Discrete spectra of semirelativistic Hamiltonians from envelope theory

TL;DR

This paper extends envelope theory to analyze the discrete spectrum of semirelativistic Hamiltonians with spherically symmetric potentials, providing bounds on eigenvalues for certain potential transformations.

Contribution

It introduces a method to estimate eigenvalues of relativistic Hamiltonians using envelope theory, applicable to convex and concave potential transformations.

Findings

Derived bounds for eigenvalues of semirelativistic Hamiltonians.

Applicable to Coulomb and harmonic oscillator potential transformations.

Established conditions for potential growth at the Coulomb critical point.

Abstract

We analyze the (discrete) spectrum of the semirelativistic ``spinless-Salpeter'' Hamiltonian H = \beta \sqrt{m^2 + p^2} + V(r), beta > 0, where V(r) represents an attractive, spherically symmetric potential in three dimensions. In order to locate the eigenvalues of H, we extend the ``envelope theory,'' originally formulated only for nonrelativistic Schroedinger operators, to the case of Hamiltonians H involving the relativistic kinetic-energy operator. If V(r) is a convex transformation of the Coulomb potential -1/r and a concave transformation of the harmonic-oscillator potential r^2, both 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 the numbers 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.