Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces

Abstract

We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

Long-Time Dynamics of the Perturbed Schrödinger Equation on Negatively Curved Surfaces

“…In [25,14], the authors looked at magnetic (or metric) perturbations of the Schrödinger operator on a general compact manifold, and they obtained some informations on the pointwise bounds of the solutions of the perturbed Schrödinger equation for a finite time, and for a typical choice of perturbations. In [24], the questions were close to the ones considered in the present article, and we will compare more precisely below our results to those from this reference. The tools used here are in fact the continuation of the ones introduced in [24].…”

“…In [24], the questions were close to the ones considered in the present article, and we will compare more precisely below our results to those from this reference. The tools used here are in fact the continuation of the ones introduced in [24]. As another application of the methods from the present article, we will also deduce some properties on the quantum Loschmidt echo below and beyond the Ehrenfest time -see section 6.…”

“…when V = 0. This question was already discussed in [24]. Precisely, we will now consider sequences of initial data (ψ ) 0< ≤1 which satisfy the following property: (5) lim…”

“…Remark 2.2. Recall that in [24], it was only required that, for every δ 0 > 0, (6) lim →0 + 1 [1−δ 0 ,1+δ 0 ] − 2 ∆ g ψ − ψ L 2 (M ) = 0, and ∀0 < ≤ 1, ψ L 2 = 1. which is more general than (5). In section 4, we will in fact consider slightly more general initial data than the ones satisfying ( 5); yet, the assumptions will still be more restricitive than (6).…”

“…Our goal is to understand the action for short logarithmic times of the "perturbed" Schrödinger propagator e − iτ P ( ) on initial data satisfying (5), and to show that imposing (5) instead of ( 6) allows to improve substantially the results from [24] -see corollary 2.6 below. In section 6, we will also show the relevance of this approach in the study of the quantum Loschmidt echo.…”