Non-autonomous quantum systems with scale-dependent interface conditions

Abstract

We consider a class of modified Schrödinger 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 $h \to 0$ , 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.

Keywords

non-autonomous quantum systems nonselfadjoint operators interface conditions

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