Superscars for Arithmetic Point Scatterers II

TL;DR

This paper studies quantum limits of eigenfunctions for a point scatterer on a 2D torus, showing they can localize or delocalize in momentum space, with limits including exotic measures like Cantor sets.

Contribution

It demonstrates that quantum limits can fully localize on various measures, including singular continuous ones, expanding understanding of possible eigenfunction behaviors.

Findings

Quantum limits can localize on lattice circle measures.

Eigenfunctions can delocalize in position while localizing in momentum.

Existence of quantum limits supported on Cantor sets.

Abstract

We consider momentum push-forwards of measures arising as quantum limits (semi-classical measures) of eigenfunctions of a point scatterer on the standard flat torus $\mathbb T^2 = \mathbb R^2/\mathbb Z^{2}$. Given any probability measure arising by placing delta masses, with equal weights, on $\mathbb Z^2$-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and is completely delocalized in position (i.e., concentration on Lagrangian states.) We also show that the mass, in momentum, can fully localize on more exotic measures, e.g. singular continous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals.