Applying Bogomolny's quantization method to generic classical systems

The quantization method of Bogomolny [Nonlinearity 5, 805 (1992)] can potentially provide semiclassical estimates for energy levels of all bound states of arbitrary systems. This approach requires the formation of the transfer matrix T_E as a function of energy E. Existing practical methods for calculating this matrix require a recalculation of many classical trajectories for each energy. This has hampered the application of Bogomolny's method to generic systems that do not possess special classical scaling properties. Generalizing earlier work [H. Barak and K. G. Kay, Phys. Rev. E 88, 062926 (2013)], we develop initial value representation formulas for T_E that overcome this problem. These expressions are obtained from a generalized Herman-Kluk formula for the propagator that allows one to easily derive a family of semiclassical integral approximations for the Green's function that are, in turn, used to form the transfer matrix. Calculations for two-dimensional systems show that Bogomolny's method with the present expressions for T_E produces accurate semiclassical energy levels from small transfer matrices.