Inverse spectral theory of finite Jacobi matrices
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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 \[ \langle e_j,(zI-J)^{-1}e_i\rangle \] 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 \[ (J,i,j)\longmapsto [\langle e_j,(zI-J)^{-1}e_i\rangle ] \] (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.
References
- William Arveson, Improper filtrations for $C^*$-algebras: spectra of unilateral tridiagonal operators, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 11–24. MR 1243265
- Daniel Boley and Gene H. Golub, A survey of matrix inverse eigenvalue problems, Inverse Problems 3 (1987), no. 4, 595–622. MR 928047
- Riccardo Benedetti and Jean-Jacques Risler, Real algebraic and semi-algebraic sets, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990. MR 1070358
- Arne Brøndsted, An introduction to convex polytopes, Graduate Texts in Mathematics, vol. 90, Springer-Verlag, New York-Berlin, 1983. MR 683612
- C. de Boor and G. H. Golub, The numerically stable reconstruction of a Jacobi matrix from spectral data, Linear Algebra Appl. 21 (1978), no. 3, 245–260. MR 504044, DOI 10.1016/0024-3795(78)90086-1
- Peter C. Gibson, Spectral distributions and isospectral sets of tridiagonal matrices, Preprint.
- Peter C. Gibson, Moment problems for Jacobi matrices and inverse problems for systems of many coupled oscillators, Ph.D. thesis, University of Calgary, 2000.
- Graham M. L. Gladwell, Inverse finite element vibration problems, Journal of Sound and Vibration 211 (1999), 309–324.
- Fritz Gesztesy and Barry Simon, $m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math. 73 (1997), 267–297. MR 1616422, DOI 10.1007/BF02788147
- Michael P. Lamoureux, Reflections on the almost Mathieu operator, Integral Equations Operator Theory 28 (1997), no. 1, 45–59. MR 1446830, DOI 10.1007/BF01198795
- Barry Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82–203. MR 1627806, DOI 10.1006/aima.1998.1728
- Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 1711536, DOI 10.1090/surv/072
- Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
Additional Information
- Peter C. Gibson
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 640454
- Email: gibson@math.washington.edu
- Received by editor(s): March 26, 2001
- Published electronically: July 15, 2002
- Additional Notes: Supported by NSERC Postdoctoral Fellowship 231108-2000
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1926834