Continuum limit of resonances for discrete Schr\"{o}dinger operators

TL;DR

This paper demonstrates that the complex resonances of discrete Schr"odinger operators converge to those of continuous operators in the continuum limit, enabling approximate computations of resonances for singular potentials.

Contribution

It extends the understanding of resonance convergence from discrete to continuous Schr"odinger operators, including cases with singular potentials, using generalized norm resolvent convergence and complex distortion methods.

Findings

Resonances of discrete models converge to continuous model resonances.

The method applies to potentials with local singularities.

Provides a way to approximate resonances for singular potentials.

Abstract

We consider complex resonances for discrete and continuous Schr\"odinger operators, and we show that the resonances of discrete models converge to resonances of continuous models in the continuum limit. The potential is supposed to be a sum of an exterior dilation analytic function and an exponentially decaying function, which may have local singularities. The proof employs a generalization of the norm resolvent convergence of discrete Schr\"odinger operators by Nakamura and Tadano (2021), combined with the complex distortion method in the Fourier space. Our results confirm that the complex resonances can be approximately computed using discrete Schr\"odinger operators. We also give a recipe for the construction of approximate discrete operators for Schr\"odinger operators with singular potentials.