Superscarred quasimodes on flat surfaces with conical singularities

TL;DR

This paper constructs a continuous family of quasimodes for the Laplace-Beltrami operator on flat surfaces with conical singularities, applying the results to rational polygonal billiards and confirming a superscar conjecture.

Contribution

It introduces a novel method to construct quasimodes on translation surfaces and rational polygons, providing evidence for the superscar conjecture in these settings.

Findings

Constructed a continuous family of quasimodes with controlled spectral width.

Showed semiclassical measures project to finite sums of Dirac measures, confirming superscar conjecture.

Applied the construction to rational polygonal billiards and Neumann Laplacian.

Abstract

We construct a continuous family of quasimodes for the Laplace-Beltrami operator on a translation surface. We apply our result to rational polygonal quantum billiards and thus construct a continuous family of quasimodes for the Neumann Laplacian on such domains with spectral width ${\mathcal{O}}_\varepsilon(\lambda^{3/8+\varepsilon})$. We show that the semiclassical measures associated with this family of quasimodes project to a finite sum of Dirac measures on momentum space, hence, they satisfy Bogomolny and Schmit's superscar conjecture for rational polygons.