Non-autonomous quantum systems with scale-dependent interface conditions

TL;DR

This paper studies modified Schrödinger operators with h-dependent interface conditions, showing they cause minimal impact on dynamics as h approaches zero, and provides stability and small-h expansion results for non-autonomous quantum systems.

Contribution

It introduces a new class of non-autonomous quantum systems with scale-dependent interface conditions and analyzes their stability and small-h behavior.

Findings

Small perturbation on dynamics as h->0

Uniform-in-h stability estimates for propagators

Small-h expansion of the non-autonomous system

Abstract

We consider a class of modified Schroedinger operators where the semiclassical Laplacian is perturbed with h-dependent interface conditions occurring at the boundaries of the potential's support. Under positivity assumptions on the potential, we show that this modification produces a small perturbation on the dynamics as h->0, independently from the time scale. In the case of a time dependent potential, this yields uniform-in-h stability estimates for products of instantaneous propagators. Then, following a standard approach, the non-autonomous dynamical system is defined as a limit of stepwise propagators and its small-h expansion is provided under suitable regularity assumptions on the potential's variations.