Random Operators

Lecture Notes byMichael Aizenman & Simone Warzel

Preliminary draft(with many misprints)

DEPARTMENT OF MATHEMATICS AND PHYSICS, PRINCETON UNI-VERSITY, PRINCETON, NJ 08544, USA

ZENTRUM MATHEMATIK, TU MUNCHEN, BOLTZMANNSTR. 3,85747 GARCHING, GERMANY

Contents

Chapter 2. Some mathematical groundwork 51. Elements of spectral theory 5

1.1. Hilbert spaces, linear operators and all that 51.2. Spectral calculus and spectral types 8

2. Spectra and dynamics 112.1. Return probabilities and recurrence 112.2. Theorems of Wiener and RAGE 122.3. Theorems of Strichartz and Last 15

3. Appendix: Herglotz functions 16Notes 17

Chapter 3. Modeling disorder 191. Stochastic processes 19

1.1. Basic notions 191.2. Ergodicity 20

2. Ergodic operators 222.1. Definitions 222.2. Deterministic spectra 232.3. Determining the spectrum 24

Notes 26

Bibliography 29

3

CHAPTER 2

Some mathematical groundwork

1. Elements of spectral theory

Since observables in quantum mechanics are described by self-adjointoperators on a Hilbert space, we will briefly and in a somewhat informalmanner review some elements of the theory of such operators. We will omitall proofs and refer the reader to the extended literature, in particular [146,41, 128, 153].

1.1. Hilbert spaces, linear operators and all that. The arena of quan-tum mechanics is a separable complex Hilbert space H, which is endowedwith a scalar product 〈·, ·〉 : H × H → C. We will take the latter tobe linear in its second entry. The induced norm of ψ ∈ H is given by‖ψ‖ :=

√〈ψ, ψ〉.

An example is the space of square-summable, complex valued se-quences over the d-dimensional square lattice

`2(Zd) :=ψ : Zd → Zd

∣∣ ∑x∈Zd

|ψ(x)|2 < ∞, (2.1)

which is rendered a Hilbert space together with the scalar product

〈ϕ, ψ〉 :=∑x∈Zd

ϕ(x)ψ(x) . (2.2)

Here (·) denotes complex conjugation.

A linear operator A : D(A) → H defined on a domain, i.e., a denselinear subspace D(A) ⊂ H, is bounded if and only if there exists somea < ∞ such that ‖Aψ‖ ≤ a‖ψ‖ for all ψ ∈ D(A). The smallest possibleconstant in this inequality defines the operator norm

‖A‖ := supψ∈D(A)‖ψ‖=1

‖Aψ‖ .

5

6 2. SOME MATHEMATICAL GROUNDWORK

An example of a bounded operator is the discrete Laplacian

∆ : `2(Zd)→ `2(Zd) ,

ψ(x) :=∑y∈Zd

|x−y|=1

(ψ(y)− ψ(x)) (2.3)

defined on `2(Zd). Its operator norm is estimated using the triangle inequal-ity for the norm on `2(Zd):

‖∆ψ‖ =(∑x∈Zd

∣∣∣∑|j|=1

(ψ(x+ j)− ψ(x))∣∣∣2)1/2

≤∑|j|=1

[(∑x∈Zd

|ψ(x+ j)|2)1/2

+(∑x∈Zd

|ψ(x)|2)1/2]

= 4d ‖ψ‖ . (2.4)

As a consequence, ‖∆‖ ≤ 4d. As we will see below, this inequality is, infact, an equality.

The space of linear operators on H may be equipped with varioustopologies. For sequences of bounded operators the most important con-cepts are those of weak, strong and norm convergence, i.e.,

An → A weakly iff 〈ϕ , (An − A)ψ〉 → 0 for all ϕ, ψ ∈ H,

An → A strongly iff ‖(An − A)ψ‖ → 0 for all ψ ∈ H,

An → A in norm iff ‖An − A‖ → 0 .

Clearly, norm convergence implies strong convergence which in turn im-plies weak convergence.

We will mainly be concerned with self-adjoint operators. Recall thatthe adjoint of A : D(A) → H is the unique operator A† : D(A†) → Hacting as

〈ϕ,Aψ〉 = 〈A†ϕ, ψ〉for allψ ∈ D(A) andϕ ∈ D(A†) := ϕ ∈ H |There is % ∈ H such that forall ψ ∈ D(A): 〈ϕ,Aψ〉 = 〈%, ψ〉. The operator A is self-adjoint if andonly if A = A†, which in particular requires D(A) = D(A†).

Any bounded, real-valued sequence, V : Zd → R, gives rise to abounded and self-adjoint Schrodinger operator H : `2(Zd) → `2(Zd)defined by

(Hψ) (x) := − (∆ψ) (x) + V (x)ψ(x) (2.5)

1. ELEMENTS OF SPECTRAL THEORY 7

on ψ ∈ `2(Zd). In fact, each of the terms individually define boundedself-adjoint operators and hence their sum is bounded and self-adjoint. Theself-adjointness of the multiplication operator V : `2(Zd) → `2(Zd) cor-responding to V is left as an easy exercise to the reader. The self-adjointnessof the discrete Laplacian follows from its representation as a quadraticform,

〈ϕ,−∆ψ〉 =1

2

∑x,y∈Zd

|x−y|=1

(ϕ(y)− ϕ(x)) (ψ(y)− ψ(x)) , (2.6)

which implies 〈ϕ,−∆ψ〉 = 〈−∆ϕ, ψ〉 for all ϕ, ψ ∈ `2(Zd).

We recall that the resolvent set of an operator A : D(A) → H isgiven by

%(A) := z ∈ C | (A− z) : D(A)→ H is bijective .

For z ∈ %(A) the operator (A− z) is hence invertible and its inverse, (A−z)−1 : H → D(A) is called the resolvent. The spectrum of A is the set

σ(A) := C \ %(A) ,

which is always closed. If A is bounded, σ(A) is compact. In case A isself-adjoint, σ(A) ⊂ R and for all z ∈ %(A):∥∥(A− z)−1

∥∥ ≤ 1

dist(z, σ(A))≤ 1

| Im z|, (2.7)

where the last inequality requires z ∈ C\R.

As an example, let us determine the spectrum of the discrete Lapla-cian. The Fourier transformation

F : `2(Zd)→ L2([0, 2π]d)

(Fψ) (k) :=1

(2π)d/2

∑x∈Zd

e−ik·x ψ(x) (2.8)

defines a unitary operator. By an explicit calculation, one checks that thediscrete Laplacian is unitarily equaivalent to a multiplication operator, i.e.

(F∆F−1ϕ

)(k) = 2

d∑ν=1

(cos(kν)− 1) ϕ(k) =: h(k)ϕ(k) , (2.9)

for all φ ∈ L2([0, 2π]d), where k = (k1, . . . , kd) ∈ [0, 2π]d. The spectrumof any multiplication operator is the range of the corresponding function,

8 2. SOME MATHEMATICAL GROUNDWORK

which in case of h, is the interval [−4d, 0]. Since the spectrum is invariantunder unitary transformation, we therefore conclude

σ(−∆) = [0, 4d] . (2.10)

This implies ‖∆‖ = 4d, since the operator norm coincides with the maxi-mal modulus of points in the spectrum.

An important tool for spectral analysis are the first and second resol-vent equation.

Proposition 2.1 (Resolvent equations) For a linear operator A : D(A)→H and z1, z2 ∈ %(A):

(A− z1)−1 − (A− z2)−1 = (z1 − z2)(A− z1)−1(A− z2)−1 . (2.11)

Moreover, if B : D(B) → H is defined on a common domain, D(A) =D(B),

(A− z)−1 − (B − z)−1 = (A− z)−1(B − A) (B − z)−1

= (B − z)−1(B − A) (A− z)−1 , (2.12)

for any z ∈ %(A) ∩ %(B).

The proof is elementary and can be found in [153].An immediate corrollary is the fact that the distance of the spectra of

two operators A,B : D(A)→ H whose difference is a bounded operator isbounded by

dist (σ(A), σ(B)) ≤ ‖A−B‖ . (2.13)Lecture 121.4.10

1.2. Spectral calculus and spectral types. The benefit of dealing withself-adjoint operators A : D(A) → H on a Hilbert space is the availablityof spectral calculus.

Recall that for any ψ ∈ H, the function given by

Fψ(z) := 〈ψ , (A− z)−1 ψ〉 (2.14)

is holomorphic for z ∈ %(A) ⊃ C+ := z ∈ C | Im z > 0. Using (2.7)we conclude:

|Fψ(z)| ≤∥∥(A− z)−1

∥∥ ‖ψ‖2 ≤ ‖ψ‖2

Im zfor any z ∈ C+

Fψ(z) = 〈(A− z)−1 ψ , ψ〉 = 〈ψ , (A− z)−1 ψ〉 = Fψ(z) .

