Time asymptotics of the Schroedinger wave function in time-periodic potentials

TL;DR

This paper analyzes the long-term behavior of quantum wave functions under time-periodic potentials, providing criteria for complete ionization and revealing that such delocalization is a generic phenomenon in these systems.

Contribution

It offers a comprehensive analytic framework for the time asymptotics of wave functions in time-periodic potentials, including a criterion for full ionization and insights into the spectral properties of the Floquet operator.

Findings

Complete ionization is shown to be a generic outcome.

A criterion for full delocalization is established.

The analysis applies to potentials with compact support and arbitrary forcing magnitude.

Abstract

We study the transition to the continuum of an initially bound quantum particle in $\RR^d$, $d=1,2,3$, subjected, for $t\ge 0$, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete analytic information on the time Laplace transform of the wave function. From this, comprehensive time asymptotic properties (Borel summable transseries) follow. We obtain in particular a criterion for whether the wave function gets fully delocalized (complete ionization). This criterion shows that complete ionization is generic and provides a convenient test for particular cases. When satisfied it implies absence of discrete spectrum and resonances of the associated Floquet operator. As an illustration we show that the parametric harmonic perturbation of a potential chosen to be any nonzero multiple of the characteristic function of a measurable compact set has this property.