Long-time dynamics of the perturbed Schrödinger equation on negatively curved surfaces

Gabriel Riviere (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

We consider perturbations of the semiclassical Schrödinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window.

Mots clés: Semiclassical Analysis; Schr\odinger equation; Negatively curved manifolds; Quantum Loschmidt echo; Strong structural stability

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]

Gabriel Riviere. Long-time dynamics of the perturbed Schrödinger equation on negatively curved surfaces. 2014. ⟨hal-01095014⟩