Discrete spectra of semirelativistic Hamiltonians from envelope theory

Abstract

We analyze the (discrete) spectrum of the semirelativistic ``spinless-Salpeter'' Hamiltonian H = β\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} [ β\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 β/π.

20 pages, 2 tables, 4 figures

Cited by in corpus (21)

• Relativistic Harmonic Oscillator
• Discrete Spectra of Semirelativistic Hamiltonians
• Energies and wave functions for a soft-core Coulomb potential
• Relativistic N-boson systems bound by pair potentials V(r_{ij}) = g(r_{ij}^2)
• Schroedinger upper bounds to semirelativistic eigenvalues
• Convexity and potential sums for Salpeter-like Hamiltonians
• The auxiliary field method in quantum mechanics
• Semirelativistic Hamiltonians and the auxiliary field method
• Some equivalences between the auxiliary field method and the envelope theory
• Semirelativistic stability of N-boson systems bound by 1/r pair potentials
• An upper bound for asymmetrical spinless Salpeter equations
• Schroedinger secant lower bounds to semirelativistic eigenvalues
• 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
• 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
• Analytical Solution of two-body spinless Salpeter Equation for Hellmann Potential
• Geometric spectral inversion for singular potentials
• Bounds for Hamiltonians with arbitrary kinetic parts