Coulomb plus power-law potentials in quantum mechanics

We study the discrete spectrum of the Hamiltonian H = −∆ + V (r) for the Coulomb plus power-law potential V (r) = −1/r + β sgn(q)r, where β > 0, q > −2 and q 6= 0 . We show by envelope theory that the discrete eigenvalues Enl of H may be approximated by the semiclassical expression Enl(q) ≈ minr>0{1/r − 1/(μr) + sgn(q)β(νr)q}. Values of μ and ν are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, ψ(r) = re q . We give detailed results for V (r) = −1/r+βrq , q = 0.5, 1, 2 for n = 1, l = 0, 1, 2, along with comparison eigenvalues found by direct numerical methods.