On resumming periodic orbits in the spectra of integrable systems

Abstract

Spectral determinants have proven to be valuable tools for resumming the periodic orbits in the Gutzwiller trace formula of chaotic systems. We investigate these tools in the context of integrable systems to which these techniques have not been previously applied. Our specific model is a stroboscopic map of an integrable Hamiltonian system with quadratic action dependence, for which each stage of the semiclassical approximation can be controlled. It is found that large errors occur in the semiclassical traces due to edge corrections which may be neglected if the eigenvalues are obtained by Fourier transformation over the long time dynamics. However, these errors cause serious harm to the spectral approximations of an integrable system obtained via the spectral determinants. The symmetry property of the spectral determinant does not generally alleviate the error, since it sometimes sheds a pair of eigenvalues from the unit circle. By taking into account the leading order asymptotics of the edge corrections, the spectral determinant method makes a significant recovery.