Quantum diffusion in the Kronig-Penney model

TL;DR

This paper analyzes quantum diffusion in a 1D periodic point interaction model, establishing optimal decay bounds for the time evolution operator and asymptotic behaviors of band functions at high energies.

Contribution

It provides the first $L^1-L^ty$ decay bounds with optimal order for the Schrf6dinger operator with periodic point interactions and detailed asymptotics of band functions at high energies.

Findings

Decay order $O(t^{-1/3})$ for the time evolution operator.

Asymptotic bounds for high energy coefficients.

Asymptotic analysis of band functions and Bloch waves.

Abstract

In this paper we consider the 1D Schr\"odinger operator $H$ with periodic point interactions. We show an $L^1-L^\infty$ bound for the time evolution operator $e^{-itH}$ restricted to each energy band with decay order $O(t^{-1/3})$ as $t\to \infty$, which comes from some kind of resonant state. The order $O(t^{-1/3})$ is optimal for our model. We also give an asymptotic bound for the coefficient in the high energy limit. For the proof, we give an asymptotic analysis for the band functions and the Bloch waves in the high energy limit. Especially we give the asymptotics for the inflection points in the graphs of band functions, which is crucial for the asymptotics of the coefficient in our estimate.