Singular potentials and annihilation
  We discuss the regularization of attractive singular potentials $-\alpha _{s}/r^{s}$, $s\geq 2$ by infinitesimal imaginary addition to interaction constant $\alpha_{s}=\alpha_{s}\pm i0$. Such a procedure enables unique definition of scattering observables and is equal to an absorption (creation) of particles in the origin. It is shown, that suggested regularization is an analytical continuation of the scattering amplitudes of repulsive singular potential in interaction constant $\alpha_{s}$. The nearthreshold properties of regularized in a mentioned way singular potential are examined. We obtain expressions for the scattering lengths, which turn to be complex even for infinitesimal imaginary part of interaction constant. The problem of perturbation of nearthreshold states of regular potential by a singular one is treated, the expressions for level shifts and widths are obtained. We show, that the physical sense of suggested regularization is that the scattering observables are insensitive to any details of the short range modification of singular potential, if there exists sufficiently strong inelastic short range interaction. In this case the scattering observables are determined by solutions of Schrodinger equation with regularized potential $-(\alpha_{s}\pm i0)/r^{s}$. We point out that the developed formalism can be applied for the description of systems with short range annihilation, in particular low energy nucleon-antinucleon scattering. 
A.Yu. Voronin