Moreover, the first resolvent equation (2.11) implies:

ImFψ(z) =1

2i

(Fψ(z)− Fψ(z)

)= Im z

∥∥(A− z)−1 ψ∥∥2

> 0 ,

1. ELEMENTS OF SPECTRAL THEORY 9

for any ψ 6= 0, hence Fψ is a Herglotz function, i.e., a holomorphic func-tion Fψ : C+ → C+ mapping the upper half plane into itself.

The representation theorem for Herglotz functions, which is summa-rized in the appendix, entails that there exists a unique finite Borel measureµψ on R, called the spectral measure of A associated with ψ, such that

Fψ(z) =

∫µψ(dλ)

λ− z, z ∈ C+ . (2.15)

By polarization, for any pair ϕ, ψ ∈ H one may identify complex finiteBorel measures µϕ,ψ such that 〈ϕ, (A − z)−1ψ〉 =

∫(λ − z)−1µϕ,ψ(dλ).

These measures enable the definition of functions f(A) of a self-adjointoperators, i.e., for any f ∈ L∞(R):

〈ϕ, f(A)ψ〉 :=

∫f(λ)µϕ,ψ(dλ) . (2.16)

This implicates the following calculus for functions f, g ∈ L∞(R) of self-adjoint operators:

(f + g)(A) = f(A) + g(A) , (fg)(A) = f(A) g(A) , f(A) = f(A)† ,

If f ≥ 0, then 〈ψ, f(A)ψ〉 ≥ 0 for all ψ ∈ H.

Moreover, ‖f(A)‖ = supλ∈σ(A) |f(λ)|.

Of particular importance are indicator functions,

1J(x) :=

1 if x ∈ J ,0 else. , J ⊂ R .

They define orthogonal projections PJ(A) := 1J(A). In fact, the map J 7→PJ(A) on the Borel sets J ⊂ R defines a projection-valued measure, i.e.

(1) PR(A) = 1 and P∅(A) = 0(2) If J =

⋃n Jn with Jn ∩ Jm = ∅ for n 6= m, then PJ(A) =∑

n PJn(A).

The support of this measure coincides with the spectrum,

σ(A) =λ ∈ R

∣∣P(λ−ε,λ+ε)(A) 6= 0 for all ε > 0.. (2.17)

The spectral theorem ensures that there is one-to-one correspondence be-tween self-adjoint operators and projections valued measures. This is sum-marized by (2.16) through the relation 〈ϕ , PJ(A)ψ〉 = µϕ,ψ(J).

Every Borel measure µ can be uniquely decomposed with respect tothe Lebesgue measure into three mutually singular parts,

µ = µpp + µsc + µac .

10 2. SOME MATHEMATICAL GROUNDWORK

Whereas µac is absolutely continuous with respect to Lebesgue measure, thesum of the first two terms is singular. It consists of a pure point component,µpp, and a singular continuous remainder, µsc.

Accordingly, for a self-adjoint operator A : D(A) → H one decom-poses the Hilbert space into closed subspaces,

H# :=ψ ∈ H |µψ = µ#

ψ

, # = pp, sc, ac

This decomposition turns out to be orthogonal,H = Hpp ⊕Hsc ⊕Hac andthe above subspaces are left invariant under the action of A.

In case of a bounded self-adjoint operator A : H → H, the restrictionto these subspaces defines the components of the spectrum,

σ#(A) := σ(A∣∣H#

), # = pp, sc, ac , (2.18)

which are called the pure point, singular continuous and absolutely con-tinuous spectrum. The point spectrum coincides with the closure of the setof eigenvalues,

σpp(A) = λ ∈ R |λ is an eigenvalue of A .

Let us conclude this subsection with two examples.

For a multiplication operator corresponding to a real-valued sequenceV ∈ `∞(Zd), one identifies the spectral measure corresponding to ψ ∈`2(Zd) as a weighted sum of Dirac measures:

µψ =∑x∈Zd

|ψ(x)|2 δV (x) ,

i.e., µψ = µppψ . The eigenvectors are given by the localized vectors δx ∈

`2(Zd),

δx(y) :=

1 if x ∈ J ,0 else. (2.19)

The corresponding eigenvalues are V (x) |x ∈ Zd, whose closure is thepure point spectrum σpp(V ).

For the Laplacian on `2(Zd) we use the unitarily equaivalent represen-tation as multiplication operator on L2([0, 2π]d) given in (2.9) to determinethe spectral measure:

〈ψ, f(∆)ψ〉 = 〈Fψ ,F f(∆)F−1Fψ〉 = 〈Fψ , f(F∆F−1

)Fψ〉

=

∫[0,2π]d|(Fψ) (k)|2 f(h(k)) dk =

∫f(λ)µψ(dλ) . (2.20)

Hence, µψ = µacψ for all ψ ∈ `2(Zd) and the spectrum is only absolutely

continuous, σac(−∆) = [0, 4d].2. lecture (incl.Herglotz App.)23.4.2010

2. SPECTRA AND DYNAMICS 11

2. Spectra and dynamics

The quantum time evolution generated by a self-adjoint operator Hon some Hilbert spaceH is given by a unitary group of operators,

ψ(t) := e−itHψ , t ∈ R . (2.21)

As we shall now see, the dynamical properties of the evolution are closelyrelated with the spectral characteristics of H .

2.1. Return probabilities and recurrence. For a relation between thetwo, consider the probability |〈ψ, ψ(t)〉|2 of a system in a state ψ ∈ Hat time t = 0 to return to itself at time t > 0. This probability may beexpressed in terms of the Fourier transform

µψ(t) :=

∫e−itEµψ(dE) = 〈ψ, ψ(t)〉 (2.22)

of the spectral measure of H associated with ψ.

More generally, quantum observables are given by self adjoint oper-ators A, with the range of possible outcomes of the measurement given byoperator’s spectrum. For a quantum system which at time t = 0 is in thestate ψ ∈ H and for which an observable A is measured at time t > 0 theprobability of observing a value in the range I ⊂ R, is given by:

Prob (A(t) ∈ I |ψ) := ‖PI(A)ψ(t)‖2

where PI(A) is the spectral projection operator. In the basic example withAa rank-one projection operator onto φ, the above reduces to the expression:∣∣〈φ, e−itHψ〉∣∣2 =

∣∣∣ ∫ e−itEµφ,ψ(dE)∣∣∣2 = |µψ,φ(t)|2 , (2.23)

which is again given by a Fourier transform, namely, of the spectral measureµψ,φ associated with φ, ψ ∈ H, which in general is complex.

It is a simple, but perhaps somewhat astounding observation that forfinite quantum systems, with dimH < ∞, the above expressions yieldquasi-periodic function of time. Denoting by (En) the eigenvalues of H ,and by (ξn) the corresponding eigenvectors:

Prob (A(t) ∈ I |ψ) =∑n,m

eit(En−Em) 〈ψ , ξn〉 〈ξn, PI(A) ξm〉〈ξm, ψ〉 .

The fact that as t → ∞ this function does not converge to an asymptoticvalue, and repeatedly assumes values arbitrarily close to ‖PI(A)ψ‖2 is rem-iniscent of the Poincare recurrence phenomenon of classical mechanics offinite systems. There, the time evolution is described by measure preserv-ing flows in a classical phase space. If taken literarily, the recurrence has

12 2. SOME MATHEMATICAL GROUNDWORK

the implication that if gas is released into a room from a bottle, then, withprobability one, there will be a moment when all the gas will be found backin the bottle. Of course, the recurrence times for a macroscopic system,for which the number of molecules can be estimated through the Avogadronumber NA ≈ 6 · 1023 mol−1, is so long that well before that the doorwill be opened, rendering the model insufficient. Actually, by the typicalrecurrence time for such an event far more greviuous deviations from theidealized description will occur (even the lab will no longer be there).

2.2. Theorems of Wiener and RAGE. Recurrence is avoidable in in-finite models, e.g., withH = `2(Zd), for which a key observation is that re-laxation in time does occur in the presence of continuous spectrum. In caseof absolutely continuous (ac) spectrum, the Riemann-Lebesgue lemmaimplies the pointwise decay of the Fourier transform of the ac componentof the spectral measure:

limt→∞

µacψ (t) = 0 . (2.24)

Relaxation in case of more general continuous measures follows from acelebrated theorem of N. Wiener.

Theorem 2.2 (Wiener) Let µ be a finite complex Borel measure on R.Then

limT→∞

1

T

∫ T

0

|µ(t)|2 dt =∑

E∈suppµpp

|µ(E)|2 . (2.25)

Proof: The assertion follows by Fubini’s theorem:

1

T

∫ T

0

|µ(t)|2 dt =

∫ ∫ (1

T

∫ T

0

ei(E−E′)t dt

)µ(dE ′)µ(dE)

=

∫ ∫eiT (E−E′) − 1

i(E − E ′)Tµ(dE ′)µ(dE) . (2.26)

Since the integrand is uniformly bounded by one and converges pointwiseto 10(E − E ′), the dominated convergence theorem implies

limT→∞

1

T

∫ T

0

|µ(t)|2 dt =

∫ ∫10(E − E ′)µ(dE ′)µ(dE)

=

∫µ(E)µ(dE) =

∑E∈suppµpp

|µ(E)|2 . (2.27)

2. SPECTRA AND DYNAMICS 13

Corollary 2.3 Let H be a self-adjoint and A be a compact operator onsome Hilbert spaceH. Then

limT→∞

1

T

∫ T

0

∥∥Ae−itHψ∥∥2dt = 0 (2.28)

for all initial states ψ ∈ Hc := Hsc ⊕Hac, for which the spectral measureof H is continuous.

