The Klein-Gordon operator on M\"obius strip domains and the Klein bottle in $\mathbb{R}^n$

TL;DR

This paper derives explicit fundamental solutions for the Klein-Gordon operator on higher-dimensional M"obius strip and Klein bottle geometries, enabling comprehensive solutions to related boundary value problems.

Contribution

It provides explicit formulas for the Klein-Gordon fundamental solution on these non-trivial manifolds using generalized Weierstra{} functions, extending prior work to higher dimensions and complex topologies.

Findings

Explicit fundamental solutions for Klein-Gordon on M"obius and Klein bottle.

Complete characterization of null solutions on Klein bottle.

Formulas expressed via generalized Weierstra{} functions.

Abstract

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the M\"obius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstra{\ss} $\wp$-function that are adapted to the context of these geometries. The explicit formulas for the kernel then allow us to express all solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given boundary data in the context of these manifolds. In the case of the Klein bottle we are able to describe all null solutions of the Klein-Gordon equation in terms of finite linear combinations of the fundamental solution and its partial derivatives.