On the adiabatic theorem when eigenvalues dive into the continuum

TL;DR

This paper investigates the behavior of bound states in a quantum dot system as its energy varies adiabatically, showing that the probability of the state surviving vanishes when the eigenvalue enters the continuum.

Contribution

It introduces a model analyzing the adiabatic evolution of bound states that dive into the continuum, demonstrating the vanishing survival probability in this scenario.

Findings

Survival probability of the bound state tends to zero in the adiabatic limit.

The model applies to a class of couplings in a two-channel quantum system.

Method suggests potential extensions to other Hamiltonian classes.

Abstract

We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.