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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 threedimensions. In order to locate the eigenvalues of H, we extend the ``envelopetheory,'' originally formulated only for nonrelativistic Schroedingeroperators, to the case of Hamiltonians H involving the relativistickinetic-energy operator. If V(r) is a convex transformation of the Coulombpotential -1/r and a concave transformation of the harmonic-oscillatorpotential r^2, both upper and lower bounds on the discrete eigenvalues of H canbe 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 hereprovided. At the critical point, the relative growth to the Coulomb potentialh(r) = -1/r must be bounded by dV/dh < 2 \beta/\pi.Comment: 20 pages, 2 tables, 4 figure

DISCRETE SPECTRA OF SEMIRELATIVISTIC HAMILTONIANS FROM ENVELOPE THEORY