19th GAMM-Seminar Leipzig on
High-dimensional problems - Numerical treatment and applications

Max-Planck-Institute for Mathematics in the Sciences
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  19th GAMM-Seminar
January, 23th-25th, 2003
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  Abstract Ognyan Kounchev, Thu, 17.30-17.55 Previous Contents Next  
  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.

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