Semiclassical quantization rising invariant tori: A gradient-descent approach

This paper presents a PDE-based, gradient-descent approach (GDA) to the EBK quantization of nearly separable Hamiltonians in the quasi-integrable regi me. The method does this by finding an optimal semiclassical basis of invar iant tori which minimizes the angular dependence of the Hamiltonian. This r epresentation of the Hamiltonian is termed an intrinsic resonance represent ation (IRR), and it gives the smallest possible basis obtainable from class ical mechanics. Because our method is PDE-based, we believe it to be signif icantly faster than previous IRR algorithms, making it possible to EBK quan tize systems of higher degrees of freedom than previously studied. In this paper we demonstrate our method by reproducing results from a two-degree-of -freedom system used to demonstrate the previous Carioli, Heller, and Molle r (CHM) implementation of the IRR approach. We then go on to show that our method can be applied to higher dimensional Hamiltonians than previously st udied by using it to EBK quantize a four- and a six-degree-of-freedom syste m.