The Complex Lagrangian Germ and the Canonical Operator

TL;DR

This paper introduces an invariant definition of the Lagrangian complex germ and constructs the canonical operator using a novel approach, broadening the scope of solvable eigenvalue problems in quantum mechanics.

Contribution

It provides a new invariant formulation of the Lagrangian complex germ and a revised quantization condition involving the universal cover, expanding the theoretical framework.

Findings

Equivalent to traditional definitions

New form of quantization condition

Enables solving wider class of eigenvalue problems

Abstract

We give a manifestly invariant definition of the Lagrangian complex germ with the minimal degree of accuracy required to define the canonical operator. The equivalence with the traditional definition is proved, and the canonical operator is constructed in new terms. A new form of the quantization condition is given, in which the volume form is assumed to be defined on the universal covering of the Lagrangian manifold rather than on the manifold itself. This allows one to solve a wider class of eigenvalue problems.