Discrete Spectra of Semirelativistic Hamiltonians

Abstract

We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form H = β\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 β\sqrt{m^2 + p^2} (where β> 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.

21 pages, 3 figures

References in corpus (3)

• Energy bounds for the spinless Salpeter equation: harmonic oscillator
• Discrete spectra of semirelativistic Hamiltonians from envelope theory
• Convexity and potential sums for Salpeter-like Hamiltonians

Cited by in corpus (21)

• The Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics
• Relativistic Harmonic Oscillator
• The relativistic massless harmonic oscillator
• Levy flights and nonlocal quantum dynamics
• The quantum N-body problem and the auxiliary field method
• An upper bound for asymmetrical spinless Salpeter equations
• Some equivalences between the auxiliary field method and the envelope theory
• The auxiliary field method in quantum mechanics
• Semirelativistic Hamiltonians and the auxiliary field method
• Eigenstates with the auxiliary field method
• Upper limit on the critical strength of central potentials in relativistic quantum mechanics
• A semirelativistic treatment of spinless particles subject to the Yukawa potential with arbitrary angular momenta
• On the coherent states for a relativistic scalar particle
• A semi-relativistic treatment of spinless particles subject to the nuclear Woods-Saxon potential
• The Spinless Relativistic Yukawa Problem
• The Spinless Relativistic Kink-Like Problem
• Two- and three-body calculations within the dominantly orbital state method
• Analytical Solution of two-body spinless Salpeter Equation for Hellmann Potential
• Geometric spectral inversion for singular potentials
• Bounds for Hamiltonians with arbitrary kinetic parts
• Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well