Proof: In case A is a rank-one operator, i.e., A = |ρ〉〈φ|, the claim followsfrom Wiener’s theorem (Theorem 2.2) applied to the spectral measure µφ,ψ,cf. (2.23). As a consequence it also holds for finite-rank operators A. Anycompact operator A may in turn be approximated by finite-rank operators.More precisely, for every ε > 0 there exists Aε of finite rank such that‖A− Aε‖ < ε and hence

‖Aψ(t)‖ ≤ ‖Aε ψ(t)‖+ ‖(A− Aε)ψ(t)‖ ≤ ‖Aε ψ(t)‖+ ε ‖ψ‖ . (2.29)

The Cesaro average of the first term goes to zero in the long-time limit. Thiscompletes the proof since ε was arbitrary.

The Wiener theorem implies the following dynamical characteriza-tion of the subspaces of H associated with the continuous and pure pointpart of the generator H of the time evolution. The result is named afterD. Ruelle [131], W. Amrein, V. Gorgescu [10] and V. Enss [46].

Theorem 2.4 (RAGE) Let H be a self-adjoint operator on some HilbertspaceH andAn be a sequence of compact operators which converge stronglyto the identity. Then

Hc =

ψ ∈ H

∣∣ limn→∞

limT→∞

1

T

∫ T

0

∥∥Ane−itHψ∥∥2dt = 0

(2.30)

Hpp =

ψ ∈ H

∣∣ limn→∞

supt∈R

∥∥(1− An)e−itHψ∥∥ = 0

(2.31)

Proof: Ifψ ∈ Hc the previous corollary implies limT→∞1T

∫ T0‖Anψ(t)‖2 dt =

0 for every n. If ψ ∈ Hpp we claim

limn→∞

supt∈R‖(1− An)ψ(t)‖ = 0 . (2.32)

14 2. SOME MATHEMATICAL GROUNDWORK

For a proof we expand ψ into eigenfunctions (ξn) of H , and split the suminto the first N terms and a remainder with norm less than ε,

ψ =N∑k=1

〈ξk, ψ〉 ξk + φN , ‖φN‖ < ε . (2.33)

Since limn→∞ ‖(1− An)ξk‖ = 0 for every k, the first term contributes zeroto limit in (2.32). This completes the proof of (2.32) since ε may be chosenarbitrarily small.

Since every ψ may be uniquely decomposed into a component ψc ∈Hc and an orthogonal one ψpp ∈ Hpp, it remains to show that (i) the limitin (2.30) does not tend to zero for ψpp and, (ii) the limit in (2.31) does nottend to zero for ψc.

The proof of the first assertion relies on (2.32) which implies that

‖Anψpp(t)‖ ≥ |‖ψ‖ − ‖(1− An)ψpp(t)‖| > 0 (2.34)

for all t ≥ 0 and sufficiently large n.

The second assertion follows by contradiction. Suppose (2.32) ap-plies to ψc, then Corollary 2.3 implies

0 =

(limn→0

limT→∞

1

T

∫ T

0

‖(1− An)ψc(t)‖2 dt

) 12

≥ ‖ψc‖ −(

limn→0

limT→∞

1

T

∫ T

0

‖Anψc(t)‖2 dt

) 12

= ‖ψc‖ (2.35)

which contradicts ψc 6= 0.

In caseH = `2(Zd) one may pick (An) as indicator functions of balls.Eqs. (2.30) and (2.31) then express the fact that extended states ψc ∈ Hc

are characterized by eventually leaving every ball,

limR→∞

limT→∞

1

T

∫ T

0

∑|x|≤R

|〈δx , ψc(t)〉|2 dt = 0 , (2.36)

and bound states ψpp ∈ Hpp by being forever confined to a sufficiently largeball

limR→∞

supt∈R

∑|x|≤R

|〈δx , ψpp(t)〉|2 = 0 . (2.37)

Similar interpretations exist for quantum systems on a continuous manifold,e.g., H = L2(Rd). In this context, it is worth noting that Corollary 2.3 aswell as the RAGE theorem can be extended to operators A which are rela-tively compact with respect to H . We refer the interested reader to [146].

2. SPECTRA AND DYNAMICS 15

2.3. Theorems of Strichartz and Last. Wiener’s theorem implies thatupon Cesaro average the return probability converges to zero for states withpurely continuous spectral measure. More can be said about the rate ofconvergence for the following class of continuous measures.

Definition 2.5 Let µ be a finite Borel measure on R and α ∈ [0, 1]. Then µis called uniformly α Holder continuous (UαH) if there is some C < ∞such that for all intervals I with |I| < 1 one has

µ(I) ≤ C |I|α . (2.38)

In case α ∈ (0, 1) the above measures are singular continuous. Inphysical applications such measures occur in models with quasi-period po-tentials. Prominent examples are the Harper Hamiltonian at criticality or theFibbonacci Hamiltonian. For these examples, the decay rate in the follow-ing theorem has been argued for by the authors of [64], who were apparentlyunaware of Strichartz’ work [143].

Theorem 2.6 (Strichartz) Let µ be UαH for some α. Then there existssome C <∞ such that for all f ∈ L2(R, µ)

1

T

∫ T

0

∣∣∣fµ(t)∣∣∣2 dt ≤ C

Tα

∫|f(E)|2µ(dE) . (2.39)

Proof: We estimate the indicator function of [0, T ] by a Gaussian, andcompute the Fourier transform

1

T

∫ T

0

∣∣∣fµ(t)∣∣∣2 ≤ e

T

∫Re−

t2

T2

∣∣∣fµ(t)∣∣∣2 dt

≤ e√π

∫|f(E)|2

[∫e−

T2

4(E−E′)2µ(dE ′)

]µ(dE) .

(2.40)

For T > 1, the integral in brackets is estimated using UαH of µψ∞∑n=0

∫nT≤|E−E′|<n+1

T

e−T2

4(E−E′)2 µ(dE ′) ≤ 2C

Tα

∞∑n=0

e−n2

4

which completes the proof.

Based on Strichartz’s theorem, Last [107] proved the following re-finement of Corollary 2.3.

16 2. SOME MATHEMATICAL GROUNDWORK

Theorem 2.7 (Last) LetH be a self-ajoint operator on some Hilbert spaceH and assume the spectral measure of ψ with respect toH is UαH for someα. Then there is C < ∞ such that for all p ≥ 1 and T > 0, and compactoperators A with tr |A|2p <∞:

limT→∞

1

T

∫ T

0

∥∥Ae−itHψ∥∥2dt ≤

(C

Tαtr |A|2p

) 1p

. (2.41)

Proof: We employ the canonical decomposition of compact opertors [128],i.e, there exist an > 0 and orthornomal basis φn and ρn in H suchthat A =

∑∞n=0 an|ρn〉〈φn|. Abbreviating the Cesaro average of a function

g ∈ L1 by AvT (g) := T−1∫ T

0g(t) dt, Holder’s inequality yields

AvT

(‖Aψ(·)‖2) =

∞∑n=0

a2n AvT

(|〈φn, ψ(·)〉|2

)≤

(∞∑n=0

a2pn

)1/p( ∞∑n=0

[AvT

(|〈φn, ψ(·)〉|2

)]q)1/q

.

(2.42)

Strichartz’ theorem guarantees 〈|〈φ, ψ(·)〉|2〉T ≤ CT−α which yields thefollowing upper bound on the above sum:

∞∑n=0

[AvT

(|〈φn, ψ(·)〉|2

)]q ≤( C

Tα

)q−1 ∞∑n=0

AvT

(|〈φn, ψ(·)〉|2

)=

(C

Tα

)q−1

. (2.43)

Inserting (q − 1)/q = 1/p completes the proof. Lecture 3 (excl.Strichatz/Last)28.4.2010

3. Appendix: Herglotz functions

A holomorphic function F : C+ → C+ is called a Herglotz func-tion. Such functions naturally arise as the Borel-Stieltjes transformationof finite Borel measures µ on R:

F (z) :=

∫µ(dλ)

λ− z, z ∈ C+ . (2.44)

The representation theorem for Herglotz functions reveals their relation.

NOTES 17

Proposition 2.8 The Borel-Stieltjes transform of any finite Borel mea-sure µ is a Herglotz function satisfying

|F (z)| ≤ µ(R)

Im z. (2.45)

Moreover, the measure µ can be recovered from F using:

1

2(µ ((λ1, λ2)) + µ ([λ1, λ2])) = lim

ε↓0

1

π

∫ λ2

λ1

ImF (λ+ iε) dλ . (2.46)

Conversely, if F is a Herglotz function satisfying

|F (z)| ≤ C

Im z, z ∈ C+ , (2.47)

then there exists a unique Borel measure µ with µ(R) ≤ C and F its Borel-Stieltjes transformation.

