4. A map U : R → L(H) satisfying the properties 1),2) of the last theorem is called a strongly continuous unitary group. Remark 4.5. If U is a strongly continuous unitary group then U(0) = I and t → U(t)ψ is continuous for all ψ ∈ H (U(t) is strongly continuous). Theorem 4.6 (Existence of Solution general case). IMPORTANT Let H : D(H) ⊂ H → H be self-adjoint. Then for all u ∈ D(H) (4) has a unique solution. The solution φt is global in time. Moreover there exists a unique strongly continuous unitary group U such that, φt = U(t)u, ∀t ∈ R, ∀u ∈ D(H). Remark 4.7. The strongly continuous group U in the last theorem is denoted by U(t) = e−iHt. From the last theorem one can see that the notion of the solution actually makes sense for all u ∈ H even if u is not in D(H).

Definition 4.8. Let U : R → L(H) be a strongly continuous unitary group. The generator of U is defined to be the linear operator A : D(A) ⊂ H → H defined through: U(t)ψ−ψ 1. D(A) := {ψ ∈ H : limt→0 t exists } U(t)ψ−ψ d 2. Aψ = i limt→ t = i dt U(t)ψ|t=0. Theorem 4.9 (Stone). Let A : D(A) ⊂ H → H be the generator of a strongly continuous unitary group U : R → L(H). Then the following statements hold: 1. U(t)D(A) ⊂ D(A) and for all φ ∈ D(A) d i U(t)φ = AU(t)φ = U(t)Aφ. dt

2. A = A∗. 3. U(t) is uniquely determined by A.

5 Observables, uncertainty principle, ground state energy

In this Chapter all proofs are short and can be asked Physically measurable quantities, called observables correspond to self-adjoint operators acting on the Hilbert space H to which the wave function ψ belongs. Definition 5.1. (i) The expectation value associated with the self-adjoint operator A in the state ψ is given by hψ, Aψi. (ii) The variance of A in ψ ∈ D(A) is defined by

2 2 2 ∆Aψ := hψ, (A − hψ, Aψi) ψi = hψ, A ψi − hψ, Aψi .

6 Definition 5.2. The commutator of two operators A, B is defined by [A, B] = AB − BA.

Theorem 5.3 (Heisenberg’s uncertainty principle). Let A, B be two self-adjoint operators acting on H. Then we have for ψ ∈ D(A) ∩ D(B) 1 ∆A ∆B ≥ hψ, [A, B]ψi. ψ ψ 4 We consider the Hamiltonian H of an atom with atomic number N defined similarly as in (2)

N X  Z  X 1 (Hφ)(x) = −∆xj − φ(x) + φ(x) (5) |xj| |xi − xj| j=1 1≤i

2 3N We now by Kato-Rellich that H is self-adjoint with domain H (R ). Each expectation value 1 3N hψ, Hψi can be extended naturally to be defined for all ψ ∈ H (R ). The ground state energy of H is defined by E = inf hψ, Hψi. 1 3N ψ∈H ( ),kψk 2 3N =1 R L (R ) If the infimum is attained then a minimizer is called a ground state.

2 3N Theorem 5.4. We always have that E ∈ σ(H). If H has a ground state ψ then ψ ∈ H (R ) and Hψ = Eψ.

Theorem 5.5. If N = 1, Z > 0 then H has up to a constant a unique ground state and its ground Z2 state energy is given by E = − 4 .

6 Some tools of

You will not be asked for any proofs in this Chapter except possibly the idea of the proof of the last two Theorems with the compactness Here we will introduce some tools of the functional analysis that we will need to handle the problem of the drum in the next section. For simplicity we will explain some of the needed tools only for Hilbert spaces.

Definition 6.1. Let (X, k.k) be a real (respectively complex) Banach space, x ∈ X and xn be a sequence in x. We say that xn converges weakly to x and we write xn * x , if f(xn) → f(x) for all f : X → R (respectively f : X → C) linear continuous.

