Perturbation of the Semiclassical Schrödinger Equation on Negatively Curved Surfaces

Abstract

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 with g * the metric induced by g on the cotangent bundle.…”

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

“…In the context of negatively curved surfaces, some equidistribution properties of the elements of N (τ, ǫ) were obtained for strong enough perturbations [16,44].…”

The quantum Loschmidt echo on flat tori

“…Finally, we emphasize that, even if the Quantum Loschmidt Echo is now a rather well studied and understood quantity in the physics literature, much less seems to be known from the mathematical perspective. For recent mathematical results we refer the reader to [5,7] for R d , to [10,6] for general compact manifolds, to [9,24] for negatively curved surfaces and to [19] for Zoll manifolds.…”