## Anderson localization and Lifshits tails for random surface potentials (2006)

Venue: | J. Funct. Anal |

Citations: | 3 - 2 self |

### BibTeX

@ARTICLE{Kirsch06andersonlocalization,

author = {Werner Kirsch and Simone Warzel},

title = {Anderson localization and Lifshits tails for random surface potentials},

journal = {J. Funct. Anal},

year = {2006},

volume = {230},

pages = {222--250}

}

### OpenURL

### Abstract

ABSTRACT. We consider Schrödinger operators on L 2 (R d) with a random potential concentrated near the surface R d1 × {0} ⊂ R d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tail rlies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of

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Citation Context ...i ∈ Z d1 . Throughout this section we assume U ∈ K(R d ) ∩ L 2 loc (R d ), which is implied, for example, by assumptions S2,S3. This ensures that H(U) is essentially self-adjoint on C ∞ 0 (R d ) (cf. =-=[CFKS87]-=-). Examples of partially periodic potentials are fully periodic ones, “surface periodic” potentials as in (1.4) or, more interestingly, commensurable combinations as in (1.5). Aspects of the spectral ... |

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Citation Context ...|ψE(x1, x2)| ≤ C e −γ|x2| (2.6) for |x2| large. x1∈[− L L 2 , 2 ]d1 Proof. Since ψE is an eigenfunction, we have ψE = exp � −t � H X SL (V ) − E�� ψE for all t ≥ 0. Using the Feynman-Kac-formula (cf. =-=[Sim79]-=-) we write the semigroup as an integral over Brownian paths β : [0, ∞[→ SL, which start at x ∈ SL for t = 0 and have either absorbing boundary conditions (in case X = D) or reflecting boundary conditi... |

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Citation Context ...C1 ψ 0(x2) ≤ ψ0(x1, x2) ≤ C2 ψ 0(x2) (3.14) Proof. By periodicity we may assume that x1 ∈ Λ1. Since ψ0 is positive and solves the Schrödinger equation Hperψ0 = E0ψ0, we may apply Harnack’s inequality =-=[AS82]-=- (see also [CFKS87, Thm. 2.5] and for explicit constants [HK90]) to obtain: C1 ψ0(x ′ 1, x2) ≤ ψ0(x1, x2) ≤ C2 ψ0(x ′ 1, x2) (3.15) for all x1, x ′ 1 ∈ Λ1. Since the local Kato norms of U ∈ K(R d ) ar... |

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Citation Context ...ides with the IDSS defined through the limit in (1.9), N(E) = N X (E) = ν( ] − ∞, E]) (2.4) for all E < 0 except countably many. In fact, by Neumann-Dirichlet-bracketing ([RS78, Prop. 3 on p. 269] or =-=[KM82a]-=-) we have HN SL (Vs) ⊕ HN ∁SL (0) ≤ H(Vs1SL) ≤ HD SL (Vs) ⊕ HD (0), which gives ∁SL N � H D SL (Vs), E � ≤ N � H(Vs1SL ), E� ≤ N � H N SL (Vs), E � , (2.5) because N � H N ∁SL (0), E� = 0 for E < 0. R... |

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Citation Context ...> 0 we have P � σ(H D SL ) ∩ ]E − ε, E + ε[ = ∅� ≤ CL 2d1 ε µ . (5.2) [Here the exponent µ was defined in S6 and the constant C may depend on E, but not on L or ε.] For a proof we refer to [BS03] and =-=[Sto00]-=-. Our Wegner estimate above has an upper bound of order L 2d1 which suffices to prove Anderson localization. However, to prove Hölder continuity of the integrated density of surface states we would ne... |

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Citation Context ...r all L ∈ N. Since (3.10) is in general wrong for the operator HN per,L with Neumann boundary consditions, Mezincescu boundary conditions were introduced in [Mez87] to be able to extend the result in =-=[KS86]-=-. The result which plays a crucial role in our Lifshits-tail estimate is the following lower bound for the gap of H χ per,L .s14 WERNER KIRSCH AND SIMONE WARZEL Theorem 3.5. Under assumptions P1–P3 th... |

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Citation Context ...iction of (3.1) to L2 (SL) with Mezincescu boundary conditions ∇n ϕ(x) = −χ(x) ϕ(x) for x ∈ ∂SL (3.9) in the domain of the Laplacian (for details of the definition of −∆ χ via quadratic forms, SL see =-=[Mez87]-=- and [KW03, Sec. 3.1]). This choice of boundary conditions ensures that inf spec H χ per,L = inf spec Hper = E0 (3.10) for all L ∈ N. Since (3.10) is in general wrong for the operator HN per,L with Ne... |

