Numerische und
instrumentelle Mathematik
Universität Münster

Frank Natterer: A discrete Gelfand - Levitan theory

Autor: Frank Natterer

Titel: A discrete Gelfand-Levitan-Theory

Stand: September 1998

Abstract:

In their pioneering paper, Gelfand and Levitan introduced an elegant and explicit method for computing the potential of a Sturm-Liouville Operator from its eigenvalues and certain values of its eigenfunctions. It has been noted by Burridge that the Gelfand-Levitan method is closely related to inversion methods for recovering the potential in a hyperbolic equation with focused initial state, thus giving a unified theory for the methods of Gelfand-Levitan, of Gopinath and Sondhi and of Parijskij and Blagoveshchenskij (see also Romanov).

In the present note we analyse discrete analogues of these problems. The spacial differential operator of the Gelfand-Levitan theory is replaced by a symmetric tridiagonal matrix. It will turn out that in this setting the Gelfand-Levitan method reduces essentially to a Cholesky decomposition. The hyperbolic equation in the other theories is replaced by a recursion relation involving an arbitrary tridiagonal matrix, the essential step for the inversion being an LU-decomposition. In our approach, (approximate) transmutation operators as originally used by Gelfand and Levitan play a paramount role.

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