Bound States for Nano-Tubes with a Dislocation

TL;DR

This paper investigates the emergence of bound states at the interface of two potentials on a cylindrical surface, showing how discrete spectrum appears and varies with a parameter, without requiring periodicity.

Contribution

It demonstrates the creation and continuous variation of interface-bound states in a non-periodic setting for potentials on a cylindrical domain.

Findings

Discrete spectrum appears in the spectral gap at certain parameter values.

Essential spectrum remains unaffected by the interface parameter.

Eigenvalues vary Lipschitz continuously with the parameter t.

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 $S=\mathbb R \times (\mathbb R/\mathbb Z)$ where we assume that there exists an interval $(a_0,b_0)$ which is free of spectrum of $-\Delta+V^{(k)}$ for $k=1,2$. We are then interested in the spectrum of $H_t = -\Delta + V_t$, for $t \in \mathbb R$, where $V_t(x,y) = V^{(1)}(x,y)$, for $x > 0$, and $V_t(x,y) = V^{(2)}(x+t,y)$, for $x < 0$. While the essential spectrum of $H_t$ 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 $-\Delta + V^{(2)}$ has some essential spectrum in $(-\infty, a_0]$. We do not require $V^{(1)}$ or $V^{(2)}$ to be periodic. We furthermore show that the discrete eigenvalues of $H_t$ are Lipschitz continuous functions of $t$ if the potential $V^{(2)}$ is locally of bounded variation.