Eigenvalues of the Schrödinger equation via the Riccati-Padé method

A method described previously for obtaining upper and lower bounds for the eigenenergies of the Schrödinger equation for parity-invariant and central potentials is extended and applied to asymmetric one-dimensional potentials. The procedure consists of transforming the Schrödinger equation into a Ricati one for the logarithmic derivative of the wave function. The solution of the latter equation is approached by a series of Padé approximants. Approximate eigenenergies are obtained from the roots of associated determinants, and such roots are proved, in some cases, to be upper or lower bounds to the actual eigenenergies. The method is illustrated by calculations for several model potentials and the results compared with those obtained by alternative procedures.