Remark 6.2. If (X, h., .i) is a Hilbert space then xn * x if and only if hxn, yi → hx, yi for all y ∈ X.

Theorem 6.3. Let (X, k.k) be a Banach space. Then every weakly convergent sequence in X is bounded. Moreover, if xn * x then kxk ≤ lim infn→∞ kxnk. Theorem 6.4 (Banach-Alaoglu, special case). Let (X, h., .i) be a Hilbert space. Then every bounded sequence in X has a weakly convergent subsequence.

Theorem 6.5. Let (X, h., .i) be a Hilbert space. If xn * x and kxnk → kxk, then xn → x. Definition 6.6 (Compact operators). Let X,Y be Banach spaces and K : X → Y a linear operator. K is called compact, if for every bounded sequence xn in X Kxn has a convergent subsequence in Y .

7 Theorem 6.7. Let X,Y be separable Hilbert spaces, and K : X → Y be a linear operator. Then K is compact if and only if for any sequence xn in X and any x ∈ X we have xn * x =⇒ Kxn → Kx.

Theorem 6.8. Let X,Y be Banach spaces spaces, and Kn : X → Y a sequence of compact operators. If K : X → Y is linear bounded and Kn → K in the operator norm, then K is compact.

1 n 1 n 2 n Theorem 6.9. Let f ∈ Cc (R ). Then the operator Tf : H (R ) → L (R ), Tf ψ = fψ is compact. 1 n Theorem 6.10. Let f be bounded with lim|x|→∞ f(x) = 0. Then the operator Tf : H (R ) → 2 n L (R ), Tf ψ = fψ is compact.

7 Decomposition of an operator

Theorem 7.1. Let X be a Banach space and consider a linear operator A : D ⊂ X 7→ X. Consider a simply closed positively oriented curve γ lying in the resolvent set ρ(A) of A. If inside of γ there R −1 are no points of the spectrum of A, then γ(z − A) dz = 0. Definition 7.2. Let X be a Banach space and A ∈ L(X). The spectral radius of A is defined by n 1 rA := lim sup kA k n . Theorem 7.3. Let A ∈ L(X). Then 1) σ(A) 6= ∅ and rA = supz∈σ(A) |z|. 2) If X is a Hilbert space and A is self-adjoint then kAk = supz∈σ(A) |z|. Theorem 7.4. Very important statement and to understand what these projections do. You might be asked about the proof of one part Let H be a Hilbert space and A : D ⊂ H → H be self-adjoint. 0 00 0 00 0 Assume that σ(A) = σ ∪ σ where σ , σ ⊂ C are disjoint and σ is compact. Let γ be a simply closed positively oriented curve such that σ0 is inside of γ but σ00 is outside of γ. We consider the 1 R −1 operator P = 2πi γ(z − A) dz. Then 1) P does not depend on the choice of γ. 2) P is an orthogonal projection. 3) PA ⊂ AP . Thus A lives the range of P and the range of 1 − P invariant. 0 00 0 0 00 00 0 00 4) Let A = A|Ran(P ), A = A|Ran(1−P ). Then σ(A ) = σ and σ(A ) = σ . Moreover A and A are both self-adjoint. 5) A0 is a bounded operator 0 6) If σ = {λ} for some λ ∈ R then AP = λP . Theorem 7.5. Important proof might be asked see and learn also exercise 26, where part of the proof was done Let H be a Hilbert space and A : D(A) ⊂ H → H be self-adjoint. Let λ ∈ R. Then λ ∈ σ(A) ⇐⇒ there exists a sequence (ψn)n∈N ⊂ D(A) with kψnk = 1 and k(A − λ)ψnk → 0.

