Static Approximation and Bounds on Single-Channel Phase Shifts

A single-channel scattering process with the set of quantum numbers $C$ is completely characterized by a phase shift, ${\eta }_C$. A common approximation in the determination of ${\eta }_C$ for the scattering of a particle by a compound system is to assume that, apart from the possibility of an exchange of the incident particle with an identical target particle, the target is unaffected by the incident particle. The incident particle is then scattered by the static potential generated by the target in its ground state. The phase shift determined in this approximation, to be called ${{\eta }_C}^P$, can be calculated for a number of scattering processes. Let $H$ represent the Hamiltonian of the entire system, incident particle plus target, let $E_{T0\mathrm{}}$ be the ground-state energy of the target, and let ${{E^{\prime }}_C}^Q$ be the smallest energy for which ${{E^{\prime }}_C}^Q+E_{T0\mathrm{}}-H$ is a negative definite operator in the space in which the given quantum numbers are $C$ and in which the ground state of the target is projected out. Utilizing the generalized optical potential formalism due to Feshbach and others, it can then be shown that ${\eta }_C>{{\eta }_C}^P$ if the incident energy is less than ${{E^{\prime }}_C}^Q$. (The bound is probably valid for higher energies, perhaps for all energies for which the process remains a single-channel process. If so, however, the difference between ${\eta }_C$ and ${{\eta }_C}^P$ for these higher energies will generally be large, of the order of a multiple of $\pi$, and the bound will not be immediately useful. We will, therefore, be concerned primarily with incident energies $E^{\prime }$ less than ${{E^{\prime }}_C}^Q$.) Furthermore, as one allows for more and more virtual excitation of the target system, the approximate phase shift is guaranteed to improve if $E^{\prime }$ is less than ${{E^{\prime }}_C}^Q$, and ${{E^{\prime }}_C}^Q$ will itself increase. Applications are given to the scattering of electrons and of positrons by hydrogen atoms.