Envelope representations for screened Coulomb potentials

We study the discrete eigenvalues $E_{\mathrm{nl}}$ of the Schrödinger Hamiltonian H=-(1/2)Δ+V(r), where V(r)=g(-1/r) is an increasing concave transformation of the Coulomb potential, and n is the principal (radial) quantum number. It is demonstrated by the method of potential envelopes that upper bounds are provided by the simple formula $E_{\mathrm{nl}}$≤ ${\min }_{\mathrm{s}>0\mathrm{}}$ {(1/2)s+V((n+l)/${\mathrm{s}}^{1/2}$)}, where s is a real variable. Numerical results are compared with previous work for two specific screened Coulomb potentials. In the case of the Yukawa potential V(r)=-(v/r)exp(-λr), it is shown that the inequality (n+l${)}^2$λ/v$E_{\mathrm{nl}}$: In the case of S states, sharp upper and lower bounds are also provided by a different method.