The inverse spectral problem for first order systems on the half line
| dc.creator | Lesch, Matthias | |
| dc.creator | Malamud, Mark M. | |
| dc.date | 1998-05-07 | |
| dc.date | 1999-04-06 | |
| dc.date.accessioned | 2026-07-07T06:21:28Z | |
| dc.date.available | 2026-07-07T06:21:28Z | |
| dc.description | 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 $σ$ and a generalized Fourier transform which diagonalizes the corresponding operator in $L^2_σ(R, C)$. We give necessary and sufficient conditions for a matrix function $σ$ 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 $σ$. 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. | |
| dc.description | LaTeX2e, 36 pages, 18 Feb 1999 completely revised version, section and several references added. 29 March 1999 final version, minor corrections and last section about systems with prescribed spectral functions improved | |
| dc.identifier | https://arxiv.org/abs/math/9805033 | |
| dc.identifier | http://arxiv.org/abs/math/9805033 | |
| dc.identifier | In: Differential operators and related topics, Vol. I (Odessa, 1997), 199--238, Oper. Theory Adv. Appl., 117, Birkhaeuser, Basel, 2000 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/95610 | |
| dc.subject | Spectral Theory | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 34A25 (Primary), 34L (Secondary) | |
| dc.title | The inverse spectral problem for first order systems on the half line | |
| dc.type | text |