Contributions to the solutions of the pion-nucleus low equation in the p-wave and particle-hole approximations with an application to the low energy π-12C scattering

The solutions of the pion-nucleus Low equation are obtained in the approximation that the pion orbital momentum is limited to p-wave and the nuclear states are restricted to the particle-hole pairs. The general formulas for the crossing matrices and cross sections are derived. The detailed derivation of the ''no-meson'' (Born) amplitudes is presented for the closed shell nuclei (in particular, with spin and isospin zero in the ground state). Three methods of numerical solutions are discussed. It is found for the low energy π-12C scattering that the (direct) iterative series diverges for the physical value of coupling constant, and the inverse amplitude method is inapplicable due to certain vanishing Born amplitudes. Numerical solutions for the π-12C scattering are obtained by using the method of inverse amplitudes with the pole approximation (of the left-hand cut), and the results are compared with the low energy experimental data and the optical model calculations. The importance of the crossing symmetry and the inadequacy of the p-wave approximation are pointed out