An Invitation to Random SchrÂ¨odinger operators - FernUniversitÃ¤t in ...

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20 If X is a random variable we call the probability measure P 0 on R defined by P 0 (A) = P ( {ω | X(ω) ∈ A } ) for any Borel set A (3.45) the distribution of X. If the distributions of the random variables X and Y agree we say that X and Y are identically distributed. We also say that X and Y have a common distribution in this case. A family {X i } i∈I of random variables is called independent if for any finite subset {i 1 , . . . , i n } of I ( {ω| P Xi1 (ω) ∈ [a 1 , b 1 ], X i2 (ω) ∈ [a 2 , b 2 ], . . . X in (ω) ∈ [a n , b n ] }) = P ( {ω| Xi1 (ω) ∈ [a 1 , b 1 ] }) · . . . · P ( {ω| Xin (ω) ∈ [a n , b n ] }) . (3.46) REMARK 3.5. If X i are independent and identically distributed (iid) with common distribution P 0 then ( {ω| P Xi1 (ω) ∈ [a 1 , b 1 ], X i2 (ω) ∈ [a 2 , b 2 ], . . . X in (ω) ∈ [a n , b n ] }) = P 0 ([a 1 , b 1 ]) · P 0 ([a 2 , b 2 ]) · . . . · P 0 ([a n , b n ]) . For the reader’s convenience we state a very useful result of elementary probability theory which we will need a number of times in this text. THEOREM 3.6 (Borel-Cantelli lemma). Let (Ω, F, P) be a probability space and {A n } n∈N be a sequence of set in F. Denote by A ∞ the set (1) If A ∞ = {ω ∈ Ω | ω ∈ A n for infinitely many n} (3.47) ∑ ∞ n=1 P(A n) < ∞, then P(A ∞ ) = 0 (2) If the sets {A n } are independent and ∑ ∞ n=1 P(A n) = ∞, then P(A ∞ ) = 1 REMARK 3.7. (1) We recall that a sequence {A n } of events (i.e. of sets from F) is called independent if for any finite subsequence {A nj } j=1,...,M P ( M ⋂ j=1 A nj ) = M ∏ j=1 (2) The set A ∞ can be written as A ∞ = ⋂ N For the proof of Theorem 3.6 see e.g. [12] or [95]. P (A nj ) (3.48) ⋃ n≥N A n. From now on we assume that the random variables {V ω (n)} n∈Z d are independent and identically distributed with common distribution P 0 . By supp P 0 we denote the support of the measure P 0 , i.e. supp P 0 = {x ∈ R | P 0 ( (x − ε, x + ε) ) > 0 for all ε > 0} . (3.49) If supp P 0 is compact then the operator H ω = H 0 + V ω is bounded. In fact, if supp P 0 ⊂ [−M, M], with probability one
21 sup |V ω (j)| ≤ M . j∈Z d Even if supp P 0 is not compact the multiplication operator V ω is selfadjoint on D = {ϕ ∈ l 2 |V ω ϕ ∈ l 2 }. It is essentially selfadjoint on l 2 0(Z d ) = {ϕ ∈ l 2 (Z d ) | ϕ(i) = 0 for all but finitely many points i} . Since H 0 is bounded it is a fortiori Kato bounded with respect to V ω . By the Kato- Rellich theorem it follows that H ω = H 0 + V ω is essentially selfadjoint on l 2 0 (Zd ) as well (see [114] for details). In a first result about the Anderson model we are now going to determine its spectrum (as a set). In particular we will see that the spectrum σ(H ω ) is (P-almost surely) a fixed non random set. First we prove a proposition which while easy is very useful in the following. Roughly speaking, this proposition tells us: Whatever can happen, will happen, in fact infinitely often. PROPOSITION 3.8. There is a set Ω 0 of probability one such that the following is true: For any ω ∈ Ω 0 , any finite set Λ ⊂ Z d , any sequence {q i } i∈Λ , q i ∈ supp P 0 and any ε > 0, there exists a sequence {j n } in Z d with || j n || ∞ → ∞ such that sup | q i − V ω (i + j n ) | < ε . i∈Λ PROOF: Fix a finite set Λ, a sequence {q i } i∈Λ , q i ∈ supp P 0 and ε > 0. Then, by the definition of supp and the independence of the q i we have for A = {ω| sup i∈Λ |V ω (i) − q i | < ε} P(A) > 0 . Pick a sequence l n ∈ Z d , such that the distance between any l n , l m (n ≠ m) is bigger than twice the diameter of Λ. Then, the events A n = A n (Λ, {q i } i∈Λ , ε) = {ω| sup |V ω (i + l n ) − q i | < ε} i∈Λ are independent and P(A n ) = P(A) > 0. Consequently, the Borel-Cantelli lemma (see Theorem 3.6) tells us that Ω Λ,{qi },ε = { ω | ω ∈ A n for infinitely many n} has probability one. The set supp P 0 contains a countable dense set R 0 . Moreover, the system Ξ of all finite subsets of Z d is countable. Thus the set Ω 0 := ⋂ Λ ∈ Ξ, {q i }∈R 0 ,n∈N Ω Λ,{qi }, 1 n
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: THEOREM 3.1 (Min-max principle). If
Page 23 and 24: 23 4.1. Ergodic stochastic processe
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
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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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