Progressive approximation of bound states by finite series of square-integrable functions

• URL: http://arxiv.org/abs/2203.17231v1
• Date: Sun, 20 Feb 2022 00:25:35 GMT
• Title: Progressive approximation of bound states by finite series of square-integrable functions
• Authors: A. D. Alhaidari
• Abstract summary: We use the "tridiagonal representation approach" to solve the time-independent Schr"odinger equation for bound states in a basis set of finite size.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We use the "tridiagonal representation approach" to solve the time-independent Schr\"odinger equation for bound states in a basis set of finite size. We obtain two classes of solutions written as finite series of square integrable functions that support a tridiagonal matrix representation of the wave operator. The differential wave equation becomes an algebraic three-term recursion relation for the expansion coefficients of the series, which is solved in terms of finite polynomials in the energy and/or potential parameters. These orthogonal polynomials contain all physical information about the system. The basis elements in configuration space are written in terms of either the Romanovski-Bessel polynomial or the Romanovski-Jacobi polynomial. The maximum degree of both polynomials is limited by the polynomial parameter(s). This makes the size of the basis set finite but sufficient to give a very good approximation of the bound states wavefunctions that improves with an increase in the basis size.

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