RATIONAL POTENTIAL USING A MODIFIED HILL DETERMINANT METHOD

The use of vector recursion relations for calculating successive appro ximants to banded secular determinants permits a considerable enlargem ent of the class of potentials whose energy eigenvalues can be evaluat ed to high precision by the method of Hill determinants. To demonstrat e this, we study the rational potential V(x) = x2 + lambdax2/(1 + gx2) in one dimension for g > 0. Using operator methods and Hill determina nts in conjunction with vector recursion relations, we display the con venience with which the energy eigenvalues of the problem can be calcu lated to great accuracy for positive as well as negative values of lam bda. The results for lambda < 0 have not previously been obtained. The energy eigenvalue spectrum of this potential is also shown to possess discontinuities when the coupling parameters lambda and g take suitab le limiting values, pointing to the need for exercising care in the co nstruction of perturbative solutions to the problem.