Virial ans\"atze for the Schr\"odinger Equation with a symmetric strictly convex potential. Part II

TL;DR

This paper tests a parameter-free ansatz method for solving the Schrödinger equation with symmetric convex potentials, analyzing its accuracy and potential for establishing error bounds across different potential degrees and coupling constants.

Contribution

It extends previous work by applying the ansatz technique to $x^{2\,\kappa}$-type potentials and evaluates its effectiveness and limitations.

Findings

The ansatz performs well for various potential degrees.

Behavior of the ansatz depends on the coupling constant.

Potential use for establishing upper bounds on errors.

Abstract

Recently was introduced in the literature a procedure to obtain ans\"atze, free of parameters, for the eigenfunctions of the time-independent Schr\"odinger equation with symmetric convex potential. In the present work, we test this technique in regard to $x^{2\kappa}$-type potentials. We study the behavior of the ans\"atze regarding the degree of the potential and to the intervening coupling constant. Finally, we discuss how the results could be used to establish the upper bounds of the relative errors in situations where intervening polynomial potentials.