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The inverse spectral problem for first order systems on the half line | | Matthias Lesch
; Mark M. Malamud
; | | Date: |
7 May 1998 | | Journal: | In: Differential operators and related topics, Vol. I (Odessa, 1997), 199--238, Oper. Theory Adv. Appl., 117, Birkhaeuser, Basel, 2000 | | Subject: | Spectral Theory; Classical Analysis and ODEs MSC-class: 34A25 (Primary), 34L (Secondary) | math.SP math.CA | | Affiliation: | Humboldt-University at Berlin) and Mark M. Malamud (Donetsk | | Abstract: | On the half line $[0,infty)$ we study first order differential operators of the form $B 1/i d/(dx) + Q(x)$, where $B:=mat{B_1}{0}{0}{-B_2}$, $B_1,B_2in M(n,C)$ are self--adjoint positive definite matrices and $Q:R_+ o M(2n,C)$, $R_+:=[0,infty)$, 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. We apply our results to show the existence of $2n imes 2n$ Dirac systems with purely absolute continuous, purely singular continuous and purely discrete spectrum of multiplicity $p$, where $1le p le n$ is arbitrary. | | Source: | arXiv, math.SP/9805033 | | Services: | Forum | Review | PDF | Favorites | |
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