Quasi-exact Solvability of the Pauli Equation

TL;DR

This paper develops a method to identify magnetic fields that make the Pauli equation quasi-exactly solvable by leveraging $sl(2)$ symmetry and supersymmetry, expanding the known classes of such systems.

Contribution

It introduces a general procedure to find magnetic fields for QES Pauli equations, covering nine out of ten $sl(2)$-based classes, enhancing understanding of solvable quantum systems.

Findings

Nine classes of $sl(2)$-based QES systems allow this construction.

The method exploits the connection between QES systems and supersymmetry.

Provides a systematic way to identify solvable magnetic field configurations.

Abstract

We present a general procedure for determining possible (nonuniform) magnetic fields such that the Pauli equation becomes quasi-exactly solvable (QES) with an underlying $sl(2)$ symmetry. This procedure makes full use of the close connection between QES systems and supersymmetry. Of the ten classes of $sl(2)$-based one-dimensional QES systems, we have found that nine classes allow such construction.