Local energy decay and smoothing effect for the damped Schr{\"o}dinger equation

TL;DR

This paper establishes local energy decay and smoothing effects for the damped Schrödinger equation on R^d, using resolvent estimates and Mourre theory, with results depending on dissipation strength.

Contribution

It introduces a novel application of the dissipative Mourre method to analyze energy decay and smoothing in damped Schrödinger equations with long-range metric perturbations.

Findings

Proves local energy decay for the damped Schrödinger equation.

Establishes smoothing effects depending on dissipation strength.

Uses uniform resolvent estimates via Mourre theory.

Abstract

We prove the local energy decay and the smoothing effect for the damped Schr{\"o}dinger equation on R^d. The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric. The proofs are based on uniform resolvent estimates obtained by the dissipative Mourre method. All the results depend on the strength of the dissipation which we consider.