Eigenfunction Statistics for a Point Scatterer on a Three-Dimensional Torus

Abstract

In this paper we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a threedimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show uniform distribution in configuration space for almost all of the perturbed eigenfunctions.

Quantum chaos for point scatterers on flat tori

“…Also note that the result holds for both rational and irrational lattices. The analogous result was proved in [15] for the standard three-dimensional torus and general three-dimensional tori which satisfy certain irrationality conditions. The proof of theorem 4.1 uses the explicit formula (2.15) for Green's function and by approximating Green's function by a sum over a polynomial size interval the problem can be translated into a number theoretical problems about the well-spacedness of lattice points in thin annuli.…”

Small Scale Equidistribution for a Point Scatterer on the Torus

“…Next, we establish small scale equidistribution for the new eigenfunctions of a point scatterer on the standard flat three-dimensional torus T 3 . For balls with radii r > λ −1/12+o (1) , our statement will hold for all new eigenfunctions, improving upon the principal result in [29].…”

“…In [29], it was shown that for a point scatterer on the standard three-dimensional torus T 3 , equidistribution in configuration space holds for all of the new eigenfunctions (and along a density one subsequence of the new eigenfunctions for point scatterers on tori with a Diophantine aspect ratio). Recently, we were able to establish equidistribution in configuration space for tori with two point scatterers [31].…”

Uniform distribution of eigenstates on a torus with two point scatterers

“…Rudnick and Ueberschär proved uniform distribution in configuration space of the perturbed eigenfunctions for a point scatterer on two-dimensional flat tori [9]. This was also proved for three-dimensional flat tori [17], both on the standard square torus and on irrational tori with a Diophantine condition on the side lengths, where in the former case of the standard torus all of the perturbed eigenfunctions equidistribute in configuration space. As for quantum ergodicity in full phase space, it was proved both on the standard two-dimensional flat torus [6] and on the standard three-dimensional torus [18].…”

“…Our results can be easily generalized to nonsquare tori with a Diophantine condition on the difference of the scatterers -see Theorem 1.5 below. In addition, using the methods of [17], Theorem 1.3 can be extended to the standard three-dimensional torus, and also to irrational tori with the same Diophantine condition on the side lengths as in [17].…”