83 research outputs found
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 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
and partial waves of the neutron-proton scattering.Comment: 6 pages, 3 figures, Revtex4, version to be publised in Physical
Review
Spectral properties of non-conservative multichannel SUSY partners of the zero potential
Spectral properties of a coupled potential model obtained with
the help of a single non-conservative supersymmetric (SUSY) transformation
starting from a system of radial Schr\"odinger equations with the zero
potential and finite threshold differences between the channels are studied.
The structure of the system of polynomial equations which determine the zeros
of the Jost-matrix determinant is analyzed. In particular, we show that the
Jost-matrix determinant has zeros which may all correspond to
virtual states. The number of bound states satisfies . The
maximal number of resonances is . A perturbation technique
for a small coupling approximation is developed. A detailed study of the
inverse spectral problem is given for the case.Comment: 17 pages, 4 figure
Toward a Spin- and Parity-Independent Nucleon-Nucleon Potential
A supersymmetric inversion method is applied to the singlet and
neutron-proton elastic phase shifts. The resulting central potential
has a one-pion-exchange (OPE) long-range behavior and a parity-independent
short-range part; it fits inverted data well. Adding a regularized OPE tensor
term also allows the reproduction of the triplet , and
phase shifts as well as of the deuteron binding energy. The potential is thus
also spin-independent (except for the OPE part) and contains no spin-orbit
term. These important simplifications of the neutron-proton interaction are
shown to be possible only if the potential possesses Pauli forbidden bound
states, as proposed in the Moscow nucleon-nucleon model.Comment: 9 pages, RevTeX, 5 ps figure
Exactly-solvable coupled-channel potential models of atom-atom magnetic Feshbach resonances from supersymmetric quantum mechanics
Starting from a system of radial Schr\"odinger equations with a vanishing
potential and finite threshold differences between the channels, a coupled exactly-solvable potential model is obtained with the help of a
single non-conservative supersymmetric transformation. The obtained potential
matrix, which subsumes a result obtained in the literature, has a compact
analytical form, as well as its Jost matrix. It depends on
unconstrained parameters and on one upper-bounded parameter, the factorization
energy. A detailed study of the model is done for the case: a
geometrical analysis of the zeros of the Jost-matrix determinant shows that the
model has 0, 1 or 2 bound states, and 0 or 1 resonance; the potential
parameters are explicitly expressed in terms of its bound-state energies, of
its resonance energy and width, or of the open-channel scattering length, which
solves schematic inverse problems. As a first physical application,
exactly-solvable atom-atom interaction potentials are constructed,
for cases where a magnetic Feshbach resonance interplays with a bound or
virtual state close to threshold, which results in a large background
scattering length.Comment: 19 pages, 15 figure
Supersymmetric transformations for coupled channels with threshold differences
The asymptotic behaviour of the superpotential of general SUSY
transformations for a coupled-channel Hamiltonian with different thresholds is
analyzed. It is shown that asymptotically the superpotential can tend to a
diagonal matrix with an arbitrary number of positive and negative entries
depending on the choice of the factorization solution. The transformation of
the Jost matrix is generalized to "non-conservative" SUSY transformations
introduced in Sparenberg et al (2006 J. Phys. A: Math. Gen. 39 L639). Applied
to the zero initial potential the method permits to construct superpartners
with a nontrivially coupled Jost-matrix. Illustrations are given for two- and
three-channel cases.Comment: 17 pages, 3 explicit examples and figures adde
Eigenphase preserving two-channel SUSY transformations
We propose a new kind of supersymmetric (SUSY) transformation in the case of
the two-channel scattering problem with equal thresholds, for partial waves of
the same parity. This two-fold transformation is based on two imaginary
factorization energies with opposite signs and with mutually conjugated
factorization solutions. We call it an eigenphase preserving SUSY
transformation as it relates two Hamiltonians, the scattering matrices of which
have identical eigenphase shifts. In contrast to known phase-equivalent
transformations, the mixing parameter is modified by the eigenphase preserving
transformation.Comment: 16 pages, 1 figur
Darboux transformations for quasi-exactly solvable Hamiltonians
We construct new quasi-exactly solvable one-dimensional potentials through
Darboux transformations. Three directions are investigated:
Reducible and two types of irreducible second-order transformations. The
irreducible transformations of the first type give singular intermediate
potentials and the ones of the second type give complex-valued intermediate
potentials while final potentials are meaningful in all cases.
These developments are illustrated on the so-called radial sextic oscillator.Comment: 11 pages, Late
Many-body approach to proton emission and the role of spectroscopic factors
The process of proton emission from nuclei is studied by utilizing the
two-potential approach of Gurvitz and Kalbermann in the context of the full
many-body problem. A time-dependent approach is used for calculating the decay
width. Starting from an initial many-body quasi-stationary state, we employ the
Feshbach projection operator approach and reduce the formalism to an effective
one-body problem. We show that the decay width can be expressed in terms of a
one-body matrix element multiplied by a normalization factor. We demonstrate
that the traditional interpretation of this normalization as the square root of
a spectroscopic factor is only valid for one particular choice of projection
operator. This causes no problem for the calculation of the decay width in a
consistent microscopic approach, but it leads to ambiguities in the
interpretation of experimental results. In particular, spectroscopic factors
extracted from a comparison of the measured decay width with a calculated
single-particle width may be affected.Comment: 17 pages, Revte
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
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