Schr\"{o}dinger equation is $\mathcal{R}$-separable in toroidal coordinates

TL;DR

This paper introduces exact solutions to the Schrödinger equation in toroidal coordinates using a novel irregular -separation method, revealing fractional angular momentum eigenvalues and broad applicability to external potentials.

Contribution

It is the first to solve the Schrödinger equation in Moon and Spencer's toroidal coordinates with irregular -separation, uncovering fractional angular momentum eigenvalues.

Findings

Exact solutions in toroidal coordinates are obtained.

Wavefunctions exhibit fractional angular momentum eigenvalues.

Solutions include external magnetic field potentials.

Abstract

We present, for the first time, exact solutions for the Schr\"{o}dinger equation in Moon and Spencer's toroidal coordinates, and in the electromagnetic toroidal--poloidal coordinate systems. Curiously, both systems present a fractional angular momentum, because of the torus's hole. We achieve these novel solutions using the irregular $\mathcal{R}$-separation of variables, an unexplored approach in Physics, which results in a wavefunction with fractional angular momentum eigenvalues. Numerous solutions for the Schr\"{o}dinger equation in a variety of external potentials are shown, including an external magnetic field. A plane-wave expansion and a Green function are also presented, setting the stage for future progress in this area.