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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 322
The inverse spectral problem for first order systems on the half line M. Lesch and M. M. Malamud
On the half line [0,infty) we study first order differential operators of the form B 1/i d/dx + Q(x), where B:= mat{B_1}{0}{0}{-B_2}, B_1, B_2 in M(n,C) are self-adjoint positive definite matrices and Q:R_+ to M(2n,C), R_+:=[0,infty), is a continuous self-adjoint off-diagonal matrix function. We determine the self-adjoint boundary conditions for these operators. We prove that for each such boundary value problem there exists a unique matrix spectral function sigma and a generalized Fourier transform which diagonalizes the corresponding operator in L^2_{sigma}(R,C). We give necessary and sufficient conditions for a matrix function sigma to be the spectral measure of a matrix potential Q. Moreover we present a procedure based on a Gelfand-Levitan type equation for the determination of Q from sigma. Our results generalize earlier results of M. Gasymov and B. Levitan. Subj-class: 34A25 (Primary), 34L (Secondary) Notes: 29 pages
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