On a rapidly converging perturbation theory for a discrete spectrum

The perturbation theory for the discrete spectrum of the radial Schrödinger equation is generalized to the case when nonperturbated function has knots. To the $k$-ih order the eigenfunction is calculated to the accuracy $\varepsilon^{2^k}$, where $\varepsilon$ is the perturbation parameter. It is possible to obtain from this eigenfunction the energy to the accuracy $\varepsilon^{2^{k+1}}$. All corrections are the quadratures of this function. The dependence on all other parts of spectrum is absent. The expressions for shiftings of the knots under the perturbation are obtained.