Phase shift effective range expansion from supersymmetric quantum mechanics

Supersymmetric or Darboux transformations are used to construct local phase equivalent deep and shallow potentials for $\ell \neq 0$ partial waves. We associate the value of the orbital angular momentum with the asymptotic form of the potential at infinity which allows us to introduce adequate long-distance transformations. The approach is shown to be effective in getting the correct phase shift effective range expansion. Applications are considered for the $^1P_1$ and $^1D_2$ partial waves of the neutron-proton scattering.Comment: 6 pages, 3 figures, Revtex4, version to be publised in Physical Review

Influence of low energy scattering on loosely bound states

Compact algebraic equations are derived, which connect the binding energy and the asymptotic normalization constant (ANC) of a subthreshold bound state with the effective-range expansion of the corresponding partial wave. These relations are established for positively-charged and neutral particles, using the analytic continuation of the scattering (S) matrix in the complex wave-number plane. Their accuracy is checked on simple local potential models for the 16O+n, 16O+p and 12C+alpha nuclear systems, with exotic nuclei and nuclear astrophysics applications in mind

Clarification of the relationship between bound and scattering states in quantum mechanics: Application to 12C + alpha

Using phase-equivalent supersymmetric partner potentials, a general result from the inverse problem in quantum scattering theory is illustrated, i.e., that bound-state properties cannot be extracted from the phase shifts of a single partial wave, as a matter of principle. In particular, recent R-matrix analyses of the 12C + alpha system, extracting the asymptotic normalization constant of the 2+ subthreshold state, C12, from the l=2 elastic-scattering phase shifts and bound-state energy, are shown to be unreliable. In contrast, this important constant in nuclear astrophysics can be deduced from the simultaneous analysis of the l=0, 2, 4, 6 partial waves in a simplified potential model. A new supersymmetric inversion potential and existing models give C12=144500+-8500 fm-1/2.Comment: Expanded version (50% larger); three errors corrected (conversion of published reduced widths to ANCs); nine references added, one remove

Equivalence of the Siegert-pseudostate and Lagrange-mesh R-matrix methods

Siegert pseudostates are purely outgoing states at some fixed point expanded over a finite basis. With discretized variables, they provide an accurate description of scattering in the s wave for short-range potentials with few basis states. The R-matrix method combined with a Lagrange basis, i.e. functions which vanish at all points of a mesh but one, leads to simple mesh-like equations which also allow an accurate description of scattering. These methods are shown to be exactly equivalent for any basis size, with or without discretization. The comparison of their assumptions shows how to accurately derive poles of the scattering matrix in the R-matrix formalism and suggests how to extend the Siegert-pseudostate method to higher partial waves. The different concepts are illustrated with the Bargmann potential and with the centrifugal potential. A simplification of the R-matrix treatment can usefully be extended to the Siegert-pseudostate method.Comment: 19 pages, 1 figur

Multi-Channel Inverse Scattering Problem on the Line: Thresholds and Bound States

We consider the multi-channel inverse scattering problem in one-dimension in the presence of thresholds and bound states for a potential of finite support. Utilizing the Levin representation, we derive the general Marchenko integral equation for N-coupled channels and show that, unlike to the case of the radial inverse scattering problem, the information on the bound state energies and asymptotic normalization constants can be inferred from the reflection coefficient matrix alone. Thus, given this matrix, the Marchenko inverse scattering procedure can provide us with a unique multi-channel potential. The relationship to supersymmetric partner potentials as well as possible applications are discussed. The integral equation has been implemented numerically and applied to several schematic examples showing the characteristic features of multi-channel systems. A possible application of the formalism to technological problems is briefly discussed.Comment: 19 pages, 5 figure

Coherent Backscattering of light in a magnetic field

This paper describes how coherent backscattering is altered by an external magnetic field. In the theory presented, magneto-optical effects occur inside Mie scatterers embedded in a non-magnetic medium. Unlike previous theories based on point-like scatterers, the decrease of coherent backscattering is obtained in leading order of the magnetic field using rigorous Mie theory. This decrease is strongly enhanced in the proximity of resonances, which cause the path length of the wave inside a scatterer to be increased. Also presented is a novel analysis of the shape of the backscattering cone in a magnetic field.Comment: 27 pages, 5 figures, Revtex, to appear in Phys. Rev.

Multi-channel phase-equivalent transformation and supersymmetry

Phase-equivalent transformation of local interaction is generalized to the multi-channel case. Generally, the transformation does not change the number of the bound states in the system and their energies. However, with a special choice of the parameters, the transformation removes one of the bound states and is equivalent to the multi-channel supersymmetry transformation recently suggested by Sparenberg and Baye. Using the transformation, it is also possible to add a bound state to the discrete spectrum of the system at a given energy $E<0$ if the angular momentum at least in one of the coupled channels $l\ge 2$.Comment: 9 pages, revtex; to be published in Phys. At. Nucl. (Oct. 2000

Single- and coupled-channel radial inverse scattering with supersymmetric transformations

The present status of the coupled-channel inverse-scattering method with supersymmetric transformations is reviewed. We first revisit in a pedagogical way the single-channel case, where the supersymmetric approach is shown to provide a complete solution to the inverse-scattering problem. A special emphasis is put on the differences between conservative and non-conservative transformations. In particular, we show that for the zero initial potential, a non-conservative transformation is always equivalent to a pair of conservative transformations. These single-channel results are illustrated on the inversion of the neutron-proton triplet eigenphase shifts for the S and D waves. We then summarize and extend our previous works on the coupled-channel case and stress remaining difficulties and open questions. We mostly concentrate on two-channel examples to illustrate general principles while keeping mathematics as simple as possible. In particular, we discuss the difference between the equal-threshold and different-threshold problems. For equal thresholds, conservative transformations can provide non-diagonal Jost and scattering matrices. Iterations of such transformations are shown to lead to practical algorithms for inversion. A convenient technique where the mixing parameter is fitted independently of the eigenphases is developed with iterations of pairs of conjugate transformations and applied to the neutron-proton triplet S-D scattering matrix, for which exactly-solvable matrix potential models are constructed. For different thresholds, conservative transformations do not seem to be able to provide a non-trivial coupling between channels. In contrast, a single non-conservative transformation can generate coupled-channel potentials starting from the zero potential and is a promising first step towards a full solution to the coupled-channel inverse problem with threshold differences.Comment: Topical review, 84 pages, 7 figures, 93 reference