One-dimensional quasi-relativistic particle in the box

TL;DR

This paper derives an asymptotic law for the eigenvalues of a one-dimensional quasi-relativistic Hamiltonian with an infinite potential well, proving eigenvalue simplicity and analyzing eigenfunction properties.

Contribution

It provides a two-term Weyl-type asymptotic law for eigenvalues of the Klein-Gordon square-root operator in a potential well, including eigenvalue simplicity and eigenfunction properties.

Findings

Eigenvalues follow a specific asymptotic law with an explicit formula.

Eigenvalues are proven to be simple.

Eigenfunctions exhibit particular L^2 and L^infinity properties.

Abstract

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional quasi-relativistic Hamiltonian (-h^2 c^2 d^2/dx^2 + m^2 c^4)^(1/2) + V_well(x) (the Klein-Gordon square-root operator with electrostatic potential) with the infinite square well potential V_well(x) is given: the n-th eigenvalue is equal to (n pi/2 - pi/8) h c/a + O(1/n), where 2a is the width of the potential well. Simplicity of eigenvalues is proved. Some L^2 and L^infinity properties of eigenfunctions are also studied. Eigenvalues represent energies of a `massive particle in the box' quasi-relativistic model.