Topology and perturbation theory

TL;DR

This paper develops a novel perturbation theory framework using topology and functional measures, enabling exact boundary accumulation and applied to the Coulomb problem for integrability insights.

Contribution

It introduces a new form of perturbation theory based on mapping quantum dynamics into collective variables space, with exact boundary accumulation and application to Coulomb problem.

Findings

Path integrals for absorption amplitudes are defined on Dirac-like measures.

Transformed perturbation theory contributions are accumulated on the boundary of the collective variables space.

The Coulomb problem is exactly solved in a specialized collective variables space.

Abstract

Paper contains description of the fields nonlinear modes successive quantization scheme. It is shown that the path integrals for absorption part of amplitudes are defined on the Dirac ($\d$-like) functional measure. This permits arbitrary transformation of the functional integral variables. New form of the perturbation theory achieved by mapping the quantum dynamics in the space $W_G$ of the ({\it action, angle})-type collective variables. It is shown that the transformed perturbation theory contributions are accumulated exactly on the boundary $\pa W_G$. Abilities of the developed formalism are illustrated by the Coulomb problem. This model is solved in the $W_C$=({\it angle, angular momentum, Runge-Lentz vector}) space and the reason of its exact integrability is `emptiness' of $\pa W_C$.