Approximation of the Schrödinger operator in the strip and half-space
Ognyan Kounchev (Bulgarian Academy of Sciences)
We consider approximations of the Schrödinger operator in domains like
strip, half-space, and the whole space, for potentials decaying at
infinity. We discretize the operator with respect to only one of the
variables and we obtain that the discretization of the scattering
problem for the Schrödinger operator is reduced to a generalized
scattering problem for higher-order elliptic operators (polyharmonic).
Respectively, we have a generalized problem for expansion in
eigenfunctions for scalar functions; in the two-dimensional case see
[N, chapter 3.6.7], about the character of the arising problem. We prove a
generalization of the Povzner-Ikebe theorem, [B], for expansion in
eigenfunctions for special generalized spectral problem with a
higher-order elliptic equation. Letting the grid size tend to zero we
obtain a new approximation of the spectral kernel and the
eigenfunctions. These approximations correspond to a Pade approximation
of an operator in the so-called polyharmonic Bergman space, [KR]. All
these concepts arise in a natural way through the so-called Polyharmonic
Paradigm in Constructive Theory of Functions, see [K, Chapter 1].
[S] B. Simon, Schroedinger semigroups, Bull. AMS, v. 7 (1982), 447-526.
[K] O. Kounchev, Multivariate Polysplines. Applications to Numerical
and Wavelet Analysis. Academic Press, San Diego-London, 2001.
[KR] O. Kounchev and H. Render, Multivariate Orthogonality, Moments,
and Schroedinger Operators, in preparation for Academic Press.
[N] M. Naimark, Linear Differential Operators, Ungar, New York, 1967.