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Ergodic Properties of Random Schrödinger Operators

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Ergodic Properties of Random Schrödinger Operators Ergodic Properties of Random Schr¨odingerOperators by Irina Y. Zhecheva A Thesis Submitted in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Honors in Mathematics WILLIAMS COLLEGE Williamstown, Massachusetts May 05, 2008 i Acknowledgements I would like to thank my thesis advisor, Professor Stoiciu, for his enormous pa- tience and understanding, and for his incredible encouragement and support. I would also like to thank professor Silva, my second reader, for giving me valuable sugges- tions. Contents 1 Introduction 1 2 Background from Ergodic Theory 2 2.1 Basic Notions from Ergodic Theory . 2 2.2 The Mean Ergodic Theorem . 3 3 The Sch¨odingerOperator 7 3.1 The Spectrum . 7 3.2 The Discrete Laplace Operator . 9 3.3 Randomizing the Potential . 13 4 Connecting Spectral Theory with Ergodicity 18 4.1 Ergodicity of the Random Schr¨odingerOperator . 18 4.2 Connections with Ergodic Theory . 22 5 More Results from Functional Analysis 25 5.1 More Definitions . 25 6 The Functional Calculus and the Spectral Theorem 30 6.1 The Functional Calculus . 30 6.2 The Spectral Theorem . 32 7 The Spectrum of the Random Schr¨odingerOperator 34 7.1 Proving that the spectrum is deterministic . 34 7.2 Decomposing the Spectrum . 36 8 The Density of States 40 9 Further Work 44 ii Chapter 1 Introduction The Schr¨odingerequation plays a central role in quantum mechanics and mathemati- cal physics in general. In this thesis we attempt to give an introduction to some of the basic theory of random Schr¨odingeroperators. In particular, we look at some major results concerning the spectral theory of the one-dimensional Schr¨odingeroperator with a random potential. The random Schr¨odinger operator has been intensively studied in the last few decades, in both its discrete and its continuous formulations. Variations of this operator arise very naturally in physics and mathematics, in the theory of electron conduction and in Hamiltonian mechanics. In the theory of electron conduction, random Schr¨odingeroperators model disordered solids. This exposition focuses on the one-dimensional Anderson model, namely the model with random operators on the Hilbert space l2(Z). The theory underlying the study of random Schrodinger¨ operators is fascinating in its own right, even abstracting from its appli- cations to physics. What is so interesting about it is that it combines several areas of mathematics and uses tools that have applications in applied probability, stochastic processes and other fields. This thesis introduces many of the tools that are needed to understand the theory of random Schr¨odinger operators. We begin by introducing fundamental results in ergodic theory, functional analysis, probability, and measure theory that we will need to be able to understand the theory of random Schr¨odingeroperators. The focus of this thesis is to examine the spectrum of the random Schr¨odinger operator and to show that, contrary to what we would expect, it is deterministic. More precisely, we show that it is non-random with probability 1. This result was proved by Pastur. A sketch of the proof can be found in [1], but a lot of the details are omitted. The goal of this thesis is to provide the missing details in the proofs of the theorems leading up to this important result by Pastur. What enables us to prove that the spectrum is deterministic is the fact that the random Schr¨odinger operator pertains to a class of operators called ergodic operators. The framework of ergodic theory turns out to be particularly suitable for investigating the properties of the random Schr¨odingeroperator. We start by giving a careful proof of the Mean Ergodic theorem, which is a very important result in ergodic theory. 