The inverse spectral problem for first order systems on the half line

Details The inverse spectral problem for first order systems on the half line The inverse spectral problem for first order systems on the half line

1998-05-07

arxiv.org/abs/math/9805033v3

In: Differential operators and related topics, Vol. I (Odessa, 1997), 199--238, Oper. Theory Adv. Appl., 117, Birkhaeuser, Bas

Mathematics

Spectral Theory

LaTeX2e, 36 pages, 18 Feb 1999 completely revised version, section and several references added. 29 March 1999 final version,

Scientific paper

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. We apply our results to show the existence of $2n\times 2n$ Dirac systems with purely absolute continuous, purely singular continuous and purely discrete spectrum of multiplicity $p$, where $1\le p \le n$ is arbitrary.

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