F. Gesztesy and B. Simon
M-Functions and Inverse Spectral Analysis for Finite and
Semi-Infinite Jacobi Matrices
(78K, amstex)
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.)