SCARS OF PERIODIC-ORBITS IN THE STADIUM ACTION BILLIARD

Compact billiards in phase space, or action billiards, are constructed by truncating the classical Hamiltonian in the action variables. The corresponding quantum mechanical system has a finite Hamiltonian matri x. In previous papers we defined the compact analog of common billiard s, i.e., straight motion in phase space followed by specular reflectio ns al the boundaries. Computation of their quantum energy spectra esta blishes that their properties are exactly those of common billiards: t he short-range statistics follow the known universality classes depend ing on the regular or chaotic nature of the motion, while the long-ran ge fluctuations are determined by the periodic orbits. In this work we show that the eigenfunctions also follow qualitatively the general ch aracteristics of common billiards. In particular, we show that the low -lying levels can be classified according to their nodal lines as usua l and that the high excited states present scars of several short peri odic orbits. Moreover, since all the eigenstates of action billiards c an be computed with great accuracy, Bogomolny's semiclassical formula For the scars can also be tested successfully.