8 Discrete and Essential spectrum

Very important Chapter Especially Weyl’s criterion, HVZ Theorem, IMS localization formula, exponential decay of eigenfunctions, ionization Threshold. Meanings and Interpretations can also be asked

Definition 8.1. Let H, be a Hilbert space and A : D ⊂ H → H be self-adjoint. The discrete spectrum σd(A) of A is defined to be the set of all isolated points of the spectrum of A that are eigenvalues of finite multiplicity. The essential spectrum σess(A) of A is defined by σess(A) = σ(A) \ σd(A)

8 Theorem 8.2. For a self-adjoint operator A as above and a λ ∈ R we have that λ ∈ σess(A) if and only if there exists a sequence ψn in D with the following properties: 1) kψnk = 1, ∀n ∈ N 2) k(H − λn)ψnk → 0 3) ψn * 0.

n Theorem 8.3. Assume that V : R → R with lim|x|→∞ V (x) = 0, H = −∆ + V is self-adjoint in 2 n H (R ) and σess(H) = [0, ∞).

n Theorem 8.4. Assume that V : R → R with lim|x|→∞ V (x) = ∞, H = −∆ + V is self-adjoint 2 n 2 n with domain D(H) = {ψ ∈ L (R ): −∆ψ + V ψ ∈ L (R )} and σess(H) = ∅. 2 n 2 n 2 n Theorem 8.5 (IMS localization formula). Let H : H (R ) ⊂ L (R ) → L (R ), H = −∆ + V be self-adjoint. n 2 α ∞ a) If f : R → R is a C function with ∂ f ∈ L , for 0 ≤ |α| ≤ 2 then 1 fHf = (f 2H + Hf 2) + |∇f|2. (6) 2

n 2 α ∞ b) If (Ja)a∈A : R → R is a finite family of C functions with ∂ Ja ∈ L , for 0 ≤ |α| ≤ 2, P 2 ∀a ∈ A and with a∈A Ja = 1 then

X X 2 H = JaHJa − |∇Ja| . a∈A a∈A

Theorem 8.6 (HVZ Theorem (Hunziker- van Winter- Zhislin)). We consider the operator HN,Z : 2 3N 2 3N 2 3N H (R ) ⊂ L (R ) → L (R )

N X  Z  X 1 (HN,Z φ)(x) = −∆xj − φ(x) + φ(x). |xj| |xi − xj| j=1 1≤i

Then σess(HN,Z ) = [inf σ(HN−1,Z ), ∞). Theorem 8.7. We consider two self-adjoint operators A, B with D(A) = D(B). If there exists a −1 −1 z ∈ C \ R such that (z − A) − (z − B) is compact then σess(A) = σess(B). Corollary 8.8. We consider two self-adjoint operators A, B with D(A) = D(B). If there exists a −1 z ∈ C \ R such that (B − A)(z − A) is compact then σess(A) = σess(B). ∗ 2 n Definition 8.9. Let H = H be an operator on L (R ) that is bounded from below and satisfies n 2 α ∞ the IMS localization formula in the sense that If f : R → R is a C function with ∂ f ∈ L , for C 0 ≤ |α| ≤ 2 then fD(H) ⊂ D(H) and (6) holds. For R > 0 let ΣR := infψ∈D(H),supp ψ⊂BR(0) . The ionization threshold of H is defined by Σ = limR→∞ ΣR.

Remark 8.10. In the next weeks we will prove that for the operator HN,Z we have Σ = inf σess(HN,Z ). Theorem 8.11 (Exponential decay of eigenfunctions to eigenvalues below Σ). Let H be as in Definition 8.9. If ψ ∈ D(H) satisfies Hψ = Eψ for an E < Σ then for all β > 0 with β2 < Σ − E β|x| 2 n we have that e ψ ∈ L (R ).