1 Chapter 2 Background from Ergodic Theory 2.1 Basic Notions from Ergodic Theory The theory of dynamical systems studies the long-term behavior of states in a system given a transformation which drives the system's evolution. A dynamical system is a transformation which maps a space into itself. Some dynamical systems are measure- preserving in the sense that the measure of a set is always the same as the measure of the set of points which map to it. More formally, a transformation is measure preserving if P(T −1(A)) = P(A) for any measurable set A. A set is called invariant if it is the same as its image. An ergodic dynamical system is one in which, with respect to some probability distribution, all invariant sets have measure either 0 or 1. Here are the formal definitions of the concepts from ergodic theory that we are going to need. Throughout we assume that we are given a triple (Ω; F; P) where Ω is a probability space, F is a σ-algebra of sets, and P is a probability measure. Definition 1. A measurable mapping T :Ω ! Ω is a measure-preserving transfor- mation if P(T −1(A) = P(A) for all A 2 F. Definition 2. Let fTigi2Z be a family of measure-preserving transformations. A set −1 A is invariant under fTig if Ti (A) = A for all i 2 Z. Definition 3. A triple (Ω; F;Ti) of an event space, probability measure and a trans- −1 formation is called stationary if P(Ti A) = P(A) for any A 2 F. Definition 4. A family fTig of measure-preserving transformations is called ergodic with respect to the probability measure P if any invariant set A 2 F has probability 0 or 1. Definition 5. A family fXigi2Z of random variables is called a stochastic process (with index set Z). This means that there is a probability space (Ω; F; P) where F is a σ-algebra on Ω and P is a probability measure on (Ω; F) such that the Xi are real-valued, measurable functions on (Ω; F). Definition 6. A stochastic process fXigi2Z is called ergodic if there exists an ergodic family of measure-preserving transformations fTigi2Z such that Xi(Tj!) = Xi−j(!). 2 CHAPTER 2. BACKGROUND FROM ERGODIC THEORY 3 In our case we are going to take Ti to be the shifting transformations. Thus, we also need the following definition. −1 Definition 7. A set A is called shift invariant if Ti A = A for all i 2 Z. Definition 8. A stationary probability measure P is called ergodic if any shift-invariant set A has probability 0 or 1. In the next section we introduce the mean ergodic theorem due to Birkhoff, which is one of the most important results in Ergodic Theory. 2.2 The Mean Ergodic Theorem The mean ergodic theorem intuitively says the following. For any integrable function of our space, if we randomly pick a point according to the ergodic distribution and calculate the time average of the function along the point's orbit, then as time goes to infinity, the time average converges to the space average, which is equal to the weighted average of the value of the function at all points in the space. More formally, consider a point x in the state space, a function f that we are averaging, a map T R and a probability measure P. The space average is f¯ = f(x)dP(x): The time- 1 Pn i average starting from x is hfix = limn!1 n i=0 f(T (x)). The ergodic theorem states that if f is integrable and T is ergodic with respect to P, then hfix exists, and ¯ P fx : hfix = fg = 1. Here we state a common formulation of the mean ergodic theorem due to Birkhoff. Theorem 2.2.1. Let (X; S; µ) be a probability measure space, and T : X ! X be a measure-preserving transformation. Let I = fg 2 L2(X): g ◦ T = g a.e.g be the subspace of T-invariant functions and let P be the projection from L2(X) to I. Then for any f 2 L2, the sequence n−1 1 X f T ix (2.1) n i=0 converges to P (f) in L2(X) . Proof. We prove the theorem for each of two orthogonal subspaces that span L2. Then since any element in L2 can be written as a linear combination of elements in the two subspaces, the theorem holds for the general case as well. When f 2 I, f ◦ T n = f a.e., so the statement of the theorem holds: n−1 1 X 1 f ◦ T i = nf = f = P (f) (2.2) n n i=0 Here f = P (f) because f 2 I, so the projection of f is also in I. Consider a subspace B of functions, B = ff : f = g ◦ T − g; g 2 L2g (2.3) CHAPTER 2. BACKGROUND FROM ERGODIC THEORY 4 Then if f = g ◦ T − g, n−1 n−1 n−1 X X X f ◦ T i = (g ◦ T − g) ◦ T i = fg ◦ T i+1 − g ◦ T ig = g ◦ T n − g (2.4) i=0 i=0 i=0 Thus, we can see that n−1 1 X 1 f ◦ T i = (g ◦ T n − g) (2.5) n n i=0 converges to 0 as n ! 1. Thus, the ergodic theorem holds for f 2 P (f). Now we are going to show that it also ¯ ¯ holds for f 2 B, where B denotes the norm closure of B. Let fj be a sequence of functions converging in norm to a function f 2 L2. Then n−1 n−1 1 X 1 X jj f ◦ T ijj =jj (f − f + f ) ◦ T ijj n 2 n j j 2 i=0 i=0 n−1 n−1 1 X 1 X ≤ jj (f − f ) ◦ T ijj + jj f ◦ T ijj n j 2 n j 2 i=0 i=0 n−1 n−1 1 X 1 X ≤ jj(f − f ) ◦ T ijj + jjf ◦ T ijj n j 2 n j 2 i=0 i=0 n−1 n−1 1 X 1 X jj(f − f )jj + jjf ◦ T ijj n j 2 n j 2 i=0 i=0 1 Pn−1 i Therefore, for functions in the norm closure of B, n i=0 f ◦ T ! 0 as n ! 1.
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