Explicit Hilbert-space techniques are reported for construction of the discrete and continuum Schrödinger states required in atomic and molecular photoexcitation and/or photoionization studies. These developments extend and clarify previously described moment-theory methods for determinations of photoabsorption cross sections from discrete basis-set calculations to include explicit construction of underlying wave functions. The appropriate Stieltjes-Tchebycheff excitation and ionization functions of $n\mathrm{th}$ order are defined as Radau-type eigenstates of an appropriate operator in an $n$-term Cauchy-Lanczos basis. The energies of these states are the Radau quadrature points of the photoabsorption cross section, and their (reciprocal) norms provide the corresponding quadrature weights. Although finite-order Stieltjes-Tchebycheff functions are $L^2$ integrable, and do not have asymptotic spatial tails in the continuous spectrum, the Radau quadrature weights nevertheless provide information for normalization in the conventional Dirac $\delta$-function sense. Since one Radau point can be placed anywhere in the spectrum, appropriately normalized convergent approximations to any of the discrete or continuum Schrödinger states are obtained from the development. Connections with matrix partitioning methods are established, demonstrating that $n\mathrm{th}$-order Stieltjes-Tchebycheff functions are optical-potential solutions of the matrix Schrödinger equation in the full Cauchy-Lanczos basis. The energies at which the $n\mathrm{th}$-order optical potential vanishes identically correspond to generalized Gaussian quadrature points, in which case the associated Stieltjes eigenfunctions provide an optimal $L^2$ representation of the Schrödinger spectrum of discrete and continuum wave functions. The spectral contributions of individual Schrödinger states to $n\mathrm{th}$-order Stieltjes-Tchebycheff functions are obtained in closed form, indicating the latter are spectrally localized in the neighborhoods of the Radau quadrature energies, and spatially localized in accordance with the extent of the target eigenstate. Illustrative studies of the dipole spectra in atomic and molecular hydrogen clarify the nature and convergence of the method. Finally, procedures are indicated for construction of photoemission anisotropies and for performing coupled-channel calculations employing Stieltjes-Tchebycheff functions, and descriptive intercomparisons are made of the present development with more conventional computational procedures.

• Received 16 December 1982

DOI:https://doi.org/10.1103/PhysRevA.28.1957