Towards the Classification of Exactly Solvable Feynman Path Integrals: $\delta$-Function Perturbations and Boundary-Problems as Miscellaneous Solvable Models

TL;DR

This paper explores a perturbative approach to solving Feynman path integrals for specific models, including boundary problems and point interactions, expanding the set of exactly solvable cases beyond classical examples.

Contribution

It introduces a perturbation expansion method for calculating path integrals involving boundary conditions and point interactions, broadening the class of solvable models.

Findings

Demonstrates perturbation expansion for boundary problems

Extends solvable models to include point interactions

Provides a framework for new exact solutions

Abstract

Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic oscillator (Besselian path integrals), and the (modified) P\"oschl-Teller potential(s) (Legendrian path integrals) are well known and can be performed explicitly by exploiting the convolution properties of the various types, a perturbative method opens other possibilities for calculating path integrals. Here I want to demonstrate the perturbation expansion method for point interactions and boundary problems in path integrals.