Bound States for Nano-Tubes with a Dislocation

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.