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Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators | | Fritz Gesztesy
; Alexander Kiselev
; Konstantin A. Makarov
; | | Date: |
19 Apr 2000 | | Subject: | Spectral Theory | math.SP | | Abstract: | Let $g(z,x)$ denote the diagonal Green’s matrix of a self-adjoint $m imes m$ matrix-valued Schrödinger operator $H= -f{d^2}{dx^2}I_m +Q(x)$ in $L^2 (bR)^{m}$, $minbN$. One of the principal results proven in this paper states that for a fixed $x_0inbR$ and all $zinbC_+$, $g(z,x_0)$ and $g^prime (z,x_0)$ uniquely determine the matrix-valued $m imes m$ potential $Q(x)$ for a.e.~$xinbR$. 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 $L^2 (bR)^{m}$. Suppose that for fixed $a>0$ and $x_0inbR$, $|g_1(z,x_0)-g_2(z,x_0)|_{bC^{m imes m}}+ |g_1^prime (z,x_0)-g_2^prime (z,x_0)|_{bC^{m imes m}} underset{|z| oinfty}{=}Oig(e^{-2Im(z^{1/2})a}ig)$ 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.~$xin [x_0-a,x_0+a]$. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators. | | Source: | arXiv, math.SP/0004120 | | Services: | Forum | Review | PDF | Favorites | |
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