Semiclassical quantification of some two degree of freedom potentials: a Differential Galois approach

TL;DR

This paper explores how Differential Galois Theory can be used to analyze the solvability of variational equations in semiclassical quantification of two-degree-of-freedom potentials, providing closed-form solutions for specific systems.

Contribution

It introduces a novel application of Differential Galois Theory to semiclassical quantization, including systems that are not Liouville integrable.

Findings

Closed-form solutions for quantum fluctuations around constant velocity solutions.

Application of Differential Galois Theory to non-integrable systems.

Insight into the solvability of variational equations in semiclassical analysis.

Abstract

In this work we explain the relevance of the Differential Galois Theory in the semiclassical (or WKB) quantification of some two degree of freedom potentials. The key point is that the semiclassical path integral quantification around a particular solution depends on the variational equation around that solution: a very well-known object in dynamical systems and variational calculus. Then, as the variational equation is a linear ordinary differential system, it is possible to apply the Differential Galois Theory to study its solvability in closed form. We obtain closed form solutions for the semiclassical quantum fluctuations around constant velocity solutions for some systems like the classical Hermite/Verhulst, Bessel, Legendre, and Lam\'e potentials. We remark that some of the systems studied are not integrable, in the Liouville-Arnold sense.