Semiclassical accuracy for billiards

The effect of the semiclassical approximation on the spectra of billiards is investigated within the context of the boundary integral method (BIM) by studying. analytically and; numerically, the changes in individual eigenvalues when an asymptotic approximation to the kernel of the BIM is used. A general formula for the shift in an eigenvalue is derived and then applied to the circle billiard where the semiclassical shift between the exact and semiclassical spectra is shown to approach a constant. It is then evaluated approximately for chaotic billiards. again showing how the semiclassical shift to be expected has a constant average behaviour. In this case though, the appearance of periodic orbits is shown (in an appendix) to impose fluctuations around the average behaviour. The numerical application of the BIM introduces numerical errors into the eigenvalues found. A general formula is derived and shown to reduce to the correct result for the circle billiard. It is also tested against two chaotic billiards, although only limited conclusions can be made. Again, both theory and computations imply a constant off-set dressed with fluctuations. The principal conclusion is that the semiclassical approximation leads to errors of O(h(cross) 2 ) with a coefficient of order 0.01, and hence it provides a very good approximation to the exact spectrum for a billiard.