A Procedure to Solve the Eigen Solution to Dirac Equation

TL;DR

This paper introduces a finite element method-based procedure for approximately solving the eigen solutions of the Dirac equation with complex potentials, ensuring convergence and effectiveness.

Contribution

It presents a novel finite element approach with rigorous convergence theory for solving Dirac eigenvalue problems approximately.

Findings

Method achieves accurate eigen solutions for complex potentials.

Proven convergence and effectiveness of the proposed approach.

Applicable to a wide range of Dirac equations with complicated potentials.

Abstract

In this paper, we provide a procedure to solve the eigen solutions of Dirac equation with complicated potential approximately. At first, we solve the eigen solutions of a linear Dirac equation with complete eigen system, which approximately equals to the original equation. Take the eigen functions as base of Hilbert space, and expand the spinor on the bases, we convert the original problem into solution of extremum of an algebraic function on the unit sphere of the coefficients. Then the problem can be easily solved. This is a standard finite element method with strict theory for convergence and effectiveness.