Gamow Vectors in a Periodically Perturbed Quantum System

TL;DR

This paper rigorously defines Gamow vectors and resonances in a time-dependent quantum system with a periodically perturbed delta potential, analyzing resonance behavior and its physical implications through theoretical and numerical methods.

Contribution

It introduces a rigorous, physically relevant framework for Gamow vectors in a time-dependent quantum system with periodic perturbation, including explicit resonance calculations.

Findings

Resonances are characterized by a Borel summable expansion involving complex exponential terms.

For small perturbation amplitude, there is typically one resonance for generic initial conditions.

The resonance position relates to multiphoton ionization phenomena.

Abstract

We analyze the behavior of the wave function $\psi(x,t)$ for one dimensional time-dependent Hamiltonian $H=-\partial_x^2\pm2\delta(x)(1+2r\cos\omega t)$ where $\psi(x,0)$ is compactly supported. We show that $\psi(x,t)$ has a Borel summable expansion containing finitely many terms of the form $\sum_{n=-\infty}^{\infty} e^{i^{3/2}\sqrt{-\lambda_{k}+n\omegai}|x|} A_{k,n} e^{-\lambda_{k}t+n\omega it}$, where $\lambda_k$ represents the associated resonance. This expression defines Gamow vectors and resonances in a rigorous and physically relevant way for all frequencies and amplitudes in a time-dependent model. For small amplitude ($|r|\ll 1$) there is one resonance for generic initial conditions. We calculate the position of the resonance and discuss its physical meaning as related to multiphoton ionization. We give qualitative theoretical results as well as numerical calculations in the general case.