Initial-value representation of the semiclassical zeta function

This work casts the semiclassical zeta function in a form suitable for practical calculations of energy levels for rather general systems. To accomplish this, the zeta function is approximated by applying an initial-value representation (IVR) treatment to the traces of the transfer matrix that appear when the function is expanded in cumulants. Because this approach does not require searches for periodic orbits or special trajectories obeying double-ended boundary conditions, it is easily applicable to multidimensional systems with smooth potentials. Calculations are presented for the energy levels of three two-dimensional systems, including one that is classically integrable, one having mixed phase space, and one that is almost fully chaotic. The results show that the present treatment is far more numerically efficient than a previously proposed IVR method for the zeta function [Barak and Kay, Phys. Rev. E 88, 062926 (2013)]. The approach described here successfully resolves nearly all energy levels in the range investigated for the first two systems as well as energy levels in spectral regions that are not too highly congested for the highly chaotic system.

Figure 1

$|{\zeta }^{-1}\left(E\right)|$ vs energy for $A_1$ states of the integrable system for $0\le E\le 12$. Vertical lines indicate quantum mechanical energy levels.

Figure 2

$|{\zeta }^{-1}\left(E\right)|$ vs energy for $A_1$ states of the integrable system for $12\le E\le 20$. Vertical lines indicate quantum mechanical energy levels.

Figure 3

$|{\zeta }^{-1}\left(E\right)|$ vs energy for $A_1$ states of the integrable system for $20\le E\le 27$. Vertical lines indicate quantum mechanical energy levels.

Figure 4

Poincaré surface of section $\left(x=0,p_y>0\right)$ for the classical Hamiltonian (18) with $\alpha =1.0,\beta =0.25$. Due to the symmetry only one quadrant is shown.

Figure 5

$|{\zeta }^{-1}\left(E\right)|$ vs energy for $A_1$ states of the system with mixed phase space. Vertical lines indicate quantum mechanical energy levels.

Figure 6

$|{\zeta }^{-1}\left(E\right)|$ vs energy for $A_1$ states of the chaotic system. Vertical lines indicate quantum mechanical energy levels.

Figure 7

$|{\zeta }^{-1}\left(E\right)|$ vs energy for $A_2$ states of the chaotic system. Vertical lines indicate quantum mechanical energy levels.

Figure 8

$|{\zeta }^{-1}\left(E\right)|$ vs energy for $B_1$ states of the chaotic system. Vertical lines indicate quantum mechanical energy levels.

Figure 9

$|{\zeta }^{-1}\left(E\right)|$ vs energy for $B_2$ states of the chaotic system. Vertical lines indicate quantum mechanical energy levels.