Abstract
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 is defined as , where is an arbitrary orthonormal set of functions. The complementary space is spanned by a set of scattering states obtained in explicit form by orthogonalizing the free continuum to the set . The free Green's function in space is constructed explicitly and the -space scattering problem is solved with the use of separable expansions of the potential. Two standard model problems—-wave scattering from the square-well potential and the -shell potential—are solved exactly, with the use of an arbitrary number of eigenstates of a particle in a spherical box to define the space. It is shown that the formalism leads to a decomposition of the exact matrix and scattering phase shift into an orthogonality scattering, a direct scattering, and a resonant scattering contribution. The pole structure of the corresponding 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.
- Received 27 December 1982
DOI:https://doi.org/10.1103/PhysRevA.28.2777
©1983 American Physical Society