Multichannel scattering with nonlocal and confining potentials. I. General theory

Abstract

A general discussion of nonrelativistic multichannel scattering for nonlocal potentials is presented. The approach is based on the extensive use of the Fredholm determinants that are associated with the integral equations occurring at various points of the theory. Special attention is paid to situations where confining potentials are present, as in nonrelativistic quark models of hadron-hadron interactions. Some standard results of multichannel scattering theory for local potentials and one-channel theory for nonlocal ones are generalized in this context, including Levinson s theorem.PACS number(s): 24. 10.Cn, 12.40.Aa

Exactly solvable coupled-channel potential models of atom-atom magnetic Feshbach resonances from supersymmetric quantum mechanics

“…Let us first summarize the notations used below for coupled-channel scattering theory [6,12,13]. We consider a multichannel radial Schrödinger equation that reads in reduced units…”

“…[11] however, an exactly-solvable coupled-channel potential with threshold differences is derived, two remarkable features of which are the compact expressions provided both for the potential and for its Jost matrix. Since the Jostmatrix completely defines the bound-and scatteringstate properties of a potential model [12,13], such an analytical expression seems very promising in the context of the scattering inverse problem. The work of Cox has however received little attention, probably because it is plagued by two problems.…”

Generalized Levinson theorem for singular potentials in two dimensions

“…We suppose that the spectral problem (4a) has at least one bound state E 0 < 0. Assuming that we start from the regular potential under conditions (11), then the eigenfunction ψ 0 ≡ ψ E 0 m (ρ) may have the following asymptotic behaviour…”

“…The Levinson theorem in 3D has been discussed for noncentral potentials [2,3,4], singular potentials [5], energy-dependent potentials [6], nonlocal interactions [7], Dirac particles [8,9], systems with coupling [10], multichannel scattering [11,12], multiparticle singlechannel scattering [13], and in the inverse scattering theory, even with singular potentials [14,15,16].…”

Levinson's theorem for the Klein-Gordon equation in one dimension

“…(16) up to a multiple of π due to the period of the tangent function. In our convention (14), the phase shift η − (k, λ), k > 0, changes continuously as λ increases from zero to one. In other words, the phase shift η − (k, λ) is determined completely in our convention, so is η + (k, λ).…”