# Small Scale Equidistribution for a Point Scatterer on the Torus

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## Abstract

We study the small scale distribution of the eigenfunctions of a point scatterer (the Laplacian perturbed by a delta potential) on two- and three-dimensional flat tori. In two dimensions, we establish small scale equidistribution for the “new” eigenfunctions holding all the way down to the Planck scale. In three dimensions, small scale equidistribution is established for all of the “new” eigenfunctions at certain scales.

Original language English 199-224 26 Communications in Mathematical Physics 377 1 https://doi.org/10.1007/s00220-019-03669-0 Published - 1 Jul 2020

### Bibliographical note

Funding Information:
The author would like to express his gratitude to Z. Rudnick and I. Wigman for useful discussions and comments. The research leading to these results was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC Grant Agreement No. 335141. 1 The normalization by 2 π \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\pi$$\end{document} is introduced to facilitate the notation below.