Convexity and potential sums for Salpeter-like Hamiltonians

TL;DR

This paper develops new energy bounds for Salpeter-like Hamiltonians by exploiting their convexity and concavity properties, extending the kinetic potential concept, and applying semi-classical approximations to various potential sums.

Contribution

It introduces a novel method to derive energy bounds for semirelativistic Hamiltonians with sum potentials, generalizing the kinetic potential approach and providing explicit bounds for specific potential forms.

Findings

Derived complementary energy bounds for the discrete spectrum.

Extended the kinetic potential concept to sum potentials.

Provided explicit bounds for power and log potentials.

Abstract

The semirelativistic Hamiltonian H = \beta\sqrt{m^2 + p^2} + V(r), where V(r) is a central potential in R^3, is concave in p^2 and convex in p. This fact enables us to obtain complementary energy bounds for the discrete spectrum of H. By extending the notion of 'kinetic potential' we are able to find general energy bounds on the ground-state energy E corresponding to potentials with the form V = sum_{i}a_{i}f^{(i)}(r). In the case of sums of powers and the log potential, where V(r) = sum_{q\ne 0} a(q) sgn(q)r^q + a(0)ln(r), the bounds can all be expressed in the semi-classical form E \approx \min_{r}{\beta\sqrt{m^2 + 1/r^2} + sum_{q\ne 0} a(q)sgn(q)(rP(q))^q + a(0)ln(rP(0))}. 'Upper' and 'lower' P-numbers are provided for q = -1,1,2, and for the log potential q = 0. Some specific examples are discussed, to show the quality of the bounds.