Complete ionization for a non-autonomous point interaction model in d = 2

TL;DR

This paper studies a 2D quantum system with a time-dependent point interaction, proving well-posedness and analyzing the decay of bound state survival probability over time.

Contribution

It establishes global well-posedness for the Schrödinger equation with a time-dependent delta potential and characterizes the asymptotic decay of bound state survival probability.

Findings

Proves global well-posedness of the model

Shows survival probability decays as (log t / t)^2

Provides conditions for no-resonance in periodic potentials

Abstract

We consider the two dimensional Schr\"odinger equation with time dependent delta potential, which represents a model for the dynamics of a quantum particle subject to a point interaction whose strength varies in time. First, we prove global well-posedness of the associated Cauchy problem under general assumptions on the potential and on the initial datum. Then, for a monochromatic periodic potential (which also satisfies a suitable no-resonance condition) we investigate the asymptotic behavior of the survival probability of a bound state of the time-independent problem. Such probability is shown to have a time decay of order $\mathcal{O}(\log t/t)^2$, up to lower order terms.