Energy bounds for the spinless Salpeter equation

TL;DR

This paper derives bounds on the eigenvalues of the spinless Salpeter Hamiltonian with certain attractive potentials, providing explicit formulas and conditions for spectral estimates in relativistic quantum mechanics.

Contribution

It introduces a method to construct upper and lower bounds for the eigenvalues of the spinless Salpeter equation using convex and concave transformations of known potentials.

Findings

Bounds are explicitly expressed as minimization problems involving potential functions.

Conditions for the potential's growth at the critical point are established.

The approach applies to potentials that are convex transformations of Coulomb and concave transformations of harmonic oscillator.

Abstract

We study the spectrum of the spinless-Salpeter Hamiltonian H = \beta \sqrt{m^2 + p^2} + V(r), where V(r) is an attractive central potential in three dimensions. If V(r) is a convex transformation of the Coulomb potential -1/r and a concave transformation of the harmonic-oscillator potential r^2, then upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form E = min_{r>0} [ \beta \sqrt{m^2 + P^2/r^2} + V(r) ] for suitable values of P here provided. At the critical point the relative growth to the Coulomb potential h(r)=-1/r must be bounded by dV/dh < 2\beta/\pi.