No. 4 (00054) Family name : Serdyukova Given name : Svetlana Affiliation : Joint Institute for Nuclear Research, Dubna, Russia Abbreviation : JINR E-mail address : sis@jinr.ru Title : Potential Reconstruction for Two-Dimensional Discrete Shcr\"odinger equation Authors : S.I.Serdyukova Abstract : \documentclass [12pt]{article} \usepackage [english,russian]{babel} \textwidth 170mm \textheight 240mm \oddsidemargin -0cm \evensidemargin -0cm \topmargin -0.4cm \begin{document} \begin{center} INVERSE PROBLEM FOR TWO-DIMENSIONAL DISCRETE SHCR\"ODINGER EQUATION S.I.Serdyukova (sis@jinr.ru) (Joint Institute for Nuclear Research, Dubna, Russia) \end{center} For two-dimensional discrete Schr\"odinger equation the boundary-value problem in rectangle $(M,N)$ with zero boundary conditions is considered. It's stated [1], that inverse problem is reduced to reconstruction of symmetric five-diagonal matrix $C$ with given spectrum and given first $k(M,N), ~ 1 \le k < N,$ components for each of basic eigenvectors. Found symmetric disturbed five diagonal matrix corresponds to discrete Schr\"odinger equation with nonlocal five-point potential. The matrix $C$ has lacuna between upper second and $(N+1)$ -th diagonals. As a result the first $N$ components of basic eigenvectors must satisfy $(N-1)2(M-1)$ additional conditions and $N$ conditions of compatibility . The elements of $C$ together with "lacking" $(N-k)$ components can be determined by solving the system of the additional conditions, the compatibility conditions and the orthonormality conditions coupled with relations determining elements of $C$ matrix by eigenvalues and components of basic eigenvectors. We succeeded to clear the statement of the problem to the end in the process of concrete calculations. Deriving and solving the huge polynomial systems had been performed on SPP by using CAS REDUCE 3.6. The developed algorithm had been used for determining defects in elements of two-dimensional arrays of tunnel junctions [2]. We discuss also the inverse problem, when symmetry of basic eigenvectors is reserved. With such symmetry discrete eigenfunctions can be prolonged from the rectangular on the whole plain. But the symmetry of basic eigenvectors leads to symmetry of found matrix with respect to both diagonals (persymmetry) [3]. As result we can't give spectrum disturbed arbitrarily. In general persymmetric nine-diagonal matrix can be constructed of given disturbed spectrum and prescribed symmetry of basic eigenvectors. Such matrix corresponds to discrete Schr\"odinger equation with nonlocal nine-point potential. The problem is reduced to solving polynomial system again. But this time the symmetry conditions are used instead of some additional conditions, orthonormality conditions and compatibility conditions, satisfied automatically. \begin{enumerate} \item S.I.Serdyukova - Russian Journal of Numerical Analysis and Mathematical Modelling 2000, Vol.15, No.5, pp. 455-468. \item L.V.Bobyleva, S.I.Serdyukova - Zh. Vychisl. Mat. i Mat. Fiz. 2002, Vol.42, No.1, pp.3-10 (in Russian) \item S.I.Serdyukova - Proceedings of the Int.Workshop "Computer Algebra and its Application to Physics". Dubna, E5, 11-2001-279, 2001, 10p. \end{enumerate} \end{document}