Jean-Marc Sparenberg

Phase-Equivalent Complex Potentials

Potentials providing the same complex phase shifts as a given complex potential but with a shallower real part are constructed with supersymmetric transformations. Successive pairs of transformations eliminate normalizable solutions corresponding to complex eigenvalues of the Schrodinger equation with the full complex potential. With respect to real potentials, a new feature is the occurrence of normalizable solutions with complex energies presenting a positive real part. Removing such solutions provides a way of suppressing narrow resonances but may lead to complicated equivalent potentials with little physical interest. We discuss the singularity of the transformed potential and its relation with the Levinson theorem, the transformation of the Jost function, and the link with the Marchenko approach. The technique is tested with the solvable Poschl-Teller potential. As physical applications, deep optical potentials for the α + 16O and 16O + 16O scatterings are transformed into l-dependent phase-equivalent shallow optical potentials.

Influence of low-energy scattering on loosely bound states

Compact algebraic equations are derived that 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 {sup 16}O+n, {sup 16}O+p, and {sup 12}C+alpha nuclear systems, with exotic nuclei and nuclear astrophysics applications in mind.

Coupled-channel R-matrix method on a Lagrange mesh

Abstract The coupled-channel R -matrix method on a Lagrange mesh is a very simple approximation of the R -matrix method with a basis. The mesh points are zeros of shifted Legendre polynomials. Bound-state energies and scattering matrices are easily calculated with small numbers of potential values at mesh points. A test with an exactly solvable two-channel potential provides an excellent accuracy over a broad energy range with only 30 mesh points. The efficiency of the method is illustrated for a single channel on α + α scattering and for two channels on the deuteron ground-state energy and on nucleon-nucleon scattering.

Inverse scattering with supersymmetric quantum mechanics

The application of supersymmetric quantum mechanics to the inverse scattering problem is reviewed. The main difference with standard treatments of the inverse problem lies in the simple and natural extension to potentials with singularities at the origin and with a Coulomb behaviour at infinity. The most general form of potentials which are phase-equivalent to a given potential is discussed. The use of singular potentials allows adding or removing states from the bound spectrum without contradicting the Levinson theorem. Physical applications of phase-equivalent potentials in nuclear reactions and in three-body systems are described. Derivation of a potential from the phase shift at fixed orbital momentum can also be performed with the supersymmetric inversion by using a Bargmann-type approximation of the scattering matrix or phase shift. A unique singular potential without bound states can be obtained from any phase shift. A limited number of bound states depending on the singularity can then be added. This inversion procedure is illustrated with nucleon?nucleon scattering.

Analysis of the R-matrix method on Lagrange meshes

The R-matrix method on a Lagrange mesh is a very simple approximation of the R-matrix method with a basis. By analysing an exactly solvable example, we observe that the mesh approximation does not reduce the accuracy of the R-matrix bound-state energies and phase shifts. This property is obtained with two different meshes, the shifted Legendre and shifted Jacobi meshes, which correspond to equivalent polynomial bases. Their comparison shows that the orthogonality of the Lagrange basis functions is not as crucial as was previously assumed: the Legendre mesh, which corresponds to a nonorthogonal Lagrange basis, is at least as accurate as the Jacobi mesh based on an orthogonal basis. We also emphasize the surprising origin of a known property of the R-matrix method: the results are much more accurate with basis functions without uniform boundary conditions because the quality of the matching is realized by a few highly excited eigenfunctions, with weak physical content, of the sum of the Hamiltonian and Bloch operators.

Supersymmetric transformations of real potentials on the line

A systematic study of supersymmetric (or Darboux) factorizations on the line is performed. All possible pairs of supersymmetric transformations with the same factorization energy are reviewed; for such pairs, there is no condition on this energy. Iterations of single transformations and of pairs both allow arbitrary modifications of the bound spectrum. Different iterative methods lead to compact analytical equations depending on the initial potential and its solutions. Iterations of single transformations are able to transform an even potential into an even potential. Particular cases of a method based on iteration of pairs conserve either the reflection coefficient at all energies or the norming constants of bound states. These results are compared with previous methods established in other contexts. The most general supersymmetric transformation of a given potential is finally described, both with and without modifications of the bound spectrum.

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.

Non-aligned hydrogen molecular ion in strong magnetic fields

The hydrogen molecular ion in a strong magnetic field is studied for arbitrary orientations of the molecular axis. A gauge preserving the parity symmetry and leading to real matrix elements for a class of basis states is introduced. The calculations are performed in prolate spheroidal coordinates with the Lagrange-mesh method. The simple resulting mesh equations provide a high accuracy with short computing times for magnetic fields γ = 1 and 10 in atomic units of 2.35 × 105 T. Less accurate results are obtained at the field γ = 100, typical of neutron stars where the size of the matrix becomes very large. The present results allow evaluation of the accuracy of published variational results. At high field strengths, the rotational motion becomes strongly hindered. Simple estimates of rotational energies are compared with other works.

Faddeev calculation of 3 alpha and alpha alpha Lambda systems using alpha alpha resonating-group method kernels

We carry out Faddeev calculations of three-alpha (3 alpha) and two-alpha plus Lambda (alpha alpha Lambda) systems, using two-cluster resonating-group method kernels. The input includes an effective two-nucleon force for the alpha alpha resonating-group method and a new effective Lambda N force for the Lambda alpha interaction. The latter force is a simple two-range Gaussian potential for each spin-singlet and triplet state, generated from the phase-shift behavior of the quark-model hyperon-nucleon interaction, fss2, by using an inversion method based on supersymmetric quantum mechanics. Owing to the exact treatment of the Pauli-forbidden states between the clusters, the present three-cluster Faddeev formalism can describe the mutually related, alpha alpha, 3 alpha and alpha alpha Lambda systems, in terms of a unique set of the baryon-baryon interactions. For the three-range Minnesota force which describes the alpha alpha phase shifts quite accurately, the ground-state and excitation energies of 9Be Lambda are reproduced within 100 - 200 keV accuracy.

Clarification of the relationship between bound and scattering states in quantum mechanics: Application toC12+α

Using phase-equivalent supersymmetric partner potentials, a general result from the inverse problem in quantum scattering theory is illustrated, i.e., bound-state properties cannot be extracted from the phase shifts of a single partial wave, as a matter of principle. In particular, recent