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We consider perturbations of the semiclassical Schr{\"o}dinger equation on acompact Riemannian surface with constant negative curvature and withoutboundary. We show that, for scales of times which are logarithmic in the sizeof the perturbation, the solutions associated to initial data in a smallspectral window become equidistributed in the semiclassical limit. As anapplication of our method, we also derive some properties of the quantumLoschmidt echo below and beyond the Ehrenfest time for initial data in a smallspectral window.

Long-time dynamics of the perturbed Schr\"odinger equation on negatively curved surfaces