Quasi-Sturm Functions in Problems of the Continuous Spectrum

Abstract

equation. In all of the enumerated approaches the Coulomb interaction enters into the unperturbed part of the Hamiltonian in the construction of the two-particle basis functions. Here, the Coulomb Green’s function was obtained in [6, 7] in terms of hypergeometric functions. In turn, the short-range potential energy operator is represented here in a finite subspace of L 2 basis functions. As a result of truncation of the basis space, the characteristics of scattering on the model potential undergo oscillations as the number of basis functions used to represent the potential is increased [8]. Thus, already in the solution of the two-particle scattering problem within the framework of the J-matrix method, additional measures are needed to hasten convergence. However, the basis functions here are found numerically, where the problem of generating the basis presents a problem almost more complex that the original scattering problem. In the present work, we propose basis functions that we call quasi-Sturm. These QS functions are formally solutions of the Schrodinger equation, whose right-hand side contains the Laguerre L 2 -functions. Thus, in contrast to the Sturm functions, the QS functions with prescribed asymptotic behavior can be obtained (with the help of the corresponding Green’s function) in closed analytical form. As our calculations show, the use of QS functions are as efficient in the solution of the two-particle scattering problem as the expansion over generalized Sturm functions. This work is organized as follows. In Section 1 we recall the necessary elements of the J-matrix method and the method of generalized Sturm functions. Section 2 gives a definition of QS functions with different asymptotic behavior