Piecewise moments method: Generalized Lanczos technique for nuclear response surfaces

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 transferred to the nucleus, respectively. 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 in which the momentum transfer is comparable with (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 Si-28. We discuss possible extensions to a variety of electroweak processes, including charged- and neutral-current neutrino scattering.