Semiclassical quantization using 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 regime. The method does this by finding an optimal semiclassical basis of invariant tori which minimizes the angular dependence of the Hamiltonian. This representation of the Hamiltonian is termed an intrinsic resonance representation (IRR), and it gives the smallest possible basis obtainable from classical mechanics. Because our method is PDE-based, we believe it to be significantly faster than previous IRR algorithms, making it possible to EBK quantize 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 Moller (CHM) implementation of the IRR approach. We then go on to show that our method can be applied to higher dimensional Hamiltonians than previously studied by using it to EBK quantize a four- and a six-degree-of-freedom system.