Energy bounds for the spinless Salpeter equation

We study the spectrum of the spinless-Salpeter Hamiltonian H = \beta \sqrt{m^2 + p^2} + V®, where V® is an attractive central potential in three dimensions. If V® is a convex transformation of the Coulomb potential -1/r and a concave transformation of the harmonic-oscillator potential r^2, then 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® ] for suitable values of P here provided. At the critical point the relative growth to the Coulomb potential h®=-1/r must be bounded by dV/dh < 2\beta/\pi.