Weakly singular perturbations of the discrete spectrum

The perturbation matrix element is calculated along with the additional point potential connected with perturbation for weakly singular perturbations λ(x)-ν, (0<ν<2). It is shown that matrix elements for even unperturbed functions exist at 0<ν<3/2. Series for Rayleigh-Schroedinger coefficients of all orders converge for the same ν values irrespective of the type of retaining potential. For the odd states matrix perturbation elements exist at 0<ν<3, while estimates for Rayleigh-Schroedinger coefficients give the boundary ν=2