### On quantum limits on flat tori.

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In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $PS{L}_{2}\left(\mathbb{Z}\right)\setminus PS{L}_{2}\left(\mathbb{R}\right)$. This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for $PS{L}_{2}\left(\mathbb{Z}\right)\setminus \mathbb{H}$. The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of $S{L}_{2}\left(\mathbb{R}\right)$. In the proof the key estimates come from applying...

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 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...

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