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 $Q=\Sigma {\nu =1\mathrm{}}^N\left|{\chi }_{\nu }〉〈{\chi }_{\nu }\right|$, where $\left\{〈\overrightarrow{\mathrm{r}}\right|{\chi }_{\nu }〉\}$ is an arbitrary orthonormal set of $L^2$ 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 $\left\{〈\overrightarrow{\mathrm{r}}\right|{\chi }_{\nu }〉\}$. 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 $\delta$-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.