Variational Principle for Scattering with Bounds on the Phase Shift

A variational principle possessing a minimum, i.e., $I=\int ({\mathfrak{L}G-X)}^2\mathrm{dr}\ge 0$, is examined for the case of an electron scattered off the static field of hydrogen with and without exchange effects. $\mathfrak{L}=H-E$, where $H$ may be either an effective one-electron Hamiltonian or the total Hamiltonian operator, and $X$ contains the exchange interactions. The positive definite character of $I$ eliminates the occurence of false resonances which appear in the Kohn and Hulthén procedures. Bounds on the phase shift based on Kato's identity and related to $\int |\mathfrak{L}G-X| \mathrm{dr}$ are derived which are useful in estimating the error in the phase shifts not only for the Kohn and Hulthén variational principles, but also in the one proposed. Through Schwartz's inequality, these bounds are also related to $\sqrt{I}$.