Calculation by scaling of highly excited states of billiards

Abstract

We propose a method which directly gives all eigenvalues and eigenfunctions of a two-dimensional billiard in a narrow energy range by solving a generalized eigenvalue problem. This is possible through the use of scaling which allows us to write the boundary norm as a function of energy. The dimension of the problem is of the order of the number of wavelengths in the perimeter, and the number of states obtained is a fraction (0.05 or 0.1 depending on the accuracy required) of it. The method can be applied to general star-shaped domains and is illustrated for the 2\ifmmode\times\else\texttimes\fi{}4 Bunimovich stadium where we obtain states around the 568 000th state with the precision of ${10}^{\mathrm{\ensuremath{-}}4}$ of the mean level spacing.