Semiclassical Fredholm determinant for strongly chaotic billiards

We investigate the 'semiclassical Fredholm determinant' for strongly chaoti c billiards derived from the semiclassical limit of the Fredholm determinan t of the boundary element method. We show that it is the same as a cycle-ex panded Gutzwiller-Voros zeta function when the bounce number of the periodi c orbit with the billiard boundary corresponds to the length of the symboli c sequence of its symbolic dynamical expression. A numerical experiment on a 'concave triangle billiard' shows that the series defining the semiclassi cal Fredholm determinant does not converge absolutely in spite of the absol ute convergence of the series defining the Fredholm determinant. However, t he series behaves like an asymptotic series, and the finite sum obtained by optimal truncation of the series defining the semiclassical Fredholm deter minant gives the semiclassical eigenenergies precisely enough such that the error of the semiclassical approximation is much smaller than the mean spa cing of the exact eigenenergies.