141 research outputs found
The quantum Loschmidt echo on flat tori
The Quantum Loschmidt Echo is a measurement of the sensitivity of a quantum
system to perturbations of the Hamiltonian. In the case of the standard
2-torus, we derive some explicit formulae for this quantity in the transition
regime where it is expected to decay in the semiclassical limit. The expression
involves both a two-microlocal defect measure of the initial data and the form
of the perturbation. As an application, we exhibit a non-concentration
criterium on the sequence of initial data under which one does not observe a
macroscopic decay of the Quantum Loschmidt Echo. We also apply our results to
several examples of physically relevant initial data such as coherent states
and plane waves.Comment: 34 page
Poincar{\'e} series and linking of Legendrian knots
On a negatively curved surface, we show that the Poincar{\'e} series counting
geodesic arcs orthogonal to some pair of closed geodesic curves has a
meromorphic continuation to the whole complex plane. When both curves are
homologically trivial, we prove that the Poincar{\'e} series has an explicit
rational value at 0 interpreting it in terms of linking number of Legendrian
knots. In particular, for any pair of points on the surface, the lengths of all
geodesic arcs connecting the two points determine its genus, and, for any pair
of homologically trivial closed geodesics, the lengths of all geodesic arcs
orthogonal to both geodesics determine the linking number of the two geodesics.Comment: Minor modifications, 78
Length orthospectrum of convex bodies on flat tori
In analogy with the study of Pollicott-Ruelle resonances on negatively curved
manifolds, we define anisotropic Sobolev spaces that are well-adapted to the
analysis of the geodesic vector field on the flat torus. Among several
applications of this functional point of view, we study properties of geodesics
that are orthogonal to two convex subsets of the torus (i.e. projection of the
boundaries of strictly convex bodies of the Euclidean space). Associated to the
set of lengths of such orthogeodesics, we define a geometric Epstein function
and prove its meromorphic continuation. We compute its residues in terms of
intrinsic volumes of the convex sets. We also prove Poisson-type summation
formulas relating the set of lengths of orthogeodesics and the spectrum of
magnetic Laplacians
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