SEMICLASSICAL ACCURACY FOR BILLIARDS

The effect of the semiclassical approximation on the spectra of billia rds is investigated within the context of the boundary integral method (BIM) by studying, analytically and numerically, the changes in indiv idual eigenvalues when an asymptotic approximation to the kernel of th e BIM is used. A general formula for the shift in an eigenvalue is der ived and then applied to the circle billiard where the semiclassical s hift 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 constan t average behaviour. In this case though, the appearence of periodic o rbits is shown (in an appendix) to impose fluctuations around the aver age behaviour. The numerical application of the BIM introduces numeric al 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 concl usions can be made. Again, both theory and computations imply a consta nt off-set dressed with fluctuations. The principal conclusion is that die semiclassical approximation leads to errors of O(h2BAR) with a co efficient of order 0.01, and hence it provides a very good approximati on to the exact spectrum for a biliard. The opposite conclusion of Pro sen and Robnik is explained away in terms of states affected by causti cs, which do not affect chaotic billiards. Bogomolny's kernel and disc retization procedure are also tested, and while the semiclassical shif ts are similar, the numerical errors are such as to make the spectrum unrecognizable from the real axis when the discretization is done in p osition space.