Spherical many-center scattering systems

High-symmetric systems of a large number of scatterers located at a sphere of radius R (the R sphere) are considered. A method of constructing such systems with point symmetries of crystalline lattices which is based on the projection of atomic shells onto the R sphere is proposed. In this way one can obtain the R sphere with the number of scatterers tending to infinity. By choosing potentials of separate centers one finds that such a system is closest to a spherically symmetric one. In this case the solution of scattering is uniquely characterized by a certain value of the orbital quantum number l. Specific shape resonances, geometric resonances, which can exist in many-center systems only in narrow energetic intervals and are destroyed as a potential becomes stronger, are examined.