Quantum mechanical inverse scattering problem at fixed energy: a constructive method

11/2011; DOI: 10.4310/MAA.2011.v18.n1.a6
Source: arXiv

ABSTRACT The inverse scattering problem of the three-dimensional Schroedinger equation
is considered at fixed scattering energy with spherically symmetric potentials.
The phase shifts determine the potential therefore a constructive scheme for
recovering the scattering potential from a finite set of phase shifts at a
fixed energy is of interest. Such a scheme is suggested by Cox and Thompson [3]
and their method is revisited here. Also some new results are added arising
from investigation of asymptotics of potentials and concerning statistics of
colliding particles. A condition is given [2] for the construction of
potentials belonging to the class L_1,1 which are the physically meaningful
ones. An uniqueness theorem is obtained [2] in the special case of one given
phase shift by applying the previous condition. It is shown that if only one
phase shift is specified for the inversion procedure the unique potential
obtained by the Cox-Thompson scheme yields the one specified phase shift while
the others are small in a certain sense. The case of two given phase shifts is
also discussed by numerical treatment and synthetic examples are given to
illustrate the results. Besides the new results this contribution provides a
systematic treatment of the CT method.

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Available from: Tamas Palmai, Jan 06, 2015
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    ABSTRACT: In order to learn more about the precision of the inversion by the Cox–Thompson method, we investigated the inversion of phase shifts of a singular potential, namely of a Coulomb potential. Using asymptotically Riccati–Bessel functions as reference functions, we could only approximately reproduce the singularity of the Coulomb potential at the origin. We also show uncertainties in the inverted potential due to different minima in the minimization solution of the nonlinear equations of the Cox–Thompson procedure. As a result, we conclude that one has to take much care with the inversion of experimental phase shifts suffering from measurement errors.
    Journal of Physics A Mathematical and Theoretical 05/2010; 43(22):225302. DOI:10.1088/1751-8113/43/22/225302 · 1.58 Impact Factor
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    ABSTRACT: The Cox–Thompson inverse scattering method at fixed energy has been generalized to treat complex phase shifts derived from experiments. New formulae for relating phase shifts to shifted angular momenta are derived. The method is applied to phase shifts of known potentials in order to test its quality and stability and, further, it is used to invert experimental n-α and n-12C phase shifts.
    Journal of Physics G Nuclear and Particle Physics 05/2006; 32(6):849. DOI:10.1088/0954-3899/32/6/008 · 2.78 Impact Factor
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    ABSTRACT: Solution of the Cox-Thompson inverse scattering problem at fixed energy [1,2,3] is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.
    Modern Physics Letters B 11/2011; 22(23). DOI:10.1142/S0217984908016972 · 0.75 Impact Factor