### Acceleration of material waves in Fermi accelerator.

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A quantum dynamical system, mimicking the classical phase doubling map $z\mapsto {z}^{2}$ on the unit circle, is formulated and its ergodic properties are studied. We prove that the quantum dynamical entropy equals the classical value log2 by using compact perturbations of the identity as operational partitions of unity.

We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on ${\mathrm{PSL}}_{2}\left(\mathbf{Z}\right)\setminus {\mathrm{PSL}}_{2}\left(\mathbf{R}\right)$. This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution on ${\mathrm{PSL}}_{2}\left(\mathbf{Z}\right)\setminus \mathbf{H}$.

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on ${S}^{2}$. We also construct a solution of the equation $\Delta u=u$ in ${\mathbb{R}}^{2}$ that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

We consider a nonlinear area preserving Anosov map $M$ on the torus phase space, which is the simplest example of a fully chaotic dynamics. We are interested in the quantum dynamics for long time, generated by the unitary quantum propagator $\widehat{M}$. The usual semi-classical Trace formula expresses $Tr\left({\widehat{M}}^{t}\right)$ for finite time $t$, in the limit $\hslash \to 0$, in terms of periodic orbits of $M$ of period $t$. Recent work reach time $t\ll {t}_{E}/6$ where ${t}_{E}=log(1/\hslash )/\lambda $ is the Ehrenfest time, and $\lambda $ is the Lyapounov coefficient. Using a semi-classical normal form...

We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $SU\left(2\right)$, providing an elementary solution of Ruziewicz problem on ${S}^{2}$ as well as giving many new examples of finitely generated subgroups of $SU\left(2\right)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $\mathbf{R}[SU(2\left)\right]$ in the $N$-th irreducible representation of $SU\left(2\right)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $\mathbf{R}[SU(2\left)\right]$, the “unfolded” consecutive spacings...