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

Abstract

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 Schrödinger situation of an energy eigenvalue depending on a coupling parameter into the socalled 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 Schrödinger eigenvalue functions are inverses of each other and that the Richardson spectrum is therefore no longer a puzzle.

Eigenvalues of Schrödinger operators with definite and indefinite weights

“…The main motivation for this paper is the work on the Richardson problem and its duality with the indefinite problems in quantum mechanics [1]; L. Cardoulis was interested in solving the Schrödinger equation with indefinite weights on the real line [3]. In this paper, a simple technique is developed to find the eigenvalues and relate these eigenvalues to zeros and poles of certain Titchmarsh-Weyl functions; this is performed with the aid of using computer algebra systems.…”

“…The above definite and indefinite problems (1) and (3) play a significate role in various applications in physics and in engineering [17]. Recently, in [7], the eigenvalues of Cut-Off Coulomb potential and the Sech-Squared problem [8] have been computed with indefinite weight and we have been able to analyze the behavior of the spectrum.…”