Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach

TL;DR

This paper introduces artificial interface conditions for 1D Schrödinger operators, enabling an adiabatic theory for shape resonances and analyzing their impact on quantum transport with controllable precision.

Contribution

It develops a novel approach using complex interface conditions to study the adiabatic evolution of shape resonances in 1D quantum systems.

Findings

Artificial interface conditions effectively unveil shape resonances.

The adiabatic theory applies to large time scales.

Stationary quantities in quantum transport are minimally affected.

Abstract

Artificial interface conditions parametrized by a complex number $\theta_{0}$ are introduced for 1D-Schr\"odinger operators. When this complex parameter equals the parameter $\theta\in i\R$ of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale $(h^N)_{N\in \N}$ as $h\to 0$, according to $\theta_{0}$.