A proof can be found in [146].

Notes

The material in the first section was taken from [128, 153]. The con-struction of the spectral measure using the representation theorem for Her-glotz functions is discussed in detailed in [146, 41]. The latter also includesmore material on Herglotz functions.

The material on the relation of spectra and dynamics was taken from[146] and [107]. Generalizations of such statements dealing with contin-uum models,H = L2(Rd) or α-continuous spectral measures can be foundin [26, 107]. Further relations concerning the speading of generalized eigen-functions and dynamical properties have been obtained in [83].

CHAPTER 3

Modeling disorder

1. Stochastic processes

As will become clear in the course of this section, random potentialsare most naturally given in terms of ergodic stochastic processes on someprobability space. We first recall some basic notions and facts, again in asomewhat informal manner. The reader is invited to consult the literature inthe notes, in particular [65, 68].

1.1. Basic notions. Any family (X(ξ))ξ∈I of random variables ,X(ξ) :

Ω → R, ω 7→ X(ξ, ω), on some probability space (Ω,A,P) is called astochastic process with index set I . We will mainly be concerned with thecase I = Zd.

An example fitting this framework are random variables attached tothe sites of a lattice, ξ ∈ Zd. In this situation, the probability space iscanonically realized as Ω = RZd with the Borel setsA = B(RZd

) serving asthe σ-algebra. The latter is the smallest σ-algebra generated by the cylindersets

Z(Iξ1 , . . . , Iξn) := ω |ωξk ∈ Iξk for all k = 1, . . . , n ,

with Iξk ⊂ R Borel. A celebrated theorem of Kolmogorov ensures thatany probability measure can be uniquely characterized by the marginals,P (Z(Iξ1 , . . . , Iξn)), provided they satisfy some compatibility conditions. Inthis setup, the family

X(ξ, ω) = ωξ ,

indexed by ξ ∈ Zd, then forms a stochastic process.

The case of independent and identically distributed (iid) randomvariables corresponds to P given by the product of a Borel measure p on R.Homogeneity of the situation is expressed through the family of shifts onΩ = RZd defined by

(Sx ω)ξ := ωξ−x , ξ, x ∈ Zd . (3.1)

19

20 3. MODELING DISORDER

They leave the probability of any cylinder set Z(Iξ1 , . . . , Iξn) invariant :

P (Z(Iξ1 , . . . , Iξn)) =n∏j=1

p(Iξj ) = P(S−1x Z(Iξ1 , . . . , Iξn)

).

Since this property extends to any set in B(RZd), the shifts are an example

of a family of measure preserving transformations, i.e.,

P(S−1x A) = P(A) for all A ∈ A.

The stochastic process X(ξ, ω) = ωξ corresponding to iid random variablesis then identified to be homogeneous (or stationary).

Definition 3.1 A stochastic process (X(ξ))ξ∈Zd is called homogenous ifthere exists a family of measure preserving transformations (Tx)x∈Zd suchthat

X(ξ, Txω) = X(ξ − x, ω) . (3.2)

Homogeneity is a physically plausible requirement. In the contextof disordered solids such as alloys it is the natural generalization of theperiodicity of the ideal cystal. Whereas the latter requires strict translationinvariance, the covariance (3.2) is a statement of translation invariance onthe average: shifted realizations are equally likely.

1.2. Ergodicity. However, in order to discover determinism in ran-domness, the stronger notion of ergodicity has proven to be vital.

Definition 3.2 A family of measure preserving transformations (Tx)x∈I ona probability space (Ω,A,P) is called ergodic if any event A ∈ A which isinvariant, i.e., T−1

x A = A for all x ∈ I , has probability zero or one.

The importance of ergodicity in connection with determinism is fur-ther exemplified in the following lemma.

Lemma 3.3 Any random variable, X : Ω→ R ∪ ∞, which is invariantunder a family (Tx)x∈I of ergodic transformations, i.e.,

X(Txω) = X(ω) , for all x ∈ I ,

is almost surely constant, i.e., there exists c ∈ R ∪ ∞ such that

P (X = c) = 1 .

1. STOCHASTIC PROCESSES 21

Proof: The distribution function t 7→ P (X ≤ t) is monotone increasingon (−∞,∞] and due to invariance of X takes values in the set 0, 1 only.A candidate for the constant thus is

c := inf t ∈ (−∞,∞] | P (X ≤ t) = 1 ,

which by construction satisfies

P (X < c) = P( ⋃n∈N

X ≤ c− 1

n

)= 0 .

This implies the assertion: P (X = c) ≥ P (X ≤ c)− P (X < c) = 1.

It is natural to refer to homogeneous stochastic processes whose mea-sure preserving family is ergodic as ergodic themselves.

Definition 3.4 A stochastic process (X(ξ))ξ∈Zd is called ergodic if thereexists a family of ergodic transformations (Tx)x∈Zd such that (3.2) holds.

Example 3.5 The canonical realization of iid random variables,X(ξ, ω) =ωξ, ξ ∈ Zd, together with the shifts (3.1). For a proof of this assertion weuse the fact that the shift operators are even mixing, i.e., for any pair ofevents A,B ∈ A:

P(A ∩ S−1

x B)−→ P(A) P(B) as |x| → ∞. (3.3)

In our situation, this is easily checked for cylinder sets and hence appliesto all events in A = B(RZd

). If E ∈ A is invariant, then (3.3) impliesP(E)2 = P(E) and hence P(E) ∈ 0, 1.

Another example fitting the framework are almost-periodic func-tions.

Example 3.6 If Ω = [0, 2π),A = B([0, 2π)) and P is uniform distributionon the torus, then for any irrational α ∈ (0, 1)

X(ξ, ω) := cos(2πα ξ + ω) , ξ ∈ Z,

defines an ergodic stochastic process. The family of ergodic transforma-tions is given by the irrational rotations, Txω := ω − 2πα x mod 2π. Wewill leave the proof as an exercise, cf. [105].

One of the most important results in ergodic theory is Birkhoff’s er-godic theorem. It states that the law of large numbers applies to ergodicstochastic processes when sampled over boxes ΛL := x ∈ Zd | |x|∞ < L.

22 3. MODELING DISORDER

Proposition 3.7 (Birkhoff) Let (X(ξ))ξ∈Zd be an ergodic stochastic pro-cess and suppose that X(0) ∈ L1(Ω,P). Then P-almost surely:

limL→∞

1

|ΛL|∑ξ∈ΛL

X(ξ) = E [X(0)] . (3.4)

We will encounter several applications of this result later on. A mod-ern proof and more information can be found in [100].

2. Ergodic operators

2.1. Definitions. Our next focus will be operator-valued maps definedon some probability space (Ω,A,P) which take values in the set of self-adjoint operators on a separable Hilbert space H. A first concern is theirmeasurability.

Definition 3.8 We will call maps ω 7→ H(ω) into to set of self-adjointoperators weakly measurable if the functions ω 7→ 〈ϕ , f(H(ω))ψ〉 aremeasurable for all f ∈ L∞(R) and all ϕ, ψ ∈ H.

It suffices to check this property for all f(x) = (x − z)−1 with z ∈C\R, cf. [25, 72]. We will not dwell on this property in the following.

Our main interest concerns the case in which the probability spacecarries a family of ergodic transformations.

Definition 3.9 Let (Ω,A,P) be a probability space with a family (Tx)x∈Iof ergodic transformations. Any weakly measurable map ω 7→ H(ω) intothe self-adjoint operators on a separable Hilbert spaceH is called a familyof ergodic operators if H(Txω) is unitarily equivalent to H(ω).

To give an example we note that the shifts (Sx)x∈Zd defined in (3.1)form a representation of the group of lattice translations on the probabilityspace. This group may also be realized by

(Uxψ) (ξ) := ψ(ξ − x) (3.5)

as unitary operators (Ux)x∈Zd on the Hilbert space `2(Zd).For any any ergodic stochastic processes (V (x))x∈Zd the correspond-

ing multiplication operator V (ω) on `2(Zd) transforms covariantly:

Ux V (ω)U−1x = V (Sxω) . (3.6)

Since the Laplacian ∆ is invariant under Ux, any random Schrodinger oper-ator of the form

H(ω) = −∆ + V (ω) (3.7)

2. ERGODIC OPERATORS 23

constitutes a family of ergodic operators on `2(Zd) for which

UxH(ω)U−1x = H(Sxω) . (3.8)

Definition 3.10 Any family of ergodic operators H(ω) in `2(Zd) satisfy-ing the covariance relation (3.8) with the translation operators Ux definedin (3.5) will be called a standard ergodic operator.

