On computing eigenvalues of the Schrodinger equation for symmetrical potentials

Publisher Website
Google Scholar
Abstract
The problem of the numerical determination of eigenvalues of the one-dimensional Schrodinger equation with symmetrical potential V(x) is considered. The problem is reduced to the definition of an 'eigenvalue function' G(E) well defined for the given potential V. It is theoretically proved that the even-parity eigenvalues are given by G(E)=0, and the odd-parity eigenvalues are given by G(E)= infinity .The method is 'canonical' in the sense that it is independent of the eigenfunction; yet, along with the eigenvalue, it allows the determination of the eigenfunction initial values. This method is applied to the potentials V=x2, V=x6-bx2, V=x2+ lambda x2/(1++gx2), where exact eigenvalues are available. The numerical results, compared with the exact ones and with those of previous confirmed methods, show that the present method is accurate and efficient.