Particle Propagation on a Circle with a Point Interaction

TL;DR

This paper analyzes particle propagation on a circle with a point interaction, deriving a Feynman kernel representation and generalizing the Poisson summation formula through trace formulas.

Contribution

It introduces a novel representation of the Feynman kernel using scattering matrix elements and generalizes trace formulas for systems with point interactions.

Findings

Feynman kernel expressed as sum over reflected and transmitted trajectories

Derived three-parameter family of trace formulas

Generalized Poisson summation formula for point interactions

Abstract

We study a particle propagation on a circle in the presence of a point interaction. We show that the one-particle Feynman kernel can be written into the sum of reflected and transmitted trajectories which are weighted by the elements of the n-th power of the scattering matrix evaluated on a line with a point interaction. As a by-product we find three-parameter family of trace formulae as a generalization of the Poisson summation formula.