Semiclassical quantization of highly excited scar states

Abstract

The semiclassical quantization of Hamiltonian systems with classically chaotic dynamics is restricted to low excited states, close to the ground state, because the number of required periodic orbits grows exponentially with energy. Nevertheless, here we demonstrate that it is possible to find eigenenergies of highly excited states scarred by a short periodic orbit. Specifically, by using 18146 homoclinic orbits (HO)s of the shortest periodic orbit of the hyperbola billiard, we find eigenenergies of the strongest scars over a range which includes 630 even eigenfunctions. The analysis of data reveals that the used semiclassical formula presents two regimes. First, when all HOs with excursion time smaller than the Heisenberg time tH are included, the error is around 3.3% of the mean level spacing. Second, in the energy region defined by , where is the maximum excursion time included in the calculation, the error is around 15% of the mean level spacing.