Spectral inversion of an indefinite Sturm-Liouville problem due to   Richardson
  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. 
Paul E. Shanley