96-395 F. Gesztesy and B. Simon

M-Functions and Inverse Spectral Analysis for Finite and Semi-Infinite Jacobi Matrices (78K, amstex) Sep 6, 96

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Abstract. We study inverse spectral analysis for finite and semi-infinite Jacobi matrices $H$. Our results include a new proof of the central result of the inverse theory (that the spectral measure determines $H$). We prove an extension of Hochstadt's theorem (who proved the result in the case $n=N$) that $n$ eigenvalues of an $N \times N$ Jacobi matrix, $H$, can replace the first n matrix elements in determining $H$ uniquely. We completely solve the inverse problem for $(\delta_n,(H-z)^{-1}\delta_n)$ in the $N<\infty$ case. (Here $\delta_n(m)=1$ if $m=n$ and $0$ otherwise.)

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