Semiclassical Fredholm determinant for strongly chaotic billiards

We investigate the `semiclassical Fredholm determinant' for strongly chaotic billiards derived from the semiclassical limit of the Fredholm determinant of the boundary element method. We show that it is the same as a cycle-expanded Gutzwiller - Voros zeta function when the bounce number of the periodic orbit with the billiard boundary corresponds to the length of the symbolic sequence of its symbolic dynamical expression. A numerical experiment on a `concave triangle billiard' shows that the series defining the semiclassical Fredholm determinant does not converge absolutely in spite of the absolute convergence of the series defining the Fredholm determinant. However, the series behaves like an asymptotic series, and the finite sum obtained by optimal truncation of the series defining the semiclassical Fredholm determinant gives the semiclassical eigenenergies precisely enough such that the error of the semiclassical approximation is much smaller than the mean spacing of the exact eigenenergies.