Inverse spectral theory of finite Jacobi matrices

Author:
Peter C. Gibson

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4703-4749

MSC (2000):
Primary 47B36; Secondary 34K29

DOI:
https://doi.org/10.1090/S0002-9947-02-03078-7

Published electronically:
July 15, 2002

MathSciNet review:
1926834

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Abstract | References | Similar Articles | Additional Information

Abstract: We solve the following physically motivated problem: to determine all finite Jacobi matrices and corresponding indices such that the Green's function

is proportional to an arbitrary prescribed function . 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

(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:
https://doi.org/10.1090/S0002-9947-02-03078-7

Received by editor(s):
March 26, 2001

Published electronically:
July 15, 2002

Additional Notes:
Supported by NSERC Postdoctoral Fellowship 231108-2000

Article copyright:
© Copyright 2002
American Mathematical Society