Zero range potential for particles interacting via Coulomb potential: application to electron positron annihilation

The zero range potential is constructed for a system of two particles interacting via the Coulomb potential. The singular part of the asymptote of the wave function at the origin which is caused by the common effect of the zero range potential singularity and of the Coulomb potential is explicitly calculated by using the Lippmann-Schwinger type integral equation. The singular pseudo potential is constructed from the requirement that it enforces the solution to the Coulomb Schr\"odinger equation to possess the calculated asymptotic behavior at the origin. This pseudo potential is then used for constructing a model of the imaginary absorbing potential which allows to treat the annihilation process in positron electron collisions on the basis of the non relativistic Schr\"odinger equation. The functional form of the pseudo potential constructed in this paper is analogous to the well known Fermi-Breit-Huang pseudo potential. The generalization of the optical theorem on the case of the imaginary absorbing potential in presence of the Coulomb force is given in terms of the partial wave series

Underthreshold resonances in three-particle molecular systems

To determine the lifetimes of Efimov states of negative two-atomic ions, the problem of resonance scattering of a light particle on a pair of identical particles has been considered. An analytic expression has been obtained for resonance widths in the limit of forces of zero radius and low binding energies in pairs. Calculations are compared with the numerical solution of the Faddeev integral equations in a wide region of masses of the light particle. It is shown that the widths of underthreshold resonances in the scattering amplitude obtained from the integral equations with the Yamaguchi potential are well described by the analytic expression, which allows this expression to be used in the mass region inaccessible for numerical calculations. It has been concluded that the lifetime of highly excited negative molecular ions with a binding energy close to the threshold of disintegration is practically unlimited.Comment: Latex, 15 page

A proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing infinite fields

Let R be a regular local ring, containing an infinite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R.Comment: Section "Formal loops and affine Grassmannians" is removed as this is now covered in arXiv:1308.3078. Exposition is improved and slightly restructured. Some minor correction