Discrete Spectra of Semirelativistic Hamiltonians

TL;DR

This paper reviews methods to bound the discrete energy spectra of semirelativistic Hamiltonians with relativistic kinetic terms and attractive potentials, emphasizing semi-analytical bounds and their relationships.

Contribution

It provides a comprehensive comparison of various approaches to derive rigorous upper and lower bounds for the energy levels of semirelativistic Hamiltonians.

Findings

Derived semi-analytical bounds for energy levels.

Compared different bounding approaches and their relationships.

Provided insights into the spectral properties of semirelativistic operators.

Abstract

We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form H = \beta \sqrt{m^2 + p^2} + V(r) (w.l.o.g. in three spatial dimensions) as entering, for instance, in the spinless Salpeter equation. Every Hamiltonian in this class of operators consists of the relativistic kinetic energy \beta \sqrt{m^2 + p^2} (where \beta > 0 allows for the possibility of more than one particles of mass m) and a spherically symmetric attractive potential V(r), r = |x|. In general, accurate eigenvalues of a nonlocal Hamiltonian operator can only be found by the use of a numerical approximation procedure. Our main emphasis, however, is on the derivation of rigorous semi-analytical expressions for both upper and lower bounds to the energy levels of such operators. We compare the bounds obtained within different approaches and present relationships existing between the bounds.