Discrete spectra of semirelativistic Hamiltonians from envelope theory

Details Discrete spectra of semirelativistic Hamiltonians from envelope theory Discrete spectra of semirelativistic Hamiltonians from envelope theory

2001-10-24

arxiv.org/abs/hep-th/0110220v1

Int.J.Mod.Phys. A17 (2002) 1931-1952

Physics

High Energy Physics

High Energy Physics - Theory

20 pages, 2 tables, 4 figures

Scientific paper

10.1142/S0217751X02010522

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.

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