On computing bound states of the Dirac and Schr\"odinger Equations

TL;DR

This paper presents a method to compute bound states of the Dirac and Schrödinger equations by transforming the problem into parameterized eigenvalue problems involving compact integral operators, ensuring the spectrum is discrete and bounded.

Contribution

It introduces a novel approach of solving bound states through auxiliary eigenvalue problems for parameterized compact operators, applicable to both relativistic and non-relativistic quantum equations.

Findings

Bound states can be found by changing parameters in the eigenvalue problems.

The spectrum of the operators is discrete and bounded, even for relativistic equations.

Starting from arbitrary initializations converges to the correct solutions.

Abstract

We cast the quantum chemistry problem of computing bound states as that of solving a set of auxiliary eigenvalue problems for a family of parameterized compact integral operators. The compactness of operators assures that their spectrum is discrete and bounded with the only possible accumulation point at zero. We show that, by changing the parameter, we can always find the bound states, i.e., the eigenfunctions that satisfy the original equations and are normalizable. While for the non-relativistic equations these properties may not be surprising, it is remarkable that the same holds for the relativistic equations where the spectrum of the original relativistic operators does not have a lower bound. We demonstrate that starting from an arbitrary initialization of the iteration leads to the solution, as dictated by the properties of compact operators.