Calculation of spectral determinants

Abstract

A method for regularizing spectral determinants is developed which facilitates their computation from a finite number of eigenvalues. This is used to calcu­late the determinant ∆ for the hyperbola billiard over a range which includes 46 quantum energy levels. The result is compared with semiclassical periodic orbit evaluations of ∆ using the Dirichlet series, Euler product, and a Riemann-Siegel-type formula. It is found that the Riemann-Siegel-type expansion, which uses the least number of orbits, gives the closest approximation. This provides explicit numerical support for recent conjectures concerning the analytic proper­ties of semiclassical formulae, and in particular for the existence of resummation relations connecting long and short pseudo-orbits.