Refined Spectral Method as an extremely accurate technique for solving 2D time-independent Schrodinger equation

TL;DR

This paper introduces a refined spectral method enhanced with optimization for highly accurate solutions to the 2D time-independent Schrödinger equation, capable of efficiently computing multiple bound states with minimal error.

Contribution

The paper develops a generalized, optimized spectral method for 2D Schrödinger equations, achieving unprecedented accuracy and efficiency in bound state calculations.

Findings

Achieves relative error of 10^(-15) in 2D problems

Simple to implement, fast, and highly accurate

Can compute multiple bound states in a single run

Abstract

We present a refinement of the Spectral Method by incorporating an optimization method into it and generalize it to two space dimensions. We then apply this Refined Spectral Method as an extremely accurate technique for finding the bound states of the two dimensional time-independent Schrodinger equation. We first illustrate the use of this method on an exactly solvable case and then use it on a case which is not so. This method is very simple to program, fast, extremely accurate (e.g. a relative error of 10^(-15) is easily obtainable in two dimensions), very robust and stable. Most importantly, one can obtain the energies and the wave functions of as many of the bound states as desired with a single run of the algorithm.