Bound states for nano-tubes with a dislocation

Abstract

Abstract. As a model for an interface in solid state physics, we consider two real-valued potentials V (1) and V (2) on the cylinder or tube S = R × (R/Z) where we assume that there exists an interval (a 0 , b 0 ) which is free of spectrum of −∆ + V (k) for k = 1, 2. We are then interested in the spectrum of Ht = −∆ + Vt, for t ∈ R, where Vt(x, y) = V (1) (x, y), for x > 0, and Vt(x, y) = V (2) (x + t, y), for x < 0. While the essential spectrum of Ht is independent of t, we show that discrete spectrum, related to the interface at x = 0, is created in the interval (a 0 , b 0 ) at suitable values of the parameter t, provided −∆ + V (2) has some essential spectrum in (−∞, a 0 ]. We do not require V (1) or V (2) to be periodic. We furthermore show that the discrete eigenvalues of Ht are Lipschitz continuous functions of t if the potential V (2) is locally of bounded variation.

Spectral properties of periodic systems cut at an angle

“…The spectral properties of such Hamiltonians have been studied e.g. by Davies and Simon [5] and Hempel and Kohlmann [7][8][9][10]. The map t → H (t ) is L-periodic.…”

“…The main tool that we use is the existence of a spectral flow when tan(θ) is rational, and a limiting argument. Apart for the last part, we mostly follow the arguments by Hempel and Kohlmann in [7][8][9][10].…”

An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential

“…We may assume that m is real-valued nonnegative, otherwise we split the operator into the sum of four such operators according to m = m Next we use these estimates in the endpoint case α = 0 for the analysis of T λ,α from (4) with α ∈ (0, 1). Up to an α-dependent prefactor, these operators may be embedded into the family of operators (28) T λ,s h := e (1−s)…”

“…We assume λ ≥ 0 and m ∈ C([a, b]). Since we are going to apply Stein's Interpolation Theorem (Theorem 1 in [38]) to the family (T λ,σs ) s∈S where S := {s ∈ C : 0 ≤ Re(s) ≤ 1} and σ ∈ [0, 1] (including the endpoint case σ = 1), we need to extend the operators from (28) to the line Re(s) = 1 in a continuous way. Only for this reason we will temporarily assume m ∈ C 1 ([a, b]), but we will see that this extra assumption is actually not necessary.…”

“…We will without loss of generality assume V 1 > V 2 in the following. Examples for elliptic problems involving interfaces modelled by potentials of this kind can be found in [14,Theorem 1], [15,Theorem 2] or [28]. To explain the motivation behind our study, we recall the interesting phenomenon called "double scattering".…”