Coulomb plus power-law potentials in quantum mechanics

We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for q the Coulomb plus power-law potential V (r) = -1/r + beta sgn(q)r(q), where beta > 0, q > -2 and q not equal 0. We show by envelope theory that the discrete eigenvalues E-nl of H may be approximated by the semiclassical expression E-nl(q) approximate to min(r>0){1/r(2) - 1/(mur) + sgn(q)beta(nur)(q)}. Values of mu and nu are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, psi(r) = r(l+1) e(-(xr)d). We give detailed results for V (r) = -1/r + betar(q), q = 0.5, 1, 2 for n = 1, l = 0, 1, 2, along with comparison eigenvalues found by direct numerical methods.