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

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48 We start the proof with the following lemma. LEMMA 5.28. If Λ 1 , Λ 2 are disjoint finite subsets of Z d then P ( dist(σ(H Λ1 ), σ(H Λ2 )) < ε ) ≤ C ‖ g ‖ ∞ ε |Λ 1 ||Λ 2 | . PROOF (Lemma) : Since Λ 1 ∩ Λ 2 = ∅ the random potentials in Λ 1 and Λ 2 are independent of each other and so are the eigenvalues E (1) 0 ≤ E (1) 1 ≤ ... of H Λ1 and the eigenvalues E (2) 0 ≤ E (2) 1 ≤ ... of H Λ2 . We denote the probability (resp. the expectation) with respect to the random variables in Λ by P Λ (resp. E Λ ). Since the random variables {V ω (n)} n∈Λ1 and {V ω (n)} n∈Λ2 are independent for Λ 1 ∩ Λ 2 = ∅ we have that for such sets P Λ1 ∪Λ 2 is the product measure P Λ1 ⊗ P Λ2 . We compute P ( dist(σ(Λ 1 ), σ(Λ 2 )) < ε ) = P ( min dist(E (1) i i , σ(H Λ2 )) < ε ) ≤ ≤ From Theorem 5.23 we know that |Λ 1 | ∑ i=1 |Λ 1 | ∑ i=1 |Λ 1 | ∑ ≤ i=1 P ( dist(E (1) i , σ(H Λ2 )) < ε ) P Λ1 ⊗ P Λ2 ( dist(E (1) i , σ(H Λ2 )) < ε ) E Λ1 ( P Λ2 ( dist(E (1) i , σ(H Λ2 )) < ε ) ) . P Λ2 ( dist(E, σ(HΛ2 )) < ε ) ≤ C ‖ g ‖ ∞ ε |Λ 2 | . (5.77) Hence, we obtain (5.77) ≤ C ‖ g ‖ ∞ ε |Λ 2 ||Λ 1 | . □ The proof of the theorem is now easy. PROOF (Theorem) : P ( There is an E ∈ R such that dist(σ(H Λ1 ), E) < ε and dist(σ(H Λ2 , E))) < ε ) ≤ P ( dist(σ(H Λ1 ), σ(H Λ2 )) < 2 ε) ≤ 2 C ‖ g ‖ ∞ ε |Λ 1 ||Λ 2 | by the lemma. □
49 Notes and Remarks General references for the density of states are [109], [63], [10] and [36], [58] [121] and [138]. A thorough discussion of the geometric resolvent equation in the context of perturbation theory can be found in [40], [47] and [127]. In the context of the discrete Laplacian Dirichlet and Neumann boundary conditions were introduced and investigated in [121]. See also [68]. For discrete ergodic operators the integrated density of states N is log-Hölder continuous, see [29]. Our proof of the continuity of N is tailored after [36]. For results concerning the Wegner estimates see [140], [59], [130] [27], [26],and [138], as well as references given there.
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 and 22: 21 sup |V ω (j)| ≤ M . j∈Z d E
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 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 −
Page 72 and 73: 72 → = ∫ T 1 |ˆµ(t)| 2 dt T 0
Page 75 and 76: 75 8.1. What physicists know. 8. An
Page 77 and 78: of a Schrödinger operator on R d c
Page 79 and 80: PROOF: From Theorem 7.14 we know
Page 81 and 82: 81 9. The Green’s function and th
Page 83 and 84: 83 |ψ(n)| ≤ ≤ ∣ ∑ (m ′ ,
Page 85 and 86: where (9.14) holds. This effect is
Page 87 and 88: 87 by C L (Λ) = {Λ L (n) | Λ L (
Page 89 and 90: 89 with some k ′ ≥ k. We conclu
Page 91 and 92: 91 We will prove LEMMA 9.16. If Res
Page 93 and 94: 93 10. Multiscale analysis 10.1. St
Page 95 and 96: 95 If an overwhelming majority of t
Page 97 and 98: THEOREM 10.5. If the cube Λ L is n
Page 99 and 100:
Thus, (10.28) and (10.30) inserted
Page 101 and 102:
101 |G Λ L E (u, n)| ≤ ∑ (q,q
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(2) no cube Λ 2l (m) in Λ L is E-
Page 105 and 106:
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
Page 114 and 115:
114 F u(n) = F n0 u(n) = e µ || n
Page 116 and 117:
116 11.3. Energies near the bottom
Page 119 and 120:
119 12. Appendix: Lost in Multiscal
Page 121 and 122:
121 References [1] M. Aizenman: Loc
Page 123 and 124:
[51] F. Germinet, A. Klein, J. Sche
Page 125 and 126:
[99] I. M. Lifshits, S. A. Gredesku
Page 127 and 128:
A(i, j), 15 A ≤ B, 19 A ≤ B, 35
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weak convergence, 31 Wegner estimat
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