Explicit Hilbert space representations of Schrödinger states: Definitions and properties of Stieltjes–Tchebycheff orbitals

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Abstract
Stieljes–Tchebycheff orbital approximations are described for the discrete and continuun Schrödinger states of self-adjoint (Hamiltonian) operators, and certain of their properties are established and clarified. The nth order orbitals are defined in accordance with corresponding moment-theory approximations to spectral densities as eigenfunctions of the appropriate operator in an n-term Cauchy–Hilbert space. Their eigenvalues are consequently the generalized Gaussian or Radau quadrature points of the associated spectral density function formed by projection with an appropriate test function on the Schrödinger states, and their norms provide the corresponding (reciprocal) quadrature weights. Stable algorithms are described for their construction employing recursive Lanczos and orthogonal-polynomial methods. In finite orders the spatial characteristics of the orbitals correspond to spectral averages in the neighborhoods of the quadrature points over the correct Schrödinger states. The spectral content of an individual orbital is obtained in closed form without reference to the correct underlying Schrödinger states. Convergence (n→∞) is obtained in the discrete spectral region to Schrödinger eigenstates of finite norm, whereas in the essential portion of the spectrum the orbitals converge to scattering states of improper (infinite) norm. Connections with matrix partitioning and optical potential theory are made, indicating that first- order orbitals can provide exact Schrödinger states over local predetermined portions of configuration space. Illustrative computational studies of the regular l-wave spectra of the Coulomb Hamiltonian are provided.

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