S. A. Rakityansky

Exact method for locating potential resonances and Regge trajectories

We propose an exact method for locating the zeros of the Jost function for analytic potentials in the complex momentum plane. We further extend the method to the complex angular - momentum plane to provide the Regge trajectories. It is shown, by using several examples, that highly accurate results for extremely wide, as well as for extremely narrow, resonances with or without the presence of the Coulomb interaction can be obtained.

A method for calculating the Jost function for analytic potentials

SummaryA combination of the variable-constant and complex coordinate rotation methods is used to solve the two-body Schrödinger equation. The latter is replaced by a system of linear first-order differential equations, which enables one to perform direct calculation of the Jost function for all complex momenta of physical interest, including the spectral points corresponding to bound and resonance states. Explicit forms of the equations, appropriate for central short-range and Coulombtailed potentials, are given. Within the proposed method, the scattering, bound, virtual, and resonance state problems can be treated in a unified way. The effectiveness of the method is demonstrated by a numerical example.

Jost function for coupled partial waves

An exact method for direct calculation of the Jost functions and Jost solutions for non-central analytic potentials which couple partial waves of different angular momenta is presented. A combination of the variable-constant method with the complex coordinate rotation is used to replace the matrix Schrodinger equation by an equivalent system of linear first-order differential equations. Solving these equations numerically, the Jost functions can be obtained to any desired accuracy for all complex momenta of physical interest, including the spectral points corresponding to bound and resonant states. The effectiveness of the method is demonstrated using the Reid soft-core and Moscow nucleon-nucleon potentials which involve tensor forces.

Analytic structure and power series expansion of the Jost function for the two-dimensional problem

For a two-dimensional quantum-mechanical problem, we obtain a generalized power series expansion of the S-matrix that can be done near an arbitrary point on the Riemann surface of the energy, similar to the standard effective-range expansion. In order to do this, we consider the Jost function and analytically factorize its momentum dependence that causes the Jost function to be a multi-valued function. The remaining single-valued function of the energy is then expanded in the power series near an arbitrary point in the complex energy plane. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This makes it possible to obtain a semi-analytic expression for the Jost function (and therefore for the S-matrix) near an arbitrary point on the Riemann surface and use it, for example, to locate the spectral points (bound and resonant states) as the S-matrix poles. The method is applied to a model similar to those used in the theory of quantum dots.

Padé approximation of the S-matrix as a way of locating quantum resonances and bound states

It is shown that the spectral points (bound states and resonances) generated by a central potential of a single-channel problem, can be found using rational parametrization of the S-matrix. To achieve this, one only needs values of the S-matrix along the real positive energy axis. No calculations of the S-matrix at complex energies or a complex rotation are necessary. The proposed method is therefore universal in that it is applicable to any potential (local, non-local, discontinuous, etc) provided that there is a way of obtaining the S-matrix (or scattering phase shifts) at real collision energies. Besides this, combined with any method that extracts the phase shifts from the scattering data, the proposed rational parametrization technique would be able to do the spectral analysis using the experimental data.

Generalized effective-range expansion

A systematic and accurate procedure has been developed for calculating the coefficients φ (in/out) �n oftheseriesexpansion f (in/out) � (k) = � ∞ n=0 (k−k0) n φ (in/out) �n of the Jost functions in the vicinity of an arbitrary point k0 in the complex momentum plane. This makes it possible to obtain an analytic expression for the S-matrix s� (k) = f (out)

Multi-channel analog of the effective-range expansion

Similar to the standard effective-range expansion that is done near the threshold energy, we obtain a generalized power-series expansion of the multi-channel Jost-matrix that can be done near an arbitrary point on the Riemann surface of the energy within the domain of its analyticity. In order to do this, we analytically factorize its momentum dependences at all the branching points on the Riemann surface. The remaining single-valued matrix functions of the energy are then expanded in the power series near an arbitrary point in the domain of the complex energy plane where it is analytic. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This means that near an arbitrary point in the domain of physically interesting complex energies it is possible to obtain a semi-analytic expression for the Jost-matrix (and therefore for the S-matrix) and use it, for example, to locate the spectral points (bound and resonant states) as the S-matrix poles.

Analytic structure of the multichannel Jost matrix for potentials with Coulombic tails

A quantum system is considered that can move in N two-body channels with the potentials that may include the Coulomb interaction. For this system, the Jost matrix is constructed in such a way that all its dependencies on the channel momenta and Sommerfeld parameters are factorized in the form of explicit analytic expressions. It is shown that the two remaining unknown matrices are single-valued analytic functions of the energy and therefore can be expanded in the Taylor series near an arbitrary point within the domain of their analyticity. It is derived a system of first-order differential equations whose solutions determine the expansion coefficients of these series. Alternatively, the unknown expansion coefficients can be used as fitting parameters for parametrizing experimental data similarly to the effective-range expansion. Such a parametrization has the advantage of preserving proper analytic structure of the Jost matrix and can be done not only near the threshold energies, but around any collision or even complex energy. As soon as the parameters are obtained, the Jost matrix (and therefore the S-matrix) is known analytically on all sheets of the Riemann surface, and thus enables one to locate possible resonances.

A method for extracting the resonance parameters from experimental cross-sections

Within the proposed method, a set of experimental data points are fitted using a multi-channel S-matrix. Then the resonance parameters are located as its poles on an appropriate sheet of the Riemann surface of the energy. The main advantage of the method is that the S-matrix is constructed in such a way that it has proper analytic structure, i.e. for any number of two-body channels, the branching at all the channel thresholds is represented via exact analytic expressions in terms of the channel momenta. The way the S-matrix is constructed makes it possible not only to locate multi-channel resonances but also to extract their partial widths as well as to obtain the scattering cross-section in the channels for which no data are available. The efficiency of the method is demonstrated by two model examples of a single-channel and a two-channel problems, where known resonance parameters are rather accurately reproduced by fitting the pseudo-data artificially generated using the corresponding potentials.

Ghost components in the Jost function and a new class of phase-equivalent potentials

We show that the introduction of components, in the Jost function, that create new bound states while leaving the S-matrix unchanged, generates potentials behaving as r-2 at large distances. We demonstrate that the modified Jost functions can be obtained by applying two successive supersymmetric transformations to the original potential. We further show that transparent potentials, with Sl(k)≡1, can also be obtained by successive supersymmetric transformations. They are characterized by the property that their SUSY-2 partners resemble centrifugal barriers. Finally, the relation of these transformations to the asymptotic normalization constants of the inverse scattering problem is discussed. We show that the two supersymmetric transformations that remove a bound state provide a potential which is the same as that obtained via the Marchenko inverse scattering procedure, when the asymptotic normalization constant is set to zero.