9 9 Spectral Theorem for self-adjoint Operators and some applica- tions

You will not be asked for any proofs of Theorems of this Chapter except possibly minimax principle and example with the hydrogen. Very important is an understanding of the spectral Theorem, in its 3 different forms. A basic understanding of the with respect to the spectral measure is needed. Remember also that its is the spectrum of the operator. Theorem 9.1 (Spectral Theorem multiplication operator form). Let A be a self-adjoint operator on a separable Hilbert space H with domain D(A). Then there is a measure space hM, µi with µ a finite measure, a unitary operator U : H → L2(M, dµ), and a real-valued function f on M which is finite a.e. so that (a) ψ ∈ D(A) ⇐⇒ f(Uψ) ∈ L2(M, µ). (b) If φ ∈ U[D(A)], then (UAU −1φ)(m) = f(m)φ(m). Theorem 9.2 (Spectral Theorem functional calculus form). Let A be a self-adjoint operator on a Hilbert space H. Then there exists a unique map φ from the bounded Borel functions on R into L(H) with the following properties: (a) φ(fg) = φ(f)φ(g), φ(λf) = λφ(f), φ(1) = I, φ(f) = φ(f)∗, that is φ is an algebraic * homomorphism. (b) kφ(h)kL(H) ≤ khkL∞ . (c) Let hn(x) be a sequence of bounded Borel functions with hn(x) → x pointwise and |hn(x)| ≤ |x| for all x and n. Then for all ψ ∈ D(A) φ(hn)ψ → Aψ. (d) If hn(x) → h(x) and khnk is bounded, then φ(hn)ψ → φ(h)ψ for all ψ ∈ H. In addition: (e) If Aψ = λψ then φ(h)ψ = h(λ)ψ. (f) If h ≥ 0 then φ(h) ≥ 0.

Theorem 9.3. Let PΩ = φ(χΩ), where Ω ⊂ R Borel and χΩ is the characteristic function of Ω. (i) The family PΩ is a projection valued measure, namely it satisfies the following properties (a) Each PΩ is an orthogonal projection. (b) P∅ = 0, PR = I. N PN (c) If Ω = ∪n=1Ωn, where N = 1, ..., ∞ with Ωn ∩ Ωm = ∅ if n 6= m, then PΩψ = n=1 PΩn ψ.

(d) PΩ1 PΩ2 = PΩ1∩Ω2 . (ii) For all Ω ⊂ R Borel we have that PΩA ⊂ APΩ. If Ω is open then

σ(A) ∩ Ω ⊂ σ(A|Ran PΩ ) ⊂ σ(A) ∩ Ω. Theorem 9.4 (Minimax Principle). Let H : D(H) ⊂ H → H be self-adjoint and bounded from below. Let µn(H) := inf ( sup hφ, HΦi) M⊂D(H),dim(M)=n φ∈M,kφk=1 Then for any n ∈ N we have either (a) There are at least n eigenvalues (counting multiplicity) below inf σess(H) and µn(H) is the n-th eigenvalue from below (counting multiplicity) or (b) µn(H) = inf σess(H) = µn+1(H) = µn+2(H) = ... and there are at most (n − 1) eigenvalues (counting multiplicity) below µn(H).

Corollary 9.5. µ1(H) = infφ∈D(H),kφk=1hφ, Hφi (ground state energy of H) is in the spectrum of H.

10 Corollary 9.6. (i) If there exists a function φ ∈ D(H) with kφk = 1 and hφ, Hφi < inf σess(H), then H has a ground state. (ii) If there exists a subspace M ⊂ D(H) of dimension N ∈ N∪{∞}, with hφ, Hφi < inf σess(H), then H has (at least) N eigenvalues below the bottom if its essential spectrum.

2 3 2 3 2 3 1 Example The operator H : H (R ) ⊂ L (R ) → L (R ), H = −∆ − |x| has infinitely many eigenvalues below the bottom of its essential spectrum.

