Application of the Method of Standard Comparison Problems to Perturbations of the Coulomb Field. The Discrete Spectrum

Abstract

The present paper is devoted to obtaining asymptotic expansions for ε → 0 for the eigen-fiinctions Ψ_n(r) and eigenvalues E_n of Sturm-Liouville problems related to the radial Schrödinger equation

$$\begin{array}{l} \psi (r) + \left[ {2(E - V(r,\varepsilon )) - \frac{{l(l + 1)}}{{{r^2}}}} \right]\psi (r) = 0 \\ l = 0,1,2,...,r \in [0,\infty ) \\ \end{array}$$

((1))

where the potential V(r, ε) is composed of the Coulomb potential and a small perturbation

$$ V(r,\varepsilon ) = - \frac{1}{r} - \varepsilon \omega (\varepsilon r) $$

((2))

.