An example for a non-standard ergodic operator are Schrodinger op-erators with magnetic fields. The simplest one is the Hofstadter operator[59] perturbed by a random potential, i.e., an operator of the form

H(ω) = −∆A + V (ω)

acting in `2(Z2), where the magnetic Laplacian ∆A corresponds to a con-stant magnetic field B ∈ [0, 2π):

(∆Aψ) (x) = e−iBx2ψ(x1 + 1, x2) + eiBx2ψ(x1 − 1, x2)

+ ψ(x1, x2 + 1) + ψ(x1, x2 − 1)− 4ψ(x) . (3.9)

Physically, −∆A generates the quantum dynamics of a particle on Z2 sub-ject to a constant perpendicular magnetic field of strength B. Clearly, thisoperator is not invariant under all Ux. We will return to its interesting spec-tral features in a later chapter. Lecture 4

30.4.10

2.2. Deterministic spectra. For ergodic operators certain questions havea predictable answer. Examples are the spectra and the density of states.The latter will be the topic of the next chapter. Concerning the spectra, thefollowing result dates back to L. Pastur [122] with extensions by H. Kunz,B. Souillard [103] and W. Kirsch and F. Martinelli [72].

Theorem 3.11 (Pastur) The spectrum of a family of ergodic operatorsH(ω) is P-almost surely non-random, i.e., there is Σ ⊂ R such that

P (σ(H) = Σ) = 1 . (3.10)

The same applies to any subset in the Lebesgue-decomposition of the spec-trum, i.e., there are Σ# ⊂ R, # = ac, sc, pp, such that

P(σ#(H) = Σ#

)= 1 , # = ac, sc, pp .

The above non-random set Σ is called the almost-sure spectrum as-sociated with H(ω) and likewise for the spectral components.

Let us stress that the set of eigenvalues of a family of ergodic opera-tors is usually heavily dependent on the realization ω. Only the closure of

24 3. MODELING DISORDER

this set is deterministic. Just think of the random multiplication operatorcorresponding to iid random variables on Zd.

Proof of (3.12): For any E1, E2 ∈ R, the functions

ω 7→ XE1,E2(ω) := dim RangeP(E1,E2) (H(ω)) ,

defined on the underlying probability space (Ω,A,P) are:

(1) measurable, i.e., random variables taking values in [0,∞].This is most easily seen by identifying them as (possibly divergent)non-negative series,XE1,E2(ω) =

∑∞k=1〈ψk , P(E1,E2) (H(ω)) ψk〉,

given in terms of an arbitrary orthonormal basis (ψk) inH.(2) invariant under the action of the ergodic transformations (Tx)x∈I ,

dim RangeP(E1,E2) (H(Txω)) = dim RangeP(E1,E2) (H(ω)) ,

by unitary equivalence.

Lemma 3.3 then ensures that there are constants cE1,E2 ∈ [0,∞] such thatP (XE1,E2 = cE1,E2) = 1. The characterization (2.17) of the spectrum asthe support of the spectral projections identifies

Σ = E ∈ R |For all E1, E2 ∈ Q with E1 < E < E2: cE1,E2 > 0

as a possible candidate for the almost-sure spectrum. The reason for choos-ing intervals with rational endpoints becomes apparent from the fact thatthe event ⋂

E1,E2∈QcE1,E2

>0

XE1,E2 = cE1,E2 ⊂ σ(H) = Σ ,

still has probability one, since it is a countable intersection of such events.

The same proof idea also applies to the components of the spectrum. Theonly subtle point is the measurability of the spectral projections associatedto the restriction of H(ω) to the subspaces H#, # = ac, sc, pp. We referthe interested reader to [72, 103].

2.3. Determining the spectrum. Determining the almost-sure spectralcomponents of an ergodic operator is in general a hard question which muchof the later chapters are devoted to. In contrast, the almost-sure spectrum ismuch easier to identify. One strategy is based on the construction of Weylsequences, i.e., sequences of approximate normalized eigenfunctions.

2. ERGODIC OPERATORS 25

Proposition 3.12 (Weyl criterion) Let A : D(A) → H be a self-adjointoperator on a Hilbert spaceH. Then

σ(A) =λ ∈ R | ∃ (ψn) ⊂ D(A), ‖ψn‖ = 1 : lim

n→∞‖(A− λ)ψn‖ = 0

.

A more general statement and a proof can be found in [146].

Let us illustrate the strategy to determine the spectrum in case of astandard ergodic operator of the form (3.7). Associated to the random po-tential (V (x))x∈Zd defined on some probability space (Ω,A,P) are two no-tions of the support of the probability distribution:

supp1(P) := λ ∈ R | ∀ε > 0 : P (|V (x)− λ| < ε) > 0supp2(P) :=

λ ∈ R | ∀ε > 0 ,Λ ⊂ Zd finite :

P(

supx∈Λ|V (x)− λ| < ε

)> 0

The first is simply the support of the distribution of the single random vari-able V (0). It is the relevant notion if one is interested in the closure ofthe range of the random function x 7→ V (x, ω). It is left as an exerciseto the reader to show that for any ergodic process (V (x))x∈Zd one has thealmost-sure equality:

supp1(P) = V (x, ω) |x ∈ Zd . (3.11)

The support supp2(P) is always contained in the first, and the inclusion isstrict if there is substantial correlation between sites. The second notionis most relevant for determining the almost-sure spectrum with the help ofWeyl sequences. For λ ∈ supp2(P) typical realizations of the random pro-cess exhibit arbitrarily large regions where x 7→ V (x, ω) is almost constant.It is those regions which accommodate approximate eigenstates of energiesin the range λ + [0, 4d]. This summarizes the proof idea of the followingresult by H. Kunz and B. Souillard [103] which was generalized in [73].

Theorem 3.13 (Kunz/Souillard) For the family H(ω) = −∆ + V (ω) ofstandard ergodic operators on `2(Zd)

[0, 4d] + supp2(P) ⊂ σ (H(ω)) ⊂ [0, 4d] + supp1(P) , (3.12)

for P-almost all ω.

Proof: The proof of the second inclusion is based on (2.13) in which weset A = H(ω) and B = V (ω) + 2d such that ‖A − B‖ ≤ 2d, cf. (2.10).

26 3. MODELING DISORDER

Since σ(B) = σ(V (ω)) + 2d, the assertion follows from

σ(V (ω)) = V (x, ω) |x ∈ Zd = supp1(P) ,

cf. (3.11).

For a proof of the first inclusion, we pick λ ∈ σ(−∆) = [0, 4d] and acorresponding Weyl sequence (ϕn) ⊂ `2(Zd). In fact, by a suitable smoothtruncation we may even assume that ϕn has compact support. For eachn ∈ N we consider the event

Ωn :=

There is j ∈ Zd: sup

x∈suppϕn

|V (x+ j)− µ| < 1n

.

If µ ∈ supp2(P) these events have a non-zero probability and are invariantunder the shifts (Sx)x∈Zd . By ergodicity we conclude that they are almostcertain, P (Ωn) = 1. Therefore their intersection

Ω0 :=⋂n∈N

Ωn

is almost certain too. By construction, for any ω ∈ Ω0 there is a sequence(jn) ⊂ Zd such that

ψn := ϕn (· − jn)

is a Weyl sequence for H(ω) and λ+ µ ∈ σ(H(ω)). This follows from

‖(H(ω)− λ− µ)ψn‖ ≤ ‖(−∆− λ)ψn‖+ ‖(V (ω)− µ)ψn‖≤ ‖(−∆− λ)ϕn‖+ sup

x∈suppϕn

|V (x+ jn)− µ| ‖ϕn‖ ,

which goes to zero as n→∞.

In case of iid random variables, it is an easy exercise to show thatsupp1(P) = supp2(P). Therefore the almost-sure spectrum may be deter-mined explicitly.

Corollary 3.14 In case (V (x))x∈Zd are iid with common distribution p(dv) :=P (V (x) ∈ dv),

σ (H(ω)) = [0, 4d] + supp p

for P-almost all ω.Lecture 55.5.10

Notes

Most of material of this chapter was taken from [65, 68].

We have skipped examples of stochastic processes with continuousindex set I = Rd. For such a notion of homogeneity and ergodicity may

NOTES 27

be defined by implementing either the full translation group or subgroupof lattice translations on the probability space. Important examples of suchprocesses like Gaussian processes are discussed in [65, 25, 123]. In physicssuch processes feature in models of amorphous material [110, 109].

The measurability issue is discussed in more detail in [25, 72].

Our definition of ergodic operators is slightly more general (and notstandard) in comparison to what is discussed there. In fact, our notion ofstandard ergodic operators agree with the usual ergodic operators. In thestandard case, much more can be said: the range of the spectral projectionsare almost surely either zero or infinity. Therefore the discrete spectrum isempty. For details see [65, 25].

The last section is a baby version of a more general framework todetermine the spectra of ergodic operators. In fact, the local supremumnorm arising in the definition of supp2(P) can be used to render the canon-ical choice of the probability space, Ω = RZd , into a polish space, i.e., acomplete metric space with a countable dense subset. The set supp2(P) isthen identified as the set of constant functions within the support of P withrespect to this topology. For more information, see [65, 25, 73].