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Citation Context ...)] 1CL ] (2.1) of expanding cubes CL = [− L L 2 , 2 ]d defines a non-random linear functional ν on ϕ ∈ C∞ 0 (R). Actually [EKSS90] considered the interface between two different random potentials and =-=[KS00]-=- remarked that the method in [EKSS90] can be used for surface potentials as well. The renormalization term ϕ(−∆) in (2.1) is needed to counterbalance the first term which diverges as soon as ϕ has sup... |

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Citation Context ...uld like to warn the reader that, in contrast to what the symbols suggests, even H(Ub) may have (generalized) eigenstates which are concentrated near R d1 × {0} (for a discussion see [DS78], and also =-=[EKSS90]-=-). 1.2. Main results. Under assumptions B1–B2 and S1–S4 we first prove the existence of the integrated density of surface states (IDSS) for negative energies, that is, below the spectrum of the bulk o... |

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Citation Context ...he above assumptions the random Schrödinger operator H(V ) is almost surely essentially self-adjoint on C ∞ 0 (R d ), the space of arbitrarily often differentiable functions with compact support (cf. =-=[KM83b]-=-). Moreover, Z d1 -ergodicity of V on the product measure space (Ωb × Ωs, Ab ⊗ As, P) with P := Pb ⊗ Ps, guarantees the validity of (cf. [KM82b, EKSS90]) Proposition 1.1. Under assumptions B1–B2 and S... |

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Citation Context ...rators was given in [KK]. This paper also deals with the case E0 = 0, a case we can not handle here. Localization of surface states by (alloy-type) random surface potentials is discussed in detail in =-=[BS03]-=-, which has been the main motivation of the present paper. In fact, in case Ub = Vb = 0 and under the assumption that P0 is Hölder continuous S6 There exist constants C, µ > 0 such that P0([a, b]) ≤ C... |

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Citation Context ...dient in our proof of Lifshits tails in Section 4, in which we have to restrict the operator Hper to strips SL and special Robin boundary conditions, which we dubbed Mezincescu boundary conditions in =-=[KW03]-=-. To introduce Mezincescu boundary conditions we additionally assume U ∈ L p loc (S) for some neighborhood S ⊂ R d of ∂S1 and some p > d. This is guaranteed by S2. We note that it is only here where w... |

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Citation Context ...ty we may assume that x1 ∈ Λ1. Since ψ0 is positive and solves the Schrödinger equation Hperψ0 = E0ψ0, we may apply Harnack’s inequality [AS82] (see also [CFKS87, Thm. 2.5] and for explicit constants =-=[HK90]-=-) to obtain: C1 ψ0(x ′ 1, x2) ≤ ψ0(x1, x2) ≤ C2 ψ0(x ′ 1, x2) (3.15) for all x1, x ′ 1 ∈ Λ1. Since the local Kato norms of U ∈ K(R d ) are uniformly bounded, the constants C1, C2 are independent of x2... |

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Citation Context ...o. However, we would like to warn the reader that, in contrast to what the symbols suggests, even H(Ub) may have (generalized) eigenstates which are concentrated near R d1 × {0} (for a discussion see =-=[DS78]-=-, and also [EKSS90]). 1.2. Main results. Under assumptions B1–B2 and S1–S4 we first prove the existence of the integrated density of surface states (IDSS) for negative energies, that is, below the spe... |

2 | F.: The band-edge behavior of the density of surfacic states
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Citation Context ...and their proof relies on a method of seperable comparison potentials developed in Subsection 3.2. We finally remark that an analysis of the Lifshits tails for discrete surface operators was given in =-=[KK]-=-. This paper also deals with the case E0 = 0, a case we can not handle here. Localization of surface states by (alloy-type) random surface potentials is discussed in detail in [BS03], which has been t... |

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Citation Context ...in the proof of the upper bound in Lemma 2.5. It seems to be folklore in Hilbert space theory. However, as we could not find it in the literature, we include it for the reader’s convenience (see also =-=[Böc03]-=- for a similar result). Lemma 2.9. Let n ∈ N and ϕ1, . . . , ϕn ∈ H be in the domain dom A of a self-adjoint operator A, which acts on a (separable) Hilbert space H. Suppose there are constants α1 ≤ ·... |

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