Lifshits Tails for Randomly Twisted Quantum Waveguides

Abstract

We consider the Dirichlet Laplacian H γ on a 3D twisted waveguide with random Anderson-type twisting γ. We introduce the integrated density of states N γ for the operator H γ , and investigate the Lifshits tails of N γ , i.e. the asymptotic behavior of N γ (E) as E ↓ inf supp dN γ . In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.

Lifshits Tails for Squared Potentials

“…In this section we state our results for the squared random potential V ω = U 2 ω as in (5). As in the conventional case (i. e. for (4)) we obtain Lifshits behavior as in (7).…”

“…In the paper [5] David Krejčiřík and the present authors investigate twisted wave guides M which emerge from the cylinder M = m × R with a cross section m ⊂ R 2 by rotation of m around the axis cylinder at an angle θ which depends on the variable along this axis. The twist function U(x) :=θ(x), x ∈ R, is supposed to be random.…”

“…This observation was the initial motivation for the present paper. In [5] we need only the one-dimensional case of (3) but here we will deal with this model in arbitrary dimension d ≥ 1 as this will not cause additional complications. Obviously, the potential V ω (x) = U ω (x) 2 is non-negative.…”