A study of periodic potentials based on quadratic splines

TL;DR

This paper introduces a quadratic spline-based variational method for approximating solutions and eigenvalues of one-dimensional periodic Schrödinger potentials, providing a practical approach to estimate energy bands with controlled numerical errors.

Contribution

It presents a novel spline-based variational technique that avoids complex algebraic equations for solving periodic Schrödinger problems, applicable to physical models.

Findings

Effective approximation of eigenvalues up to a certain level.

Accurate estimation of lowest energy bands.

Analysis of numerical errors in the method.

Abstract

We discuss a method based on a segmentary approximation of solutions of the Schr\"odinger by quadratic splines, for which the coefficients are determined by a variational method that does not require the resolution of complicated algebraic equations. The idea is the application of the method to one dimensional periodic potentials. We include the determination of the eigenvalues up to a given level, and therefore an approximation to the lowest energy bands. We apply the method to concrete examples with interest in physics and discussed the numerical errors.