New stationary bounds on matrix elements including positron--atom scattering lengths

In a previous study of bound-state matrix elements of a Hermitian operator $W$, it was possible to obtain at most an upper (or lower) stationary bound. The possibility arose only for the diagonal matrix element case, and only for $W$ nonpositive (or nonnegative). In the present treatment, both upper and lower stationary bounds are obtained, for diagonal and off-diagonal matrix elements, and, though some restrictions on $W$ remain, the requirement that $W$ be of well-defined sign can be dropped. The derivation also improves upon that given previously in that the possibility of any difficulty with near singularities in the equation defining the trial auxiliary (or Lagrange) function is unambiguously avoided. As an example, the method is applied to the problem of the zero-energy scattering of positrons by atoms or ions, and an expression is derived which provides a rigorous stationary upper bound on the scattering length; the target ground-state wave function need not be known exactly. Crude but rigorous numerical results are obtained quite simply in the Born approximation.