Quantum Ergodicity for a Point Scatterer on the Three-Dimensional Torus

Abstract

Consider a point scatterer (the Laplacian perturbed by a delta-potential) on the standard three-dimensional flat torus. Together with the eigenfunctions of the Laplacian which vanish at the point, this operator has a set of new, perturbed eigenfunctions. In a recent paper, the author was able to show that all of the perturbed eigenfunctions are uniformly distributed in configuration space. In this paper we prove that almost all of these eigenfunctions are uniformly distributed in phase space, i.e. we prove quantum ergodicity for the subspace of the perturbed eigenfunctions. An analogue result for a point scatterer on the two-dimensional torus was recently proved by Kurlberg and Ueberschär.

On the phase-space distribution of Bloch eigenmodes for periodic point scatterers

“…In light of this it suffices to prove our theorem only for these finite polynomials -the extension to a wider class of functions can be performed by expanding the function in a basis of these polynomials, truncating at some finite order, and controlling the error term (see [19] for details). We are now able to state the main results.…”

“…To deal with phase space we first need to define quantisation. We follow the approach used in [19]. Consider a classical symbol a ∈ C ∞ (S * T 3 ), where S * T 3 T 3 × S 2 .…”

“…For zero quasimomentum the problem of limiting phase space distributions has been studied in two dimensions by Rudnick and Ueberschär [11], and Ueberschär and Kurlberg [17,8], who showed that almost all eigenfunctions equidistribute in position space for all tori, and that in momentum space almost all eigenfunctions either equidistribute for square tori, or localise for tori with a diophantine ratio of side lengths. These results were partially generalised to three dimensions by Yesha [18,19] who showed that for the cubic torus, all eigenfunctions equidistribute in position space, and that almost all eigenfunctions equidistribute in phase space. These results are further complimented by Kurlberg and Rosenzweig [7] who show the existence of localisation in position representation in 2 dimensions and momentum representation in both 2 and 3 dimensions.…”

Small Scale Equidistribution for a Point Scatterer on the Torus

“…Recently, we were able to establish equidistribution in configuration space for tori with two point scatterers [31]. Equidistribution in full phase space (along a density one subsequence) was established both on the standard two-dimensional torus T 2 by Kurlberg and Ueberschär [19], and on the standard three-dimensional torus T 3 [30]. The quantum limits of a point scatterer on a torus with an irrational aspect ratio (also known as the Šeba billiard [23]) were further studied by Kurlberg-Ueberschär [20], who proved the existence of "scars", i.e., localized quantum limits.…”

Uniform distribution of eigenstates on a torus with two point scatterers

“…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].…”