Improved eigenvalue sums for inferring quantum billiard geometry

Abstract

An algorithm is described for obtaining successive approximations to geometric properties Kj of a closed boundary B (such as its length L or the area A within it), given the lowest N eigenvalues (En) of some wave operator defined on the domain bounded by B. The technique is based on the asymptotic expansion of the partition function for small t: Phi (t)= Sigma n=1infinity exp(-Ent) approximately t-1 Sigma j=0infinity Kjtj/2. Four different billiards are employed to illustrate the method. The first is the rectangular membrane, which is classically integrable; for the other three, B is an Africa shape, which is classically chaotic: Africa membrane, Africa Aharonov-Bohm billiard and Africa neutrino (massless Dirac) billiard. A typical result is that A and L can be reconstructed from 125 eigenvalues to a few parts in 104 (for the rectangle the accuracy is even higher). The efficiency of the reconstruction algorithm appears to be independent of the classical chaology of B.