bibkey: teschl2000jacobi authors: Gerald Teschl year: 2000 title: Jacobi Operators and Completely Integrable Nonlinear Lattices doi: null url: https://www.mat.univie.ac.at/~gerald/ftp/book-jac/jacop.pdf claim: Bounded selfadjoint Jacobi operators admit half-line Green functions, a Riccati relation, and cyclic spectral-moment descriptions under the specified boundary and coefficient hypotheses. strata_touched: [] license: citation-only triage: anchor

Half-line endpoint response

The source is Mathematical Surveys and Monographs 72, AMS, using the author-corrected online version of 24 August 2017. Section 1.1, Hypothesis H.1.4 and Theorem 1.5, p. 13, gives the bounded selfadjoint operator setting. Lemma 1.8, p. 15, and equations (1.90)–(1.95), p. 16, treat half-line Green functions. Section 2.1, pp. 27–29, equations (2.1), (2.7)–(2.11), supplies the Weyl function and Riccati relation. Section 2.5, pp. 40–41, equations (2.94)–(2.98) and Theorem 2.12, connects cyclic vectors, spectral moments and polynomial Gram data. The coefficients are real and bounded, with nonzero off-diagonals and a specified half-line boundary.

The Static continuation, Theorems 29.8–29.9 uses the half-line Dirichlet operator with $a=-1$, $b=3$, $z=-s$, $s\ge 0$. Its independently displayed coercivity puts $z$ outside the spectrum. Alternating-sign conjugation changes the off-diagonal signs and fixes the endpoint. The application selects the smaller Riccati root by $0<r\le 1/(s+1)\le 1$; positivity alone would not select it. The finite tridiagonal determinant recurrence and continued fraction are intermediate computations, not new general Jacobi results.

The finite spectral decomposition in Theorem 29.2 groups repeated rates by whole spectral projections. Extracting coefficients at distinct rates uses finite Vandermonde algebra. The cited spectral method does not assert that one static cancellation removes every cross response. The application’s positive path comparison and sharp all-time input error are supplied in its own proof; they are not attributed to a quoted theorem of this book. No pages or figures are reproduced.