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Inverse spectral theory of finite Jacobi matrices

Author(s): Peter C. Gibson
Journal: Trans. Amer. Math. Soc. 354 (2002), 4703-4749.
MSC (2000): Primary 47B36; Secondary 34K29
Posted: July 15, 2002
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Abstract: We solve the following physically motivated problem: to determine all finite Jacobi matrices $J$ and corresponding indices $i,j$ such that the Green's function

\begin{displaymath}\langle e_j,(zI-J)^{-1}e_i\rangle \end{displaymath}

is proportional to an arbitrary prescribed function $f(z)$. Our approach is via probability distributions and orthogonal polynomials.

We introduce what we call the auxiliary polynomial of a solution in order to factor the map

\begin{displaymath}(J,i,j)\longmapsto [\langle e_j,(zI-J)^{-1}e_i\rangle] \end{displaymath}

(where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.

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Additional Information:

Peter C. Gibson
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: gibson@math.washington.edu

DOI: 10.1090/S0002-9947-02-03078-7
PII: S 0002-9947(02)03078-7
Received by editor(s): March 26, 2001
Posted: July 15, 2002
Additional Notes: Supported by NSERC Postdoctoral Fellowship 231108-2000
Copyright of article: Copyright 2002, American Mathematical Society

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