Straight quantum layer with impurities inducing resonances

TL;DR

This paper studies a 3D quantum layer with a wire-like impurity, showing the emergence of embedded eigenvalues and their transformation into resonances upon adding surface impurities, with analysis of their asymptotic behavior.

Contribution

It demonstrates the existence of infinitely many embedded eigenvalues and their transition to resonances due to surface impurities, providing asymptotic analysis of these phenomena.

Findings

Infinite embedded eigenvalues in the quantum layer.

Surface impurities induce resonances from embedded eigenvalues.

Asymptotic behavior of resonance imaginary parts analyzed.

Abstract

We consider a straight three dimensional quantum layer with singular potential supported on a straight wire which is localized perpendicularly to the walls and connects them. We prove that the infinite number of embedded eigenvalues appears in this system. Furthermore, we show that after introducing a small surface impurity to the layer, the embedded eigenvalues turn to the second sheet resolvent poles which state resonances. We discuss the asymptotics of the imaginary component of the resolvent pole with respect to the surface area.