Discrete vs. continuous in the semiclassical limit

TL;DR

This paper investigates the relationship between discrete and continuous Schrödinger operators with periodic potentials in the semiclassical limit, establishing their spectral convergence and exploring the spectrum's dependence on the semiclassical parameter.

Contribution

It provides explicit convergence rates for the spectral comparison and demonstrates the optimality of these results using Bohr-Sommerfeld quantization and numerical experiments.

Findings

Spectral levels of discrete and continuous operators coincide in the semiclassical limit.

Explicit rate of convergence for the spectral comparison is established.

Spectrum of the discrete operator can be discontinuous with respect to the semiclassical parameter.

Abstract

We compare the bottom of the spectrum of discrete and continuous Schr\"odinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the semiclassical limit and we provide an explicit rate of convergence. We demonstrate the optimality of our results by using Bohr-Sommerfeld quantization conditions for potentials exhibiting non-degenerate wells, and by numerical experiments for more general potentials. We also investigate the dependence of the spectrum of the discrete semiclassical Schr\"odinger operator on the semiclassical parameter $h$ and show that it can be discontinuous.