Resonances for Thin Barriers on the Circle

TL;DR

This paper investigates high energy resonances for a quantum operator with frequency-dependent delta potentials on a circular domain, providing bounds on resonance-free regions, resonance bands, and resonance counts.

Contribution

It offers new sharp bounds on resonance-free regions, resonance band locations, and resonance counts for operators with strongly frequency-dependent potentials, specifically in a circular domain.

Findings

Sharp bounds on resonance-free regions for  q  q 1.

Identification of resonance bands for  q  q 1.

Lower bounds on the number of resonances in logarithmic strips.

Abstract

We study high energy resonances for the operator $-\Delta_{V,\partial\Omega}:=-\Delta+\delta_{\partial\Omega}\otimes V $ when $V$ has strong frequency dependence. The operator $-\Delta_{V,\partial\Omega}$ is a Hamiltonian used to model both quantum corrals and leaky quantum graphs. Since highly frequency dependent delta potentials are out of reach of the more general techniques in previous work, we study the special case where $\Omega=B(0,1)\subset \mathbb{R}^2$ and $V\equiv h^{-\alpha }V_0>0$ with $\alpha\leq 1$. Here $h^{-1}\sim \Re \lambda$ is the frequency. We give sharp bounds on the size of resonance free regions for $\alpha\leq 1$ and the location of bands of resonances when $5/6\leq \alpha\leq 1$. Finally, we give a lower bound on the number of resonances in logarithmic size strips: $-M\log \Re \lambda\leq \Im \lambda \leq 0$.