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22 has probability one. It is a countable intersection of sets of probability one. By its definition, Ω 0 satisfies the requirements of the assertion. □ We now turn to the announced theorem THEOREM 3.9. For P-almost all ω we have σ(H ω ) = [0, 4d ] + supp P 0 . PROOF: The spectrum σ(V ) of the multiplication operator with V (n) is given by the closure of the set R(V ) = {V (n)|n ∈ Z d }. Hence σ(V ω ) = supp P 0 almost surely. Since 0 ≤ H 0 ≤ 4d we have σ(H 0 + V ω ) ⊂ σ(V ω ) + [0, ‖H 0 ‖] = supp P 0 + [0, 4d] . Let us prove the converse. We use the Weyl criterion (see [117] or [141]): λ ∈ σ(H ω ) ⇔ ∃ ϕ n ∈ D 0 , ||ϕ n || = 1 : ||(H ω − λ)ϕ n || → 0 , where D 0 is any vector space such that H ω is essentially selfadjoint on D 0 . The sequence ϕ n is called a Weyl sequence. In a sense, ϕ n is an ‘approximate eigenfunction’. Let λ ∈ [0, 4d] + supp P 0 , say λ = λ 0 + λ 1 , λ 0 ∈ σ(H 0 ) = [0, 4d], λ 1 ∈ supp P 0 . Take a Weyl sequence ϕ n for H 0 and λ 0 , i. e. ||(H 0 − λ 0 )ϕ n || → 0, ||ϕ n || = 1. Since H 0 is essentially selfadjoint on D 0 = l 2 0 (Zd ) (in fact H 0 is bounded), we may suppose ϕ n ∈ D 0 . Setting ϕ (j) (i) = ϕ(i − j), we easily see H 0 ϕ (j) = (H 0 ϕ) (j) . Due to Proposition 3.8 there is (with probability one) a sequence {j n }, || j n || ∞ → ∞ such that sup i∈supp ϕ n |V ω (i + j n ) − λ 1 | < 1 n . (3.50) Define ψ n = ϕ jn n . Then ψ n is a Weyl sequence for H ω and λ = λ 0 + λ 1 . This proves the theorem. □ The above result tells us in particular that the spectrum σ(H ω ) is (almost surely) non random. Moreover, an inspection of the proof shows that there is no discrete spectrum (almost surely), as the constructed Weyl sequence tends to zero weakly, in fact can be chosen to be orthonormal. Both results are valid in much bigger generality. They are due to ergodicity properties of the potential V ω . We will discuss this topic in the following chapter. Notes and Remarks For further information see [23] and [30] or consult [94] and [64, 65].
23 4.1. Ergodic stochastic processes. 4. Ergodicity properties Some of the basic questions about random Schrödinger can be formulated and answered most conveniently within the framework of ‘ergodic operators’. This class of operators comprises many random operators, such as the Anderson model and its continuous analogs, the Poisson model, as well as random acoustic operators. Moreover, also operators with almost periodic potentials can be viewed as ergodic operators. In these notes we only briefly touch the topic of ergodic operators. We just collect a few definitions and results we will need in the following chapters. We recommend the references cited in the notes at the end of this chapter for further reading. Ergodic stochastic processes are a certain generalization of independent, identically distributed random variables. The assumption that the random variables X i and X j are independent for |i − j| > 0 is replaced by the requirement that X i and X j are ‘almost independent’ if |i − j| is large (see the discussion below, especially (4.2), for a precise statement). The most important result about ergodic processes is the ergodic theorem (see Theorem 4.2 below), which says that the strong law of large numbers, one of the basic results about independent, identically distributed random variables, extends to ergodic processes. At a number a places in these notes we will have to deal with ergodic processes. Certain important quantities connected with random operators are ergodic but not independent even if the potential V ω is a sequence of independent random variables. A family {X i } i∈Z d of random variables is called a stochastic process (with index set Z d ). This means that there is a probability space (Ω, F, P) (F a σ-algebra on Ω and P a probability measure on (Ω, F)) such that the X i are real valued, measurable functions on (Ω, F). The quantities of interest are the probabilities of events that can be expressed through the random variables X i , like 1 {ω| lim N→∞ |Λ N | ∑ || i|| ∞≤N X i (ω) = 0} . The special way Ω is constructed is irrelevant. For example, one may take the set R Zd as Ω. The corresponding σ-algebra F is generated by cylinder sets of the form {ω | ω i1 ∈ A 1 , . . . , ω in ∈ A n } (4.1) where A 1 , . . . , A n are Borel subsets of R. On Ω the random variables X i can be realized by X i (ω) = ω i . This choice of (Ω, F) is called the canonical probability space . For details in connection with random operators see e.g. [58, 64]. Given a probability space (Ω, F, P) we call a measurable mapping T : Ω → Ω a measure preserving transformation if P(T −1 A) = P(A) for all A ∈ F. If {T i } i∈Z d is a family of
