Superscarred quasimodes on flat surfaces with conical singularities
Volume 264 / 2022
Abstract
For any given momentum vector $\xi _0\in \mathbb {S}^1$ 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 which are supported on the images of $\xi _0$ under the action of the dihedral group associated with the rational polygon. Hence, we show that these measures are of the form predicted by Bogomolny and Schmit’s superscar conjecture for rational polygons.
