Löwdin partitioning in a Lanczos basis: Applications to scattering states

Methods are reported for construction of closed‐form optical potentials that provide useful L 2 ‐basis‐set approximations to the discrete and continuum Schrödinger states of self‐adjoint Hamiltonian operators. The potentials are obtained employing information from a finite (Lanczos) reference space only, but nevertheless correspond to explicit summation over an infinite‐dimensional remainder space. Connections are indicated between the Stieltjes–Tchebycheff orbital solutions of the resulting optical‐potential Schrödinger problem and previously described corresponding moment‐theory approximations to spectral densities and distributions. Use of a Lanczos basis insures that the orbital eigenvalues are generalized Gaussian or Radau quadrature points of the spectral density, and that their (reciprocal) norms provide the associated quadrature weights. Convergence of the orbitals in the limit of high order is obtained to Schrödinger eigenstates of finite norm in the discrete spectral region, and to scattering states of improper (infinite) norm in the essential portion of the spectrum. In finite orders the spatial characteristics of the Stieltjes–Tchebycheff orbitals correspond to spectral averages in the neighborhoods of the quadrature points over the correct Schrödinger states. Explicit closed‐form expressions are obtained for the spectral content of individual orbitals in terms of orthogonal polynomials without reference to the correct Schrödinger states. A computational application to regular Coulomb l waves illustrates the nature and convergence of the development.