Mathematical Models for Transport in Macroscopic and Mesoscopic Systems - Abstract

Exner, Pavel

Scattering and resonances in leaky quantum-wire systems

We discuss a model of quantum-wire systems which takes tunneling into account, being formally described by Hamiltonians of the type $-Delta-alphadelta(x-Gamma)$ in $L^2(mathbbR^2)$ where $Gammasubset mathbbR^2$ is a metric graph with asymptotically straight “leads”. Such systems can exhibit interesting spectral and scattering properties provided we exclude the trivial case when $Gamma=Gamma_0$ is just a straight line. We will analyze negative-energy scattering in the situation when $Gamma$ is a local deformation of $Gamma_0$. We will also recall an approximation result for these operators which gives numerical hints for existence of resonances due to the global geometry, and mention some open problems.