ABSTRACT
Using ensembles of two, three, and four spheres immersed in a fermionic background we evaluate the (integrated) density of states and the Casimir energy. We thus infer that for sufficiently smooth objects, whose various geometric characteristic lengths are larger then the Fermi wave length one can use the simplest semiclassical approximation (the contribution due shortest periodic orbits only) to evaluate the Casimir energy. We also show that the Casimir energy for several objects can be represented fairly accurately as a sum of pairwise Casimir interactions between pairs of objects.
ABSTRACT
The high-lying resonances in the quantum mechanical scattering problem of a point particle from two or three equally sized (and spaced) circular hard disks in the two-dimensional plane are predicted quite well by the classical cycle expansion. There are, however, noticeable deviations for the lowest resonances. Therefore, the leading corrections from creeping paths to the cycle expansion in the two-disk scattering problem are constructed. Generalizations to the three-disk problem are indicated. The size of the corrections are estimated. They are shown to be too small to account for the deviations mentioned above. Finally, arguments are given that, for the two- and three-disk problem, the semiclassical predictions of the low-lying resonance poles are bound to fail.
