Inverse spectral theory of finite Jacobi matrices
Author:
Peter C. Gibson
Journal:
Trans. Amer. Math. Soc. 354 (2002), 47034749
MSC (2000):
Primary 47B36; Secondary 34K29
Published electronically:
July 15, 2002
MathSciNet review:
1926834
Fulltext PDF Free Access
Abstract 
References 
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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, semialgebraic 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.
 [Arv93]
William Arveson, Improper filtrations for algebras: Spectra of unilateral tridiagonal operators, Acta Sci. Math. (Szeged) 57 (1993), no. 14, 1124. MR 94i:46071
 [BG87]
Daniel Boley and Gene H. Golub, A survey of matrix inverse eigenvalue problems, Inverse Problems 3 (1987), 595622. MR 89m:65036
 [BR90]
Riccardo Benedetti and JeanJacques Risler, Real algebraic and semialgebraic sets, Actualités Mathématiques, Hermann, Paris, 1990. MR 91j:14045
 [Brø83]
Arne Brøndsted, An introduction to convex polytopes, Graduate Texts in Mathematics, vol. 90, SpringerVerlag, New YorkBerlin, 1983. MR 84d:52009
 [dBG78]
Carl de Boor and Gene H. Golub, The numerically stable reconstruction of a Jacobi matrix from spectral data, Linear Algebra and its Applications 21 (1978), 245260. MR 80i:15007
 [Gib]
Peter C. Gibson, Spectral distributions and isospectral sets of tridiagonal matrices, Preprint.
 [Gib00]
Peter C. Gibson, Moment problems for Jacobi matrices and inverse problems for systems of many coupled oscillators, Ph.D. thesis, University of Calgary, 2000.
 [Gla99]
Graham M. L. Gladwell, Inverse finite element vibration problems, Journal of Sound and Vibration 211 (1999), 309324.
 [GS97]
Fritz Gesztesy and Barry Simon, Mfunctions and inverse spectral analysis for finite and semiinfinite Jacobi matrices, Journal d'Analyse Mathématique 73 (1997), 267297. MR 99c:47039
 [Lam97]
Michael P. Lamoureux, Reflections on the almost Mathieu operator, Integral Equations and Operator Theory 28 (1997), 4559. MR 98d:47068
 [Sim98]
Barry Simon, The classical moment problem as a selfadjoint finite difference operator, Advances in Mathematics 137 (1998), 82203. MR 2001e:47020
 [Tes00]
Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 2001b:39019
 [Var62]
Richard S. Varga, Matrix iterative analysis, PrenticeHall, New Jersey, 1962. MR 28:1725
 [Arv93]
 William Arveson, Improper filtrations for algebras: Spectra of unilateral tridiagonal operators, Acta Sci. Math. (Szeged) 57 (1993), no. 14, 1124. MR 94i:46071
 [BG87]
 Daniel Boley and Gene H. Golub, A survey of matrix inverse eigenvalue problems, Inverse Problems 3 (1987), 595622. MR 89m:65036
 [BR90]
 Riccardo Benedetti and JeanJacques Risler, Real algebraic and semialgebraic sets, Actualités Mathématiques, Hermann, Paris, 1990. MR 91j:14045
 [Brø83]
 Arne Brøndsted, An introduction to convex polytopes, Graduate Texts in Mathematics, vol. 90, SpringerVerlag, New YorkBerlin, 1983. MR 84d:52009
 [dBG78]
 Carl de Boor and Gene H. Golub, The numerically stable reconstruction of a Jacobi matrix from spectral data, Linear Algebra and its Applications 21 (1978), 245260. MR 80i:15007
 [Gib]
 Peter C. Gibson, Spectral distributions and isospectral sets of tridiagonal matrices, Preprint.
 [Gib00]
 Peter C. Gibson, Moment problems for Jacobi matrices and inverse problems for systems of many coupled oscillators, Ph.D. thesis, University of Calgary, 2000.
 [Gla99]
 Graham M. L. Gladwell, Inverse finite element vibration problems, Journal of Sound and Vibration 211 (1999), 309324.
 [GS97]
 Fritz Gesztesy and Barry Simon, Mfunctions and inverse spectral analysis for finite and semiinfinite Jacobi matrices, Journal d'Analyse Mathématique 73 (1997), 267297. MR 99c:47039
 [Lam97]
 Michael P. Lamoureux, Reflections on the almost Mathieu operator, Integral Equations and Operator Theory 28 (1997), 4559. MR 98d:47068
 [Sim98]
 Barry Simon, The classical moment problem as a selfadjoint finite difference operator, Advances in Mathematics 137 (1998), 82203. MR 2001e:47020
 [Tes00]
 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 2001b:39019
 [Var62]
 Richard S. Varga, Matrix iterative analysis, PrenticeHall, New Jersey, 1962. MR 28:1725
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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:
http://dx.doi.org/10.1090/S0002994702030787
PII:
S 00029947(02)030787
Received by editor(s):
March 26, 2001
Published electronically:
July 15, 2002
Additional Notes:
Supported by NSERC Postdoctoral Fellowship 2311082000
Article copyright:
© Copyright 2002
American Mathematical Society
