  For some years Lanczos moments methods have been combined with large-scale
shell-model calculations in evaluations of the spectral distributions of
certain operators. This technique is of great value because the alternative, a
state-by-state summation over final states, is generally not feasible. The most
celebrated application is to the Gamow-Teller operator, which governs beta
decay and neutrino reactions in the allowed limit. The Lanczos procedure
determines the nuclear response along a line q=0 in the (omega,q) plane, where
omega and q are the energy and three-momentum transfered to the nucleus.
However, generalizing such treatments from the allowed limit to general
electroweak response functions at arbitrary momentum transfers seems
considerably more difficult: the response function must be determined over the
entire (omega,q) plane for an operator O(q) that is not fixed, but depends
explicitly on q. Such operators arise in any semileptonic process where the
momentum transfer is comparable to (or larger than) the inverse nuclear size.
Here we show, for Slater determinants built on harmonic oscillator basis
functions, that the nuclear response for any multipole operator O(q) can be
determined efficiently over the full response plane by a generalization of the
standard Lanczos moments method. We describe the Piecewise Moments Method and
thoroughly explore its convergence properties for the test case of
electromagnetic responses in a full sd-shell calculation of 28Si. We discuss
possible extensions to a variety of electroweak processes, including charged-
and neutral-current neutrino scattering.