Bibliography

[1] M. Aizenman: Localization at weak disorder: Some elementary bounds. Rev. Math.Phys. 6, 1163-1182 (1994).

[2] M. Aizenman, A. Elgart, S. Naboko, J. Schenker, G. Stolz: Moment Analysis forLocalization in Random Schrodinger Operators, Preprint mp-arc 03-377

[3] M. Aizenman, G. M. Graf: Localization bounds for an electron gas. J. Phys. A,Math. Gen. 31, No.32, 6783-6806 (1998).

[4] M. Aizenman, S. Molchanov: Localization at large disorder and at extreme energies:An elementary derivation. Commun. Math. Phys. 157, 245-278 (1993).

[5] M. Aizenman, J. H. Schenker, R. M. Friedrich, D. Hundertmark: Finite-volumefractional-moment criteria for Anderson localization. Commun. Math. Phys. 224,No.1, 219-253 (2001).

[6] M. Aizenman, R. Sims, S. Warzel: Stability of the absolutely continuous spectrumof random Schrodinger operators on tree graphs, Probab. Theory Relat. Fields 136:363-394 (2006)

[7] M. Aizenman, R. Sims, S. Warzel: Absolutely continuous spectra of quantum treegraphs with weak disorder, Commun. Math. Phys. 264: 371-389 (2006)

[8] M. Aizenman, S. Warzel, The canopy graph and level statistics for random operatorson trees, Math. Phys. Anal. Geom. 9: 291-333 (2007)

[9] M. Aizenman, S. Warzel: Localization bounds for multiparticle systems, Commun.Math. Phys. 290: 903-934 (2009).

[10] W. Amrein, V. Georgescu: On the characterization of bound states and scatteringstates. Helv. Phys. Acta 46, 635-658 (1973).

[11] P. W. Anderson: Absence of diffusion in certain random lattices. Phys. Rev. 109.1492-1505 (1958).

[12] J. Avron, B. Simon: Almost periodic Schrodinger operators. II: The integrated den-sity of states. Duke Math. J. 50, 369-391 (1983).

[13] J. M. Barbaroux, J. M. Combes, P. D. Hislop: Localization near band edges forrandom Schrodinger operators. Helv. Phys. Acta 70, No.1-2, 16-43 (1997).

[14] H. Bauer: Measure and integration theory, de Gruyter (2001).[15] Yu. M. Berezanskij: On an eigenfunction expansion for selfadjoint operators. Am.

Math. Soc., Transl., II. Ser. 93, 227-238 (1970).[16] S. de Bievre, F. Germinet: Dynamical localization for discrete and continuous ran-

dom Schrodinger operators. Commun. Math. Phys. 194, 323-341 (1998).[17] M. Biskup, W. Konig: Long-time tails in the parabolic Anderson model, Ann. of

Prob. 29, No.2, 636–682 (2001).[18] V. L. Bonch-Bruevich, R. Enderlein, B. Esser, R. Keiper, A. G. Mironov, I. P. Zvya-

gin: Elektronentheorie ungeordneter Halbleiter. Berlin: VEB Deutscher Verlagder Wissenschaften (1984).

29

30 BIBLIOGRAPHY

[19] J.-M. Bouclet, F. Germinet, A. Klein, J. Schenker: Linear response theory formagnetic Schrodinger operators in disordered media. J. Funct. Anal. 226, 301–372(2005)

[20] J. Bourgain: On random Schrodinger operators on Zd. Discrete Contin. Dyn. Syst.8, 1–15 (2002).

[21] J. Bourgain, C. Kenig: On localization in the continuous Anderson-Bernoulli modelin higher dimension. Invent. Math. 161, No.2, 389-426 (2005).

[22] K. Broderix, D. Hundertmark, W. Kirsch, H. Leschke: The fate of Lifshitz tails inmagnetic fields J. Stat. Phys. 80, 1-22 (1995).

[23] R. Carmona: Exponential localization in one dimensional disordered systems. DukeMath. J. 49, 191-213 (1982).

[24] R. Carmona, A. Klein, F. Martinelli: Anderson localization for Bernoulli and othersingular potentials. Commun. Math. Phys. 108, 41-66 (1987).

[25] R. Carmona, J. Lacroix: Spectral theory of random Schrodinger operators.Probability and Its Applications. Basel etc.: Birkhauser Verlag (1990).

[26] J.-M. Combes: Connections between quantum dynamics and spectral propertiesof time-evolution operators. W.F. Ames, E.M. Harrel, J.V. Herod (eds.) Differen-tial equations with applications to mathematical physics, 59–68, Academic Press,Boston, MA, 1993.

[27] J. M. Combes, P. D. Hislop: Localization for some continuous, random Hamiltoni-ans in d-dimensions. J. Funct. Anal. 124, No.1, 149-180 (1994).

[28] J. M. Combes , P. D. Hislop: Landau Hamiltonians with random potentials: Local-ization and the density of states. Commun. Math. Phys. 177, No.3, 603-629 (1996).

[29] J. M. Combes, P. D. Hislop, F. Klopp: An optimal Wegner estimate and its ap-plication to the global continuity of the integrated density of states for randomSchrodinger operators, Preprint, http://www.arxiv.org/pdf/math-ph/0605029

[30] J. M. Combes, P. D. Hislop, E. Mourre: Spectral averaging, perturbation of singularspectra, and localization. Trans. Am. Math. Soc. 348, No.12, 4883-4894 (1996).

[31] J. M. Combes, L. Thomas: Asymptotic behaviour of eigenfunctions for multiparticleSchrodinger operators. Commun. Math. Phys. 34, 251-270 (1973).

[32] W. Craig, B. Simon: Log Holder continuity of the integrated density of states forstochastic Jacobi matrices. Commun. Math. Phys. 90, 207–218 (1983).

[33] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon: Schrodinger operators, withapplication to quantum mechanics and global geometry. Texts and Monographsin Physics. Berlin etc.: Springer-Verlag. (1987).

[34] D. Damanik, R. Sims, G. Stolz: Localization for one-dimensional, continuum,Bernoulli-Anderson models. Duke Math. J. 114, No.1, 59-100 (2002).

[35] D. Damanik, P. Stollmann: Multi-scale analysis implies strong dynamical localiza-tion. Geom. Funct. Anal. 11, 11-29 (2001).

[36] E. B. Davies: Heat kernels and spectral theory. Cambridge Tracts in Mathematics,92. Cambridge etc.: Cambridge University Press. (1989).

[37] E. B. Davies: Spectral theory and differential operators. Cambridge studies inadvanced mathematics, 42. Cambridge: Cambridge University Press. (1995).

[38] R. del Rio, S. Jitomirskaya, Y. Last, B. Simon: Operators with singular continu-ous spectrum IV: Hausdorff dimensions, rank one perturbations and localization. J.d’Analyse Math. 69, 153–200 (1996).

[39] F. Delyon, B. Souillard: Remark on the continuity of the density of states of ergodicfinite difference operators. Commun. Math. Phys. 94, 289-291 (1984).

BIBLIOGRAPHY 31

[40] F. Delyon, Y. Levy, B. Souillard: Anderson localization for multi-dimensional sys-tems at large disorder or large energy. Commun. Math. Phys. 100, 463-470 (1985).

[41] M. Demuth, M. Krishna, Determining spectra in quantum theory, Progress inMathematical Physics, 44, Birkhauser, Boston (2005).

[42] M. D. Donsker, S. R. S. Varadhan: Asymptotic for the Wiener sausage. Commun.Pure Appl. Math. 28, 525-565 (1975).

[43] H. von Dreifus: On the effects of randomness in ferromagnetic models andSchrodinger operators, NYU PhD thesis (1987).

[44] H. von Dreifus, A. Klein: A new proof of localization in the Anderson tight bindingmodel. Commun. Math. Phys. 124, No.2, 285-299 (1989).

[45] M. S. P. Eastham: The spectral theory of periodic differential equations. Texts inMathematics. Edinburgh-London: Scottish Academic Press. (1973).

[46] V. Enss: Asymptotic completeness for quantum-mechanical potential scattering, I.Short range potentials. Commun. Math. Phys. 61, 285-291 (1978).

[47] L. Erdos: Lifschitz tail in a magnetic field: The nonclassical regime. Probab. TheoryRelat. Fields 112, No.3, 321-371 (1998).

[48] A. Figotin, A. Klein: Localization of classical waves I: Acoustic waves. Commun.Math. Phys. 180, 439-482 (1996).

[49] R. Froese, D. Hasler, W. Spitzer: Absolutely continuous spectrum for the Andersonmodel on a tree: a geometric proof of Klein’s theorem, Preprint math-ph/0511050

[50] J. Frohlich, F. Martinelli, E. Scoppola, T. Spencer: Constructive proof of localizationin the Anderson tight binding model. Commun. Math. Phys. 101, 21-46 (1985).

[51] J. Frohlich, T. Spencer: Absence of diffusion in the Anderson tight binding modelfor large disorder or low energy. Commun. Math. Phys. 88, 151-184 (1983).

