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.

29 pages

References in corpus (2)

• A variational approach to dislocation problems for periodic Schrödinger operators
• Schrödinger operator with a junction of two 1-dimensional periodic potentials

Cited by in corpus (5)

• Characterization of edge states in perturbed honeycomb structures
• Spectral properties of periodic systems cut at an angle
• Periodic Dirac operator on the half-line
• Mean--field stability for the junction of quasi 1D systems with Coulomb interactions
• Eigenvalues of periodic difference operators on lattice octant