Projection-operator approach to potential scattering

The projection-operator approach of Feshbach is applied to potential scattering. The aim is to describe single-particle or shape resonances in a mathematically rigorous manner as discrete states interacting with a continuum, in analogy to the well-known description of closed-channel resonances in scattering from targets with internal degrees of freedom. A projection operator Q is defined as Q=Σν=1N|χ_ν〉〈χ_ν|, where 〈→r|χ_ν〉 is an arbitrary orthonormal set of L² functions. The complementary P space is spanned by a set of scattering states obtained in explicit form by orthogonalizing the free continuum to the set 〈→r|χ_ν〉. The free Green's function in P space is constructed explicitly and the P-space scattering problem is solved with the use of separable expansions of the potential. Two standard model problems---s-wave scattering from the square-well potential and the δ-shell potential---are solved exactly, with the use of an arbitrary number of eigenstates of a particle in a spherical box to define the Q space. It is shown that the formalism leads to a decomposition of the exact T matrix and scattering phase shift into an orthogonality scattering, a direct scattering, and a resonant scattering contribution. The pole structure of the corresponding S matrices in the complex momentum plane is analyzed. Finally, the question of how to construct the appropriate discrete state, which projects out a given resonance, is briefly discussed.