Improved eigenvalue sums for inferring quantum billiard geometry

Abstract

An algorithm is described for obtaining successive approximations to geometric properties K_j of a closed boundary B (such as its length L or the area A within it), given the lowest N eigenvalues (E_n) 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=1^infinity exp(-E_nt) approximately t^-1 Sigma _j=0^infinity K_jt^j/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.