Wave operators, similarity and dynamics for a class of Schroedinger operators with generic non-mixed interface conditions in 1D

TL;DR

This paper studies a modified 1D Schrödinger operator with non-mixed interface conditions, developing wave operators to compare its dynamics with a standard Hamiltonian, and providing uniform-in-time estimates of the associated semigroup.

Contribution

It introduces a method to construct stationary wave operators for non-selfadjoint Schrödinger operators with interface conditions, enabling dynamical comparison with physical Hamiltonians.

Findings

Uniform-in-time estimates of the perturbed semigroup are obtained.

The distance between the semigroups is controlled by the deformation parameter |θ|.

The approach facilitates analysis of quantum transport in heterostructures.

Abstract

We consider a simple modification of the 1D-Laplacian where non-mixed interface conditions occur at the boundaries of a finite interval. It has recently been shown that Schr\"odinger operators having this form allow a new approach to the transverse quantum transport through resonant heterostructures. In this perspective, it is important to control the deformations effects introduced on the spectrum and on the time propagator by this class of non-selfadjont perturbations. In order to obtain uniform-in-time estimates of the perturbed semigroup, our strategy consists in constructing stationary waves operators allowing to intertwine the modified non-selfadjoint Schr\"odinger operator with a 'physical' Hamiltonian. For small values of a deformation parameter '{\theta}', this yields a dynamical comparison between the two models showing that the distance between the corresponding semigroups is dominated by |{\theta}| uniformly in time in the L^2-operator norm.