Inverse Problem Numerical Solution for Schrödinger Equation and Different Thesholds of Partial Channels

Purpose of research. We consider a problem of setting up and solution the inverse scattering problem for systems of Schrödinger equations. We consider a connected problem (in Gelfand–Levitanand- Marchenko approaches) of solution of systems of Fredholm integral equations of the second kind.

Methods. Based on the performed an analysis the setting up of the scattering theory inverse problem for radial Schrödinger equations is presented in Marchenko approach for case of a number of coupled channels with different thresholds (by the example of two channels with obvious generalization). 

Results. Specific properties of the corresponding scattering S-matrix and asymptotics of possible bound states are obtained necessary and sufficient for explicit solution of the considered inverse scattering problem and its physical adequacy. A quasirational approximation of the S-matrix elements (Pade type approximant) for the scattering theory inverse problem for system of Schrödinger type equations is presented. The obtained approximation has explicitly all the necessary and sufficient properties for solution of the considered problem. The presented approximation allows to solve the considered inverse problem (system of coupled Marchenko equations – Fredholm integral equations of the second kind analytically, in principle.  

Conclusion. Possible areas of application of the presented algorithm and developed method of numerical solution of the inverse problem of equations (systems of equations) of Schrödinger type and other similar problems. Analytical solution of the scattering theory inverse problem for the system of Schrödinger type equations with different thresholds and for case of specific kind S-matrix. Designed specific kind of S-matrix allows to approximate (interpolate) any physically adequate S-matrices. 

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