Exponential decay estimates for the damped defocusing Schrödinger equation in exterior domains

Abstract

We are concerned with the existence as well as the exponential stability in H1-level for the damped defocusing Schrödinger equation posed in a two-dimensional exterior domain Ω with smooth boundary ∂Ω. The proofs of the existence are based on the properties of pseudo-differential operators introduced in Dehman et al. [Math. Z. 254, 729–749 (2006)] and a Strichartz estimate proved by Anton [Bull. Soc. Math. Fr. 136, 27–65 (2008)], while the exponential stability is achieved by combining arguments firstly considered by Zuazua [J. Math. Pures Appl. 9, 513–529 (1991)] for the wave equation adapted to the present context and a global uniqueness theorem. In addition, we proved propagation results for the linear Schrödinger equation for any dimensional and for any (reasonable) boundary conditions employing microlocal analysis.