Symbolic Algorithm of the Functional-Discrete Method for a Sturm-Liouville Problem with a Polynomial Potential

TL;DR

This paper introduces a symbolic, algebraic algorithm for solving Sturm-Liouville problems with polynomial potentials, achieving exponential convergence without solving boundary value problems or integrals.

Contribution

It develops a new symbolic algorithm for the functional-discrete method applied to Schrödinger equations with polynomial potentials, simplifying computations and improving efficiency.

Findings

Algorithm operates with decomposition coefficients of eigenfunction corrections.

Method achieves exponential convergence for the Sturm-Liouville problem.

Numerical example confirms theoretical effectiveness.

Abstract

A new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete (FD-) method is developed and justified for the Sturm-Liouville problem on a finite interval for the Schr\"odinger equation with a polynomial potential and the boundary conditions of Dirichlet type. The algorithm of the general scheme of our method is developed when the potential function is approximated by the piecewise-constant function. Our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis. The number of summands in these decompositions depends on the degree of the potential polynomial and on the correction number. Our method uses the algebraic operations only and does not need solutions of any boundary value problems and computations of any integrals unlike the previously version. The numerical example illustrates the theoretical results.