Lifshitz tails for Anderson models with sign‐indefinite single‐site potentials

Abstract

We study the spectral minimum and Lifshitz tails for continuum random Schrödinger operators of the formwhere V 0 is the periodic potential, {ω i } i∈Z d are i.i.d random variables and u is the sign-indefinite impurity potential. Recently, this model has been proven to exhibit Lifshitz tails near the bottom of the spectrum under the small support assumption of u and the reflection symmetry assumption of V 0 and u. We here drop the reflection symmetry assumption of V 0 and u. We first give characterizations of the bottom of the spectrum. Then, we show the existence of Lifshitz tails in the regime where the characterization of the bottom of the spectrum is explicit. In particular, this regime covers the reflection symmetry case.

Lifshits Tails for Randomly Twisted Quantum Waveguides

“…Remarks: (i) Proposition 4.3 also follows from the results of the article [27] which extends [18]. More precisely, (4.12) follows from [ (ii) If ǫ ≤ 0 and hence v ǫ does not change sign, (4.12) and (4.13) have been known since long ago (see [13] and [16] respectively).…”

“…Remarks: (i) Proposition 4.3 also follows from the results of the article [27] which extends [18]. More precisely, (4.12) follows from [27, Theorem 1.1], while (4.13) follows from [27,Theorem 1.2].…”

“…In Section 4, we formulate and prove our main results on the Lifshits tails for N, applying the estimates obtained in Section 3, as well as certain results on the Lifshits tails for the operator h θ,ǫ . Some of these necessary results turned out to be available in the literature (see [18,27]) and some of them are borrowed from our companion paper [15] where Lifshits tails for Schrödinger operators with squared Anderson-type potentials are investigated in any dimension d ≥ 1.…”

Lifshits Tails for Squared Potentials

“…For a review (as of 2006) see e.g. [9] (see also [1], [4]), some more recent developments are [3], [11], [12] and [16].…”