Energy bounds for the spinless Salpeter equation: harmonic oscillator

TL;DR

This paper derives energy bounds for the eigenvalues of the spinless Salpeter equation with a harmonic oscillator potential using geometric and semi-classical methods.

Contribution

It introduces a geometric approach to establish upper and lower bounds for eigenvalues of the Salpeter Hamiltonian with harmonic oscillator potential.

Findings

Derived semi-classical energy bounds for all eigenvalues.

Validated bounds using geometric arguments.

Applicable to three-dimensional Salpeter Hamiltonian.

Abstract

We study the eigenvalues E_{n\ell} of the Salpeter Hamiltonian H = \beta\sqrt(m^2 + p^2) + vr^2, v>0, \beta > 0, in three dimensions. By using geometrical arguments we show that, for suitable values of P, here provided, the simple semi-classical formula E = min_{r > 0} {v(P/r)^2 + \beta\sqrt(m^2 + r^2)} provides both upper and lower energy bounds for all the eigenvalues of the problem.