Quantum Chaos for point scatterers on flat tori

This survey article deals with a delta potential - also known as a point scatterer - on flat 2D and 3D tori. We introduce the main conjectures regarding the spectral and wave function statistics of this model in the so- called weak and strong coupling regimes. We report on recent progress as well as a number of open problems in this field.Comment: 13 pages, 1 figure, added some references and further explanations, Phil. Trans. R. Soc. A to appea

Spectral geometry of flat tori with random impurities

We discuss new results on the geometry of eigenfunctions in disor- dered systems. More precisely, we study tori $R^d/LZ^d$, $d=2,3$, with uniformly distributed Dirac masses. Whereas at the bottom of the spectrum eigenfunctions are known to be localized, we show that for sufficiently large eigenvalue there exist uniformly distributed eigenfunctions with positive probability. We also study the limit $L\to\infty$ with a positive density of random Dirac masses, and deduce a lower polynomial bound for the localization length in terms of the eigenvalue for Poisson distributed Dirac masses on $R^d$ . Finally, we discuss some results on the breakdown of localization in random displacement models above a certain energy threshold.Comment: 8 pages, 1 figure, to appear in the Proceedings of the Journees E. D.

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