Matthias Lesch, Mark Malamud
The Inverse Spectral Problem for First Order Systems on the Half Line
Preprint series: ESI preprints

MSC:

34A55 Inverse problems

34L05 General spectral theory

Abstract: On the half line $[0,\infty)$ we study
first order differential operators of the form
\[ B\frac 1i \frac{d}{dx} + Q(x),\]
where $B:=\mat{B_1}{0}{0}{-B_2},$
$B_1,B_2\in {\rm M}(n,\C)$ are self--adjoint
positive definite matrices and $Q:\R_+\to {\rm M}(2n,\C),$
$\R_+:=[0,\infty),$ 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 $L^2_{\sigma }(\Bbb R,\Bbb C)$.

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.
Keywords: half line, first order differential operator, boundary value problem