[52] F. Germinet, P. Hislop, A. Klein: On localization for the Schrodinger operator witha Poisson random potential, Preprint math-ph/0506012

[53] F. Germinet, A. Klein: Bootstrap multiscale analysis and localization in randommedia. Commun. Math. Phys. 222, 415–448 (2001).

[54] F. Germinet, A. Klein: A characterization of the Anderson metal-insulator transporttransition. Duke Math. J. 124, 309-351 (2004).

[55] F. Germinet, A. Klein, J. Schenker: Dynamical delocalization in random LandauHamiltonians. Annals Math., to appear.

[56] I. Goldsheid, S. Molchanov, L. Pastur: A pure point spectrum of the stochastic one-dimensional Schrodinger equation. Funct. Anal. Appl. 11, 1-10 (1977).

[57] G. M. Graf: Anderson localization and the space-time characteristic of continuumstates. J. Stat. Phys. 75, 337-346 (1994).

[58] I. Guarneri: On an estimate concerning quantum diffusion in the presence of a fractalspectrum. Europhys. Lett. 21, 729–733 (1993).

[59] D. Hofstadter: Energy levels and wave functions of Bloch electrons in rational andirrational magnetic field. Phys. Rev. B 14, 2240–2249 (1976)

[60] D. Hundertmark, W. Kirsch: Spectral theory for sparse potentials. CMS Conf. Proc.28, 213-238 (2000).

[61] S. Jitomirskaya: Ergodic Schrodinger operators (on one foot), in: F. Gesztesy, P.Deift, C. Galvez, P. Perry, W. Schlag (Editors): Spectral Theory andMathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, 613–647 (2007).

[62] S. Jitomirskaya, H. Schulz-Baldes, G. Stolz: Delocalization in random polymermodels. Commun. Math. Phys. 233, 27–48 (2003).

32 BIBLIOGRAPHY

[63] T. Kato: Perturbation theory for linear operators. 2nd corr. print. of the 2nd ed.Grundlehren der Mathematischen Wissenschaften, 132. Berlin etc.: Springer-Verlag(1984).

[64] R. Ketzmerick, G. Petschel, and T. Geisel: Slow decay of temporal correlations inquantum systems with Cantor spectra. Phys. Rev. Lett. 69, 695 - 698 (1992).

[65] W. Kirsch: Random Schrodinger operators. Schrodinger operators, Proc. Nord.Summer Sch. Math., Sandbjerg Slot, Sonderborg/Denmark 1988, Lect. Notes Phys.345, 264-370 (1989).

[66] W. Kirsch: Wegner estimates and Anderson localization for alloy-type potentials.Math. Z. 221, No.3, 507-512 (1996).

[67] W. Kirsch: Scattering theory for sparse random potentials. Random Oper. Stoch.Equ. 10, 329-334 (2002).

[68] W. Kirsch: An invitation to random-Schrodinger operators, Panoramas et Syntheses25, 1-119

[69] W. Kirsch, M. Krishna, J. Obermeit: Anderson model with decaying randomness:Mobility edge. Math. Z. 235, 421-433 (2000).

[70] W. Kirsch, O. Lenoble, L. Pastur: On the Mott formula for the ac conductivity andbinary correlators in the strong localization regime of disordered systems. J. Phys.A, Math. Gen. 36, No.49, 12157-12180 (2003).

[71] W. Kirsch, F. Martinelli: On the density of states of Schrodinger operators with arandom potential. J. Phys. A 15, 2139-2156 (1982).

[72] W. Kirsch, F. Martinelli: On the ergodic properties of the spectrum of general ran-dom operators. J. Reine Angew. Math. 334, 141-156 (1982).

[73] W. Kirsch, F. Martinelli: On the Spectrum of Schrodinger Operators with a RandomPotential, Commun. Math. Phys. 85, 329-350 (1982).

[74] W. Kirsch, F. Martinelli: Large deviations and Lifshitz singularity of the integrateddensity of states of random Hamiltonians. Commun. Math. Phys. 89, 27-40 (1983).

[75] W. Kirsch, B. Metzger: The Integrated Density of States for Random SchrodingerOperators, in: F. Gesztesy, P. Deift, C. Galvez, P. Perry, W. Schlag (Editors): Spec-tral Theory andMathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, 649–696 (2007).

[76] W. Kirsch, P. Muller: Spectral properties of the Laplacian on bond-percolationgraphs. Math. Z. 252, 899–916 (2006).

[77] W. Kirsch, B. Simon: Lifshitz tails for periodic plus random potential. J. Stat. Phys.42, 799 - 808 (1986).

[78] W. Kirsch, P. Stollmann, G. Stolz: Localization for random perturbations of periodicSchrodinger operators. Random Oper. Stoch. Equ. 6, No.3, 241-268 (1998).

[79] W. Kirsch, P. Stollmann, G. Stolz: Anderson localization for random Schrodingeroperators with long range interactions. Commun. Math. Phys. 195, 495-507 (1998).

[80] W. Kirsch, S. Warzel: Lifshits tails caused by anisotropic decay: the emergence ofa quantum-classical regime. Math. Phys. Anal. Geometry 8, 257–285 (2005).

[81] W. Kirsch, S. Warzel: Anderson localization and Lifshits tails for random surfacepotentials. Journal of Functional Analysis 230, 222-250 (2006).

[82] W. Kirsch, O. Lenoble, L. Pastur: On the Mott formula for the a.c. conductivity andbinary correlators in the strong localization regime of disordered systems, J. Phys.A Math. Gen. 36, 12157–12180 (2003)

BIBLIOGRAPHY 33

[83] A. Kiselev, Y. Last: Solutions, spectrum, and dynamics for Schrodinger operatorson infinite domains. Duke Math. J. 102, 125–150 (2000).

[84] A. Klein: Absolutely continuous spectrum in the Anderson model on the Bethelattice. Math. Res. Lett. 1, No.4, 399-407 (1994).

[85] A. Klein: The Anderson metal-insulator transition on the Bethe lattice. Iagol-nitzer, Daniel (ed.), Proceedings of the XIth international congress on mathemat-ical physics, Paris, France, July 18-23, 1994. Cambridge, MA: International Press.383-391 (1995).

[86] A. Klein: Extended states in the Anderson model on the Bethe lattice. Adv. Math.133, No.1, 163-184 (1998).

[87] A. Klein: Multiscale analysis and localization of random operators, to appear in:Panoramas et Synthese

[88] A. Klein, O. Lenoble, P. Muller: On Mott’s formula for the ac-conductivity in theAnderson model, to appear in: Ann. Math.

[89] F. Klopp, Localization for some continuous random Schrodinger operators. Com-mun. Math. Phys. 167, No.3, 553-569 (1995).

[90] F. Klopp: Lifshitz tails for random perturbations of periodic Schrodinger operators.Proc. Indian Acad. Sci., Math. Sci. 112, No.1, 147-162 (2002).

[91] F. Klopp: Precise high energy asymptotics for the integrated density of states of anunbounded random Jacobi matrix. Rev. Math. Phys. 12, No.4, 575-620 (2000).

[92] F. Klopp: Lifshitz tails for random perturbations of periodic Schrodinger operators.Proc. Indian Acad. Sci., Math. Sci. 112, No.1, 147-162 (2002).

[93] F. Klopp: Weak disorder localization and Lifshitz tails. Commun. Math. Phys. 232,No.1, 125-155 (2002).

[94] F. Klopp: Weak disorder localization and Lifshitz tails: Continuous Hamiltonians.Ann. Henri Poincare 3, No.4, 711-737 (2002).

[95] F. Klopp: Internal Lifshitz tails for Schrodinger operators with random potentials. J.Math. Phys. 43, No.6, 2948-2958 (2002).

[96] F. Klopp, T. Wolff: Lifshitz tails for 2-dimensional random Schrodinger operators.J. Anal. Math. 88, 63-147 (2002).

[97] S. Kotani: Ljapunov indices determine absolutely continuous spectra of station-ary random one-dimensional Schrodinger operators. North-Holland Math. Libr. 32,225-247 (1984).

[98] S. Kotani: Lyapunov exponents and spectra for one-dimensional randomSchrodinger operators. Contemp. Math. 50, 277-286 (1986).

[99] S. Kotani, B. Simon: Localization in general one-dimensional random systems. II.Continuum Schrodinger operators. Commun. Math. Phys. 112, 103-119 (1987).

[100] U. Krengel: Ergodic theorems. With a supplement by Antoine Brunel. Walter deGruyter, Berlin-New York (1985).

[101] M. Krishna: Anderson models with decaying randomness: Existence of extendedstates. Proc. Indian Acad. Sci. Math. 100, 285- 294 (1990).

[102] M. Krishna: Absolutely continuous spectrum for sparse potentials. Proc. IndianAcad. Sci. Math. 103, 333- 339 (1993).

[103] H. Kunz, B. Souillard: Sur le spectre des operateurs aux differences finies aleatoires.Commun. Math. Phys. 78, 201-246 (1980).

