Stark resonances in a quantum waveguide with analytic curvature

TL;DR

This paper studies how an electric field affects trapped modes in a curved quantum waveguide with analytic curvature, revealing the emergence of resonances near bound states due to the field.

Contribution

It demonstrates that electric fields induce resonances near eigenvalues of the Laplacian in a curved waveguide with analytic curvature decay.

Findings

Electric field causes resonances near bound state eigenvalues.

Resonances depend on the strength and direction of the electric field.

Analytic curvature decay conditions are crucial for the results.

Abstract

We investigate the influence of an electric field on trapped modes arising in a two-dimensional curved quantum waveguide ${\bf \Omega}$ i.e. bound states of the corresponding Laplace operator $-\Delta\_{{\bf \Omega}}$. Here the curvature of the guide is supposed to satisfy some assumptions of analyticity, and decays as $O(|s|^{-\varepsilon}), \varepsilon > 3$ at infinity. We show that under conditions on the electric field $ \bf F$, ${\bf H}(F):= -\Delta\_{{\bf \Omega}} + {\bf F}. {\bf x} $ has resonances near the discrete eigenvalues of $-\Delta\_{{\bf \Omega}}$.