Nonconcentration in partially rectangular billiards

TL;DR

This paper investigates how eigenfunctions distribute their mass in certain partially rectangular billiards at high energies, using adiabatic ansatz and control estimates to establish nonconcentration results.

Contribution

It introduces a novel approach combining adiabatic ansatz and spectral analysis to estimate eigenfunction mass distribution in partially rectangular billiards.

Findings

Eigenfunctions do not concentrate in the non-rectangular regions at high energy.

The method applies sharp one-dimensional control estimates.

Results depend on the energy not resonating with the rectangular spectrum.

Abstract

In specific types of partially rectangular billiards we estimate the mass of an eigenfunction of energy $E$ in the region outside the rectangular set in the high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy form with a second quadratic form for which separation of variables applies. This allows us to use sharp one-dimensional control estimates and to derive the bound assuming that $E$ is not resonating with the Dirichlet spectrum of the rectangular part.