[104] J. W. Lamperti: Probability. A survey of the mathematical theory. 2nd ed. WileySeries in Probability and Mathematical Statistics. New York, NY: Wiley (1996).

34 BIBLIOGRAPHY

[105] J. W. Lamperti: Stochastic processes. A survey of the mathematical theory. Ap-plied Mathematical Sciences. 23. New York - Heidelberg - Berlin: Springer-Verlag(1977).

[106] R. Lang: Spectral theory of random Schrodinger operators. A genetic introduc-tion. Lecture Notes in Mathematics. 1498. Berlin: Springer-Verlag (1991).

[107] Y. Last: Quantum dynamics and decompositions of singular continuous spectra. J.Funct. Anal. 142, 406–445 (1996).

[108] I. M. Lifshitz: Energy spectrum structure and quantum states of disordered con-densed systems, Sov. Phy. Usp. 7, 549 (1965).

[109] H. Leschke, P. Muller, S. Warzel: A survey of rigorous results on randomSchrodinger operators for amorphous solids, Markov Proc. Relat. Fields 9, 729D760(2003)

[110] I. M. Lifshits, S. A. Gredeskul, L. A. Pastur: Introduction to the theory of disor-dered systems. Wiley, New York (1988).

[111] F. Martinelli, H. Holden: On absence of diffusion near the bottom of the spectrumfor a random Schrodinger operator on L2(Rν). Commun. Math. Phys. 93, 197-217(1984).

[112] F. Martinelli, E. Scoppola: Remark on the absence of absolutely continuous spec-trum for d-dimensional Schrodinger operators with random potential for large dis-order or low energy. Commun. Math. Phys. 97, 465-471 (1985).

[113] B. Metzger: Asymptotische Eigenschaften im Wechselspiel von Diffusion undWellenausbreitung in zufalligen Medien. Dissertation, TU Chemnitz (2005).

[114] G. A. Mezincescu: Internal Lifschitz singularities of disordered finite-differenceSchrodinger operators. Commun. Math. Phys. 103, 167-176 (1986).

[115] S. A. Molchanov: The structure of eigenfunctions of one-dimensional unorderedstructures. Math. USSR Izv. 12, 69 (1978).

[116] S. A. Molchanov: Ideas in the theory of random media. Acta Appl. Math. 22,No.2/3, 139-282 (1991).

[117] S. A. Molchanov: Lectures on random media. Bakry, Dominique (ed.) et al., Lec-tures on probability theory. Ecole d’Ete? de Probabilites de Saint-Flour XXII-1992.Summer School, 9th- 25th July, 1992, Saint-Flour, France. Berlin: Springer-Verlag.Lect. Notes Math. 1581, 242-411 (1994).

[118] H. Najar: Lifshitz tails for random acoustic operators, J. Math. Phys. 44, 1842–1867(2003).

[119] S. Nakao: On the spectral distribution of the Schrodinger operator with randompotential. Jap. J. Math., new Ser. 3, 111-139 (1977).

[120] L. A. Pastur: Spectra of random self adjoint operators. Russ. Math. Surv. 28, No.1,1-67 (1973).

[121] L. A. Pastur: Behaviour of some Wiener integrals for t → ∞ and the density ofstates of the Schrodinger equation with random potential. Teor. Mat. Fiz. 32, 88-95(1977).

[122] L. A. Pastur: Spectral properties of disordered systems in the one-body approxima-tion. Commun. Math. Phys. 75, 179-196 (1980).

[123] L. Pastur, A. Figotin: Spectra of random and almost-periodic operators.Grundlehren der Mathematischen Wissenschaften. 297. Berlin etc.: Springer-Verlag. (1992).

[124] T. Poerschke , G. Stolz , J. Weidmann: Expansions in generalized eigenfunctions ofselfadjoint operators. Math. Z. 202, No.3, 397-408 (1989).

BIBLIOGRAPHY 35

[125] M. Reed, B. Simon: Methods of modern mathematical physics. II: Fourier anal-ysis, self- adjointness. New York - San Francisco - London: Academic Press, asubsidiary of Harcourt Brace Jovanovich, Publishers (1975).

[126] M. Reed, B. Simon: Methods of modern mathematical physics. IV: Analysis ofoperators. New York - San Francisco - London: Academic Press (1978).

[127] M. Reed, B. Simon: Methods of modern mathematical physics. III: Scatteringtheory. New York, San Francisco, London: Academic Press (1979).

[128] M. Reed, B. Simon: Methods of modern mathematical physics. I: Functionalanalysis. Rev. and enl. ed. New York etc.: Academic Press, A Subsidiary of Har-court Brace Jovanovich, Publishers (1980).

[129] I. Rodnianski, W. Schlag: Classical and quantum scattering for a class of long rangerandom potentials. Int. Math. Res. Not. 5, 243–300 (2003).

[130] W. Rudin: Functional Alnalysis. McGraw-Hill (1973)[131] D. Ruelle: A remark on bound states in potential scattering theory. Rivista Nuevo

Cimento 61A, 655-662 (1969).[132] B. Simon: Lifshitz Tails for the Anderson Model. Journal of Statisticl Physics 38,

No.1, 65-76 (1985).[133] B. Simon: Internal Lifshitz Tails, Journal of Statisticl Physics 46, 911 - 918 (1987).[134] B. Simon: Schrodinger semigroups. Bull. Am. Math. Soc., New Ser. 7, 447-526

(1982).[135] B. Simon: Spectral analysis of rank one perturbations and applications. Feldman,

J. (ed.) et al., Mathematical quantum theory II: Schrodinger operators. Providence,RI: American Mathematical Society. CRM Proc. Lect. Notes. 8, 109-149 (1995).

[136] B. Simon, T. Spencer: Trace class perturbations and the absence of absolutely con-tinuous spectra. Commun. Math. Phys. 125, No.1, 113-125 (1989).

[137] B. Simon, T. Wolff: Singular continuous spectrum under rank one perturbationsand localization for random Hamiltonians. Commun. Pure Appl. Math. 39, 75-90(1986).

[138] T. Spencer: The Schrodinger equation with a random potential. A mathematicalreview. Critical phenomena, random systems, gauge theories, Proc. Summer Sch.Theor. Phys., Sess. 43, Les Houches/France 1984, Pt. 2, 895-942 (1986).

[139] P. Stollmann: Caught by disorder. Bound states in random media. Progress inMathematical Physics. 20. Boston: Birkhauser (2001).

[140] P. Stollmann: Lifshitz asymptotics via linear coupling of disorder. Math. Phys. Anal.Geom. 2, No.3, 279-289 (1999).

[141] P. Stollmann: Wegner estimates and localization for continuum Anderson modelswith some singular distributions. Arch. Math. 75, 307-311 (2000).

[142] G. Stolz: Localization for random Schrodinger operators with Poisson potential.Ann. Inst. Henri Poincare, Phys. Theor. 63, No.3, 297-314 (1995).

[143] R. Strichartz: Fourier asymptotics of fractal measures. J. Funct. Anal. 89, 154–187(1990).

[144] A. S. Sznitman: Brownian motion, obstacles and random media. Springer Mono-graphs in Mathematics. Berlin: Springer (1998).

[145] G. Teschl: Jacobi Operators and Completely Integrable Nonlinear Lattices;AMS 1999

[146] G. Teschl: Mathematical Methods in Quantum Mechanics, AMS 2009.[147] D. J. Thouless: Phys. Repts. 13 93- (1974).

36 BIBLIOGRAPHY

[148] D. J. Thouless: Introduction to disordered systems. Critical phenomena, ran-dom systems, gauge theories, Proc. Summer Sch. Theor. Phys., Sess. 43, LesHouches/France 1984, Pt. 2, 685-722 (1986).

[149] I. Veselic: Localization for random perturbations of periodic Schrodinger operatorswith regular Floquet eigenvalues Annal. H. Poinc. 3, 389-409 (2002).

[150] I. Veselic: Integrated density and Wegner estimates for random Schrodinger opera-tors. del Rio, Rafael (ed.) et al., Spectral theory of Schrodinger operators. Lecturenotes from a workshop, Mexico, December 3-7, 2001. Providence, RI: AmericanMathematical Society (AMS). Contemp. Math. 340, 97-183 (2004).

[151] S. Warzel: On Lifshitz tails in magnetic fields. Dissertation. Universitat Erlangen-Nurnberg, Logos (2001).

[152] F. Wegner: Bounds on the density of states in disordered systems. Zeit. fur Phys.B44, 9-15 (1981).

[153] J. Weidmann: Linear operators in Hilbert spaces. Graduate Texts in Mathematics,Vol. 68. New York - Heidelberg -Berlin: Springer-Verlag (1980).

[154] R. Zallen: The physics amorphous solids. Wiley-Interscience (1983).[155] Ya. B. Zeldovich, A. A. Ruzmaikin, D. D. Sokoloff: Almighty chance. World Sci-

entific Lecture Notes in Physics. 20. Singapore: World Scientific (1990).