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A new approach is proposed to the solution of the quantum mechanical inverse scattering problem at a fixed energy. The method relates the fixed energy phase shifts to those arising in an auxiliary Sturm-Liouville problem via the interpolation theory of the Weyl-Titchmarsh m-function. Then a Marchenko equation is solved to obtain the potential. As an illustration test results are shown including the reconstructions of the piecewise constant and Coulombtype potentials. A byproduct of the new framework is a demonstration of the connection found between the Gelfand-Levitan and Marchenko formalisms. [ABSTRACT FROM AUTHOR]

We establish a condition for obtaining nonsingular potentials using the Cox–Thompson inverse scattering method with one phase shift. The anomalous singularities of the potentials are avoided by maintaining unique solutions of the underlying Regge–Newton integral equation for the transformation kernel. As a by-product, new inequality sequences of zeros of Bessel functions are discovered. [ABSTRACT FROM AUTHOR]

Based on the relation between the m-function and the spectral function we construct an inverse quantum scattering procedure at fixed energy which can be applied to spherical radial potentials vanishing beyond a fixed radius a. To solve the Gelfand–Levitan–Marchenko integral equation for the transformation kernel, we determine the input symmetrical kernel by using a minimum norm method with moments defined by the input set of scattering phase shifts. The method applied to the box and Gauss potentials needs further practical developments regarding the treatment of bound states. [ABSTRACT FROM AUTHOR]

The article gives a concise review on methods for solving the inverse scattering problem of coupled reaction channels in quantum mechanics at fixed energy and presents specific applications to nuclear reactions. Two methods are considered, namely the Newton–Sabatier method and an approximate method based on the first Born approximation. Examples with artificial coupling potentials are shown to provide successful tests of both the methods. Further developments of the coupled channel inverse problem at fixed energy are needed for manifold applications in various fields of physics. [ABSTRACT FROM AUTHOR]

Solution of the Cox–Thompson inverse scattering problem at fixed energy1–3 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. [ABSTRACT FROM AUTHOR]

Rapidly developing techniques of optical cooling and trapping have opened the way for studying atomic interactions in the sub-millikelvin regime. Recently, two clouds of rubidium condensate were accelerated towards each other and were let to collide. The measured phase shifts allow us to explore the nature of interaction in a collisional energy range between 100 and 1400 μK. Using the modified Newton-Sabatier and the extended Cox-Thompson inverse scattering methods we construct effective interaction potentials from the measured quantities in a model-independent way. [Copyright &y& Elsevier]