DISCRETE SPECTRA OF SEMIRELATIVISTIC HAMILTONIANS FROM ENVELOPE THEORY

We analyze the (discrete) spectrum of the semirelativistic "spinless-Salpeter" Hamiltonian

where V(r) is 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 Schrödinger operators, to the case of Hamiltonians 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 r², both upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form 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 d V/ dh < 2β/π.