# Scarred quasimodes on translation surfaces

## TL;DR

This paper constructs quasimodes for quantum billiards on rational polygons, providing evidence for a conjecture that eigenfunctions localize along specific momentum directions as eigenvalues grow large.

## Contribution

It introduces a method to construct quasimodes on rational polygonal billiards, supporting the conjecture of eigenfunction localization in momentum space.

## Key findings

- Constructed continuous families of quasimodes for rational polygon billiards.
- Produced semi-classical measures supported on dihedral group orbits.
- Provided evidence for eigenfunction localization conjecture.

## Abstract

Rational polygonal billiards are one of the key models among the larger class of pseudo-integrable billiards. Their billiard flow may be lifted to the geodesic flow on a translation surface. Whereas such classical billiards have been much studied in the literature, the analogous quantum billiards have received much less attention. This paper is concerned with a conjecture of Bogomolny and Schmit who proposed in 2004 that the eigenfunctions of the Laplacian on rational polygonal billiards ought to become localized along a finite number of vectors in momentum space, as the eigenvalue tends to infinity. For any given momentum vector $\xi_0\in\mathbb{S}^1$ we construct a continuous family of quasimodes which gives rise to a semi-classical measure whose projection on momentum space is supported on the orbit $D\xi_0$, where $D$ denotes the dihedral group associated with the rational polygon.