Perturbation theory for one-dimensional Schrodinger equations that can be used in a~region where the wave function is small

V. S. Pekar

Teoretičeskaâ i matematičeskaâ fizika, Tome 9 (1971) no. 3, pp. 440-444

Résumé

The usual perturbation theory series converges badly in the region where the wavefunction $\psi$ is small and the relative correction to $\psi$ is great. The new simple perturbation method is proposed, which is valid, in particular, in the region where $\psi$ is small. The method is based on expanding in the perturbation theory series not the function $\psi$ itself, but its logarithmic derivative,$\frac{d}{dx}\ln\psi$. Corrections of any order to eigen-functions and eigen-values are expressed in quadratures instead of infinite seria. The examples are considered which demonstrate the rapid convergence of the method proposed in cases when the series of the usual theory converges badly.