Details

Autor(en) / Beteiligte

• Cornille, Henri

Titel

Differential equations satisfied by Fredholm determinants and application to the inversion formalism for parameter dependent potentials

Ist Teil von

• Journal of mathematical physics, 1976-12, Vol.17 (12), p.2143-2158

Erscheinungsjahr

1976

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• We consider the two first order differential operators A x =μ (x) ∂/∂x+λ (x), B x =−(∂/∂x) μ (x)+λ (x), associate two kernels f, g satisfying both well‐defined boundary conditions and A x f=B y g, A y f=B x g, and construct the Fredholm determinants corresponding to these kernels. From these determinants we can build up solutions of second order differential equations. These solutions have an interpretation in the Schrödinger inversion formalism. For instance, for the inversion at fixed angular momentum l, these solutions for k=0 correspond to the classical Gel’fand–Levitan and Marchenko equations, whereas, for k≠0, they correspond to k dependent potentials. Similarly, for the inversion at fixed k, these solutions for l=0 correspond to the classical Regge–Newton equations, whereas, for l≠0, they correspond to l‐dependent potentials. More generally we show, in a generalized inversion formalism, how parameter dependent potentials appear very naturally in the theory.