Polyelliptic coordinates for solving the Schr\"odinger and Helmholtz equations

TL;DR

The paper introduces a new polyelliptic coordinate system that is orthogonal and allows separation of variables, enabling exact solutions to previously unsolved problems in quantum mechanics and diffraction theory.

Contribution

It develops a novel polyelliptic coordinate system that extends local elliptic coordinates, facilitating exact solutions in complex physical problems.

Findings

The coordinate system is orthogonal and admits separation of variables.

It enables exact solutions for certain quantum mechanics problems.

It provides new tools for diffraction theory analysis.

Abstract

Several local elliptic coordinates are used to build a new polyelliptic coordinate system which is orthogonal and admits the separation of variables. Such coordinate systems can give the exact solutions of some unsolved problems in quantum mechanics and diffraction theory.