Newton polygons and resonances of multiple delta-potentials

TL;DR

This paper analyzes the asymptotic distribution of semiclassical scattering resonances for multiple delta-function potentials, revealing resonance lines and bounding their parameters using Newton polygons, with numerical evidence for complex configurations.

Contribution

It introduces a method using Newton polygons to explicitly describe resonance locations for multiple delta potentials, extending understanding beyond two or three poles.

Findings

Resonances occur along specific lines with imaginary parts proportional to -h log(1/h).

The number of such resonance lines is finite and bounded.

Numerical evidence suggests more resonance lines appear with increasing delta poles.

Abstract

We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb{R}$. In the cases of two or three delta poles, we are able to show that resonances occur along specific lines of the form $\Im z \sim -\gamma h \log(1/h).$ More generally, we use the method of Newton polygons to show that resonances near the real axis may only occur along a finite collection of such lines, and we bound the possible number of values of the parameter $\gamma.$ We present numerical evidence of the existence of more and more possible values of $\gamma$ for larger numbers of delta poles.