Spectral Extrema and Lifshitz Tails for Non-Monotonous Alloy Type Models

Abstract

In the present note, we determine the ground state energy and study the existence of Lifshitz tails near this energy for some non monotonous alloy type models. Here, non monotonous means that the single site potential coming into the alloy random potential changes sign. In particular, the random operator is not a monotonous function of the random variables.Résumé. Cet article est consacréà la détermination de l'énergie de l'état fondamental età l'étude de possibles asymptotiques de Lifshitz au voisinage de cetteénergie pour certains modèles d'Anderson continus non monotones. Ici, non monotone signifie que le potentiel de simple site entrant dans la composition du potentiel aléatoire change de signe. En particulier, l'opérateur aléatoire n'est pas une fonction monotone des variables aléatoires. Introduction and resultsIn this paper, we consider the continuous alloy type (or Anderson) random Schrödinger operator:where V is the site potential, and (ω γ ) γ∈Z d are the random coupling constants. Throughout this paper, we assume Let Σ be the almost sure spectrum of H ω and E − = inf Σ. When V has a fixed sign, it is well known that theMoreover, in this case, it is well known that the integrated density of states of the Hamiltonian (see e.g. (0.3)) admits a Lifshitz tail near E − , i.e., that the integrated density of states at energy E decays exponentially fast as E goes to E − from above. We refer to [9,7,22,20,6,5,11] for precise statements.In the present paper, we address the case when V changes sign, i.e., there may exist x + = x − such thatThe basic difficulty this property introduces is that the variations of the potential V ω as a function of ω are not monotonous. In the monotonous case, to get the minimum, one can simply minimize with respect to each of the random variables individually. In the non monotonous case, this uncoupling between the different random variables may fail. Our results concern reflection symmetric potentials since, as we will see, for these potentials we also have an analogous decoupling between the different random variables. Thus, we make the following symmetry assumption on V :We now consider the operator H N λ = −∆ + λV with Neumann boundary conditions on the cube [−1/2, 1/2] d . Its spectrum is discrete, and we let E − (λ) be its ground state energy. It is a simple eigenvalue and λ → E − (λ) is a real analytic concave function defined on R. We first observe:Proposition 0.1. Under the above assumptions (H1) and (H2),For a and b sufficiently small, this result was proven in [17] without the assumption (H2) but with an additional assumption on the sign ofThe method used by Najar relies on a small coupling constant expansion for the infimum of Σ. These ideas were first used in [3] to treat other non 2 monotonous perturbations, in this case magnetic ones, of the Laplace operator. In [1], the authors study the minimum of the almost sure spectrum for a random displacement model i.e. the random potential is defined as V ω (x) = γ∈Z d V (x − γ − ξ γ ) where (ξ γ ) γ are i.i.d. rando...

Low Lying Spectrum of Weak-Disorder Quantum Waveguides

“…It is maybe worthwhile to point out some differences to the recent paper [KN09] of Klopp and Nakamura which is devoted to the proof of Lifshitz tails for alloy-type Schrödinger operators with single site potentials which are allowed to change sign. There are two aspect in common between this work and ours: both of them concern the analysis of the low lying eigenvalues of finite volume random Hamiltonians and both of them deal with non-monotone parameter dependence.…”

“…[BLS08,KN09]. For more intricate properties, like the regularity of the density of states or the analysis of spectral fluctuation boundaries, the difference between monotone and non-monotone models is even more striking.…”

“…On the other hand we assume no reflection symmetry for the individual perturbations. A crucial assumption of [KN09] is that the single site potential obeys such an condition. It allows Klopp and Nakamura to use, after some work and ingenious ideas, an effective decoupling between different random variables (similarly as in the case of fixed-sign single site potentials).…”

“…This aspect of the proof of [KN09] is discussed on page 1134 before the statement of hypothesis (H2) there. In our model there is no such symmetry assumption.…”

“…This means that the analysis of the single parameter random Hamiltonian on a unit cell with Neumann b.c. does not give us the crucial information which was instrumental in the proof strategy of [KN09]. Consequently, we need to analyze the fully interacting model, which leads to an eigenvalue perturbation problem with respect to many parameters.…”