Non-autonomous quantum systems with scale-dependent interface conditions

Article type: Research Article

Authors: Mantile, Andrea

Affiliations: Laboratoire de Mathématiques, Université de Reims – FR3399 CNRS, Moulin de la Housse BP 1039, 51687 Reims, France. E-mail: [email protected]

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→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

DOI: 10.3233/ASY-161359

Journal: Asymptotic Analysis, vol. 98, no. 1-2, pp. 1-30, 2016