Adiabatic theorems for quantum resonances

TL;DR

This paper investigates the adiabatic evolution of quantum resonances across different scenarios, employing advanced mathematical frameworks to understand their behavior over short time scales relative to their lifetimes.

Contribution

It introduces new adiabatic theorems tailored for quantum resonances, including shape, complex eigenvalue, and embedded spectrum cases, extending existing theory without requiring spectral gaps.

Findings

Established adiabatic theorems for shape resonances using time-energy uncertainty.

Extended adiabatic theory to resonances as complex eigenvalues in deformed Hamiltonians.

Analyzed adiabatic evolution for embedded eigenvalues without spectral gap assumptions.

Abstract

We study the adiabatic time evolution of quantum resonances over time scales which are small compared to the lifetime of the resonances. We consider three typical examples of resonances: The first one is that of shape resonances corresponding, for example, to the state of a quantum-mechanical particle in a potential well whose shape changes over time scales small compared to the escape time of the particle from the well. Our approach to studying the adiabatic evolution of shape resonances is based on a precise form of the time-energy uncertainty relation and the usual adiabatic theorem in quantum mechanics. The second example concerns resonances that appear as isolated complex eigenvalues of spectrally deformed Hamiltonians, such as those encountered in the N-body Stark effect. Our approach to study such resonances is based on the Balslev-Combes theory of dilatation-analytic Hamiltonians and an adiabatic theorem for nonnormal generators of time evolution. Our third example concerns resonances arising from eigenvalues embedded in the continuous spectrum when a perturbation is turned on, such as those encountered when a small system is coupled to an infinitely extended, dispersive medium. Our approach to this class of examples is based on an extension of adiabatic theorems without a spectral gap condition. We finally comment on resonance crossings, which can be studied using the last approach.