Page 1: An Invitation to Random Schrödinge
Page 4 and 5: 4 10.2. Analytic estimate - first t
Page 7 and 8: 7 2. Introduction: Why random Schr
Page 9 and 10: is the random point measure ν =
Page 11 and 12: symmetric subspace is the correct o
Page 13 and 14: 13 3. Setup: The Anderson model 3.1
Page 15 and 16: orthonormal basis of l 2 (Z d ). Th
Page 17 and 18: 17 (αf + βg) (A) = αf(A) + βg(A
Page 19 and 20: THEOREM 3.1 (Min-max principle). If
Page 21: 21 sup |V ω (j)| ≤ M . j∈Z d E
Page 25 and 26: For a proof of this fundamental res
Page 27: holds for all . Thus the measures
Page 30 and 31: 30 REMARK 5.3. The right hand side
Page 32 and 33: 32 (2) In the continuous case the d
Page 34 and 35: 34 PROOF: If λ /∈ Σ then there
Page 36 and 37: 36 The easiest version of boundary
Page 38 and 39: 38 DEFINITION 5.18. The Dirichlet L
Page 40 and 41: 40 5.3. The geometric resolvent equ
Page 42 and 43: 42 We define the eigenvalue countin
Page 44 and 45: 44 PROOF (Corollary 5.25) : By the
Page 46 and 47: 46 Hence ( ∂ E tr ( ϱ (H Λ (V
Page 48 and 49: 48 We start the proof with the foll
Page 51 and 52: 51 6.1. Statement of the Result. 6.
Page 53 and 54: 53 In fact (H 0 ) N Λ L ψ 0 = 0.
Page 55 and 56: 55 This estimate is the desired upp
Page 57 and 58: for any ψ with ||ψ|| = 1. Now we
Page 59: For an approach using periodic appr
Page 62 and 63: 62 µ n,m (∆) = 〈δ n , µ(∆)
Page 64 and 65: 64 So far, the sequence α n has on
Page 66 and 67: 66 mean in what follows a complex-v
Page 68 and 69: 68 Let us look at the time evolutio
Page 70 and 71: 70 lim t→∞ ( ∑ x∈Λ | e −
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72 → = ∫ T 1 |ˆµ(t)| 2 dt T 0
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75 8.1. What physicists know. 8. An
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of a Schrödinger operator on R d c
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PROOF: From Theorem 7.14 we know
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81 9. The Green’s function and th
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83 |ψ(n)| ≤ ≤ ∣ ∑ (m ′ ,
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where (9.14) holds. This effect is
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87 by C L (Λ) = {Λ L (n) | Λ L (
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89 with some k ′ ≥ k. We conclu
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91 We will prove LEMMA 9.16. If Res
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93 10. Multiscale analysis 10.1. St
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95 If an overwhelming majority of t
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THEOREM 10.5. If the cube Λ L is n
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Thus, (10.28) and (10.30) inserted
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101 |G Λ L E (u, n)| ≤ ∑ (q,q
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(2) no cube Λ 2l (m) in Λ L is E-
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105 ∑ i,j∈Λ L Λ l (i)∩Λ l
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connection with a Wegner-type bound
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109 If n j belongs to one of the M
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≤ ∑ C i ∈ C l (Λ L ) pairwis
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114 F u(n) = F n0 u(n) = e µ || n
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116 11.3. Energies near the bottom
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119 12. Appendix: Lost in Multiscal
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121 References [1] M. Aizenman: Loc
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[51] F. Germinet, A. Klein, J. Sche
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[99] I. M. Lifshits, S. A. Gredesku
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A(i, j), 15 A ≤ B, 19 A ≤ B, 35
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weak convergence, 31 Wegner estimat
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