Definition 9.7. Let A : D(A) ⊂ H → H be self-adjoint with functional calculus φ. The map P : B → L(H) with P (Ω) = PΩ = φ(χΩ)) = χΩ(A). P is called the spectral measure of A. Its support is defined by supp(P ) = {x ∈ R : P (U) 6= 0 for all neighborhoods U of x}. Theorem 9.8. Let A = A∗ with spectral measure P . Then 1. S := supp(P ) is closed.

2. PS = 1 and PR\S = 0. 3. For all f bounded continuous we have f(A) := φ(f) = supx∈S |f(x)|. 4. S = σ(A), k(z − A)−1k = dist(z, σ(A))−1, ∀z ∈ ρ(A).

Theorem 9.9. (Spectral Theorem projection valued form) There is a 1-1 correspondence between self-adjoint operators and projection valued measures on H the correspondence given by Z ∞ Z A = λdPλ = λdPλ. −∞ σ(A)

If g is a real valued Borel function on R then Z ∞ g(A) := g(λ)dPλ −∞ defined on Z ∞ 2 Dg := {φ : |g(λ)| d(φ, Pλφ) < ∞}, −∞ is self-adjoint.

10 Newton’s Theorem and Zhislin’s Theorem

IMPORTANT Chapter meanings and interpretations can all be asked

3 Theorem 10.1 (Newton’s Theorem). We consider a spherically symmetric function ρ : R → R. 3 ρ(y) 1 3 Assume that for all x ∈ R the function g(y) = |x−y| is in L (R ). Then

Z ρ(y) Z ρ(y) 1 Z Z ρ(y) dy = dy = ρ(y)dy = dy R3 |x − y| R3 max{|x|, |y|} |x| |y|≤|x| |y|≥|x| |y|

2 3N 2 3N Theorem 10.2 (Zhislin’s Theorem). We consider the operator HN,Z : H (R ) ⊂ L (R ) → 2 3N L (R ) N X  Z  X 1 (HN,Z φ)(x) = −∆xj − φ(x) + φ(x). |xj| |xi − xj| j=1 1≤i

If Z > N − 1 then HN,Z has infinitely many eigenvalues below the bottom of the essential spectrum.

11 11 Continuous spectrum, pure point spectrum and RAGE Theo- rem

IMPORTANT CHAPTER Only Chapter 4 and Chapter 11 are about dynamics and you will defi- nitely asked about dynamics

Definition 11.1. Let H : D(H) ⊂ H → H be self-adjoint. We define

⊥ Hpp := span{ψ : ψ is eigenvalue of H}, Hc := Hpp.

We define as the pure point spectrum σpp(H) of H as its set of eigenvalues. We also define the continuous spectrum of H by σc(H) := σ(H|Hc ).

Remark 11.2. We have σ(H) = σpp(H) ∪ σc(H), where the union is not necessarily disjoint and

σpp(H) = σ(H|Hpp ). If P is the spectral measure of H then φ ∈ Hpp, if and only if the measure Ω → φ, PΩφ has discrete support and φ ∈ Hc if and only if the measure is continuous (namely every set with one element has measure zero).

Theorem 11.3 (RAGE Theorem). Let Let H : D(H) ⊂ H → H be self-adjoint. Assume that the −1 operator χR(H + i) is compact for all R > 0, where χR = χBR(0). Then

−itH φ ∈ Hpp ⇐⇒ lim sup k(1 − χR)e φk = 0 R→∞ t≥0

Z T 1 −itH 2 φ ∈ Hc ⇐⇒ lim kχRe φk dt = 0, for all R > 0. T →∞ T 0 Lemma 11.4. Let µ be a complex valued finite measure on . Define µˆ(t) := R e−itλdµ(λ). Then R R

1 Z T X lim |µˆ(t)|2dt = |µ({y})|2. (7) T →∞ T 0 y∈R

Lemma 11.5. Let Let H : D(H) ⊂ H → H be self-adjoint. If φ ∈ H and ψ ∈ Hc then

1 Z T lim |hφ, e−iHtψi|2dt = 0. (8) T →∞ T 0

12