Energy bounds for the spinless Salpeter equation

We study the spectrum of the Salpeter Hamiltonian H=beta rootm(2)+p(2)+V(r) , where V(r) is an attractive central potential in three dimensions. If V(r ) is a convex transformation of the Coulomb potential -1/r and a concave tr ansformation of the harmonic-oscillator potential r(2), then both upper and lower bounds on the discrete eigenvalues of H can be constructed, which ma y all be expressed in the form E=min(r >0)[beta rootm(2)+P-2/r(2)+V(r)] for suitable values of P here provided. At the critical point r=(r) over cap t he relative growth to the Coulomb potential h(r)=-1/r must be bounded by dV /dh <2 beta/pi. (C) 2001 American Institute of Physics.