Spectral inversion of an indefinite Sturm-Liouville problem due to Richardson

TL;DR

This paper reveals that the Richardson indefinite Sturm-Liouville problem can be inverted to relate its eigenvalues to those of a corresponding Schrödinger problem, clarifying its complex eigenvalue dependence.

Contribution

It demonstrates that the Richardson problem is of Sturmian type, allowing the inversion of eigenvalue dependence and resolving a long-standing puzzle.

Findings

Richardson eigenvalues are inverses of Schrödinger eigenvalues.

The spectrum of the Richardson problem is clarified through spectral inversion.

The complex eigenvalue dependence becomes understandable via Sturmian inversion.

Abstract

We study an indefinite Sturm-Liouville problem due to Richardson whose complicated eigenvalue dependence on a parameter has been a puzzle for decades. In atomic physics a process exists that inverts the usual Schrodinger situation of an energy eigenvalue depending on a coupling parameter into the so-called Sturmian problem where the coupling parameter becomes the eigenvalue which then depends on the energy. We observe that the Richardson equation is of the Sturmian type. This means that the Richardson and its related Schrodinger eigevalue functions are inverses of each other and that the Richardson spectrum is therefore no longer a puzzle.