On separable Schr\"odinger equations

TL;DR

This paper classifies all (1+3)-dimensional Schrödinger equations with electromagnetic interactions that are solvable by separation of variables, identifying eleven classes of electromagnetic potentials and establishing conditions on the magnetic field.

Contribution

It provides a complete classification of separable Schrödinger equations with electromagnetic fields and links the separability conditions to properties of the magnetic field.

Findings

Eleven classes of electromagnetic potentials allow separation of variables.

Magnetic field independence of spatial variables is necessary for separability.

Any separable Schrödinger equation can be reduced to one of the classified eleven forms.

Abstract

We classify (1+3)-dimensional Schr\"odinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables. As a result, we get eleven classes of the electromagnetic vector potentials of the electromagnetic field $A(t, \vec x)=(A_0(t, \vec x)$, $\vec A(t, \vec x))$ providing separability of the corresponding Schr\"odinger equations. It is established, in particular, that the necessary condition for the Schr\"odinger equation to be separable is that the magnetic field must be independent of the spatial variables. Next, we prove that any Schr\"odinger equation admitting variable separation into second-order ordinary differential equations can be reduced to one of the eleven separable Schr\"odinger equations mentioned above and carry out variable separation in the latter. Furthermore, we apply the results obtained for separating variables in the Hamilton-Jacobi equation.