Lifshits Tails for Squared Potentials

Abstract

We consider Schrödinger operators with a random potential which is the square of an alloy-type potential. We investigate their integrated density of states and prove Lifshits tails. Our interest in this type of models is triggered by an investigation of randomly twisted waveguides.

Dimensional ensemble and (topological) fracton thermodynamics: the slow route to equilibrium

“…Decades of study uncovered the thermal properties of glasses, particularly their exponentially slow time behaviour in reaching equilibrium at low-temperatures 1 . This behaviour can be natural in weakly disordered systems where, in the core of the Lifshitz tails 2–5 , the Helmholtz free energy behaves like as well as for some gases at low temperatures 6 ( σ amount of disorder and T temperature). However, as pointed out by Chamon 7 , thermodynamic glass behaviour may exist in ordered systems.…”

Lifshits Tails for Randomly Twisted Quantum Waveguides

“…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 hθ ,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.…”

“…In Subsection 4.1 we consider single-site twisting w of power-like decay while in Subsection 4.2 we handle the case of compactly supported w.4.1.Single-site twisting w of power-like decay. The following proposition contains results from[15] on the Lifshits tails for 1D Schrödinger operators with squared random Anderson-type potentials.Proposition 4.1 ([15, Theorem 1]). Assume (3.3).…”