Perspective on semiclassical quantization: How periodic orbits converge to quantizing tori

Abstract

This paper investigates the connection between quantization of periodic orbits and invariant tori satisfying Einstein-Brillouin-Keller (EBK) quantization conditions. We define Bohr-Sommerfeld (BS) conditions on periodic orbits, EBK conditions on the invariant torus, and a hybrid condition on periodic orbits, first introduced by Takatsuka, and here called BS-EBK quantization. It is found that as BS-EBK periodic orbits converge to the quantizing torus, their energies and actions converge to the torus exponentially. These periodic orbits contribute to the semiclassical wave function coherently as they converge to the quantizing torus. Together, the BS-EBK condition with the demand of collective phase coherence appear to be a powerful criterion for determining approximate semiclassical energies and wave functions from periodic orbits, with possible applications to situations where EBK tori do not exist, including chaotic dynamics.