Mathematics – Spectral Theory
Scientific paper
2000-04-19
Mathematics
Spectral Theory
LaTeX, 38 pages, this is a revised and updated version (to appear in Math. Nachr.)
Scientific paper
Let $g(z,x)$ denote the diagonal Green's matrix of a self-adjoint $m\times m$ matrix-valued Schr\"odinger operator $H= -\f{d^2}{dx^2}I_m +Q(x)$ in $L^2 (\bbR)^{m}$, $m\in\bbN$. One of the principal results proven in this paper states that for a fixed $x_0\in\bbR$ and all $z\in\bbC_+$, $g(z,x_0)$ and $g^\prime (z,x_0)$ uniquely determine the matrix-valued $m\times m$ potential $Q(x)$ for a.e.~$x\in\bbR$. We also prove the following local version of this result. Let $g_j(z,x)$, $j=1,2$ be the diagonal Green's matrices of the self-adjoint Schr\"odinger operators $H_j=-\f{d^2}{dx^2}I_m +Q_j(x)$ in $L^2 (\bbR)^{m}$. Suppose that for fixed $a>0$ and $x_0\in\bbR$, $\|g_1(z,x_0)-g_2(z,x_0)\|_{\bbC^{m\times m}}+ \|g_1^\prime (z,x_0)-g_2^\prime (z,x_0)\|_{\bbC^{m\times m}} \underset{|z|\to\infty}{=}O\big(e^{-2\Im(z^{1/2})a}\big)$ for $z$ inside a cone along the imaginary axis with vertex zero and opening angle less than $\pi/2$, excluding the real axis. Then $Q_1(x)=Q_2(x)$ for a.e.~$x\in [x_0-a,x_0+a]$. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators.
Gesztesy Fritz
Kiselev Alexander
Makarov Konstantin A.
No associations
LandOfFree
Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-314150