An explicit model for the adiabatic evolution of quantum observables driven by 1D shape resonances

TL;DR

This paper develops an explicit model for the adiabatic evolution of quantum observables in a 1D system with shape resonances, using a semiclassical island and delta interaction to analyze charge accumulation.

Contribution

It introduces a solvable model combining complex deformation and interface conditions to describe adiabatic quantum dynamics with shape resonances.

Findings

Derived a reduced equation for charge density evolution.

Analyzed the impact of shape resonances on quantum observables.

Provided explicit solutions for the model's adiabatic behavior.

Abstract

This paper is concerned with a linearized version of the transport problem where the Schr\"{o}dinger-Poisson operator is replaced by a non-autonomous Hamiltonian, slowly varying in time. We consider an explicitly solvable model where a semiclassical island is described by a flat potential barrier, while a time dependent 'delta' interaction is used as a model for a single quantum well. Introducing, in addition to the complex deformation, a further modification formed by artificial interface conditions, we give a reduced equation for the adiabatic evolution of the sheet density of charges accumulating around the interaction point.