A new approximation method for the Schrodinger equation

In the framework of nonrelativistic quantum mechanics in 3 dimensions, we propose a new way of calculating the energies of the 1t-stares from the properties of the 1s-state. The method is based on the generalized Bertlmann-Martin inequalities; corrected to obtain approximative relationships relating the moments of the ground state density to the (E-1l-E-1s) energy differences for a large class of potentials. Three specific examples are studied: Hulthen, Poschl-Teller and the square well. The results clearly establish the advantages and the limitations of the method. It is most efficient for confining potentials. We also discuss two other inequalities. The first one concerns the kinetic energy of the is-slate, and the second one arises From the monopole transition sum rule. (C) 1999 Academic Press.