Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators

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dc.creatorGesztesy, Fritz
dc.creatorKiselev, Alexander
dc.creatorMakarov, Konstantin A.
dc.date2000-04-19
dc.date2001-02-07
dc.date.accessioned2026-07-07T04:34:47Z
dc.date.available2026-07-07T04:34:47Z
dc.descriptionLet $g(z,x)$ denote the diagonal Green's matrix of a self-adjoint $m\times m$ matrix-valued Schrödinger 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 Schrödinger 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 $π/2$, 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.
dc.descriptionLaTeX, 38 pages, this is a revised and updated version (to appear in Math. Nachr.)
dc.identifierhttps://arxiv.org/abs/math/0004120
dc.identifierhttp://arxiv.org/abs/math/0004120
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/59044
dc.subjectSpectral Theory
dc.titleUniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators
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