A Novel Method for the Solution of the Schroedinger Eq. in the Presence of Exchange Terms

In the Hartree-Fock approximation the Pauli exclusion principle leads to a Schroedinger Eq. of an integro-differential form. We describe a new spectral noniterative method (S-IEM), previously developed for solving the Lippman-Schwinger integral equation with local potentials, which has now been extended so as to include the exchange nonlocality. We apply it to the restricted case of electron-Hydrogen scattering in which the bound electron remains in the ground state and the incident electron has zero angular momentum, and we compare the acuracy and economy of the new method to three other methods. One is a non-iterative solution (NIEM) of the integral equation as described by Sams and Kouri in 1969. Another is an iterative method introduced by Kim and Udagawa in 1990 for nuclear physics applications, which makes an expansion of the solution into an especially favorable basis obtained by a method of moments. The third one is based on the Singular Value Decomposition of the exchange term followed by iterations over the remainder. The S-IEM method turns out to be more accurate by many orders of magnitude than any of the other three methods described above for the same number of mesh points.Comment: 29 pages, 4 figures, submitted to Phys. Rev.

Efficient Resolution of Anisotropic Structures

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in $L_2$. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For frequent query problems, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings