Theory ofJ-matrix Green's functions with applications to atomic polarizability and phase-shift error bounds

The recently introduced Jacobi or $J$-matrix techniques for quantum scattering are developed to include the construction of exact analytic matrix elements of regular and Coulomb partial-wave zeroth-order and full Green's functions. Very simple results obtain for the unperturbed Green's functions, while full Green's functions require a single diagonalization of an $N\times N$ Hamiltonian matrix, where $N$ is the number of basis functions coupled by the matrix truncated potential. In an application of the $J$-matrix Green's functions to the theory of atomic dynamic polarizabilities, the analytic result for hydrogen is derived, and it is shown how more general systems may be treated in a way which is superior to the usual $N$-term variational approach. In an application to error bounds for phase shifts, we show how the full Green's functions can be used to demonstrate the absence of false pseudoresonances in $J$-matrix scattering calculations, and bound the possible errors in computed phase shifts.