Vecteurs propres de matrices de Jacobi

Michèle Audin

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 5, page 1505-1517
  • ISSN: 0373-0956

Abstract

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It is shown that the set of symmetric tridiagonal periodic Jacobi matrices of given spectrum has a preferred tangent vector field, constructed using the eigenvectors of the matrices and the Jacobian of a hyperelliptic curve. It turns out that this preferred vector field is the infinitesimal operator of the celebrated periodic Toda flow.

How to cite

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Audin, Michèle. "Vecteurs propres de matrices de Jacobi." Annales de l'institut Fourier 44.5 (1994): 1505-1517. <http://eudml.org/doc/75107>.

@article{Audin1994,
abstract = {On montre que l’ensemble des matrices tridiagonales périodiques symétriques de spectre fixé possède une direction tangente privilégiée, construite à l’aide des vecteurs propres des matrices et de la jacobienne d’une courbe hyperelliptique. Il se trouve que cette direction est celle du célèbre flot de Toda périodique.},
author = {Audin, Michèle},
journal = {Annales de l'institut Fourier},
keywords = {integrable systems; Toda lattices; Jacobi matrices; Lax equations; real Abelian varieties},
language = {fre},
number = {5},
pages = {1505-1517},
publisher = {Association des Annales de l'Institut Fourier},
title = {Vecteurs propres de matrices de Jacobi},
url = {http://eudml.org/doc/75107},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Audin, Michèle
TI - Vecteurs propres de matrices de Jacobi
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 5
SP - 1505
EP - 1517
AB - On montre que l’ensemble des matrices tridiagonales périodiques symétriques de spectre fixé possède une direction tangente privilégiée, construite à l’aide des vecteurs propres des matrices et de la jacobienne d’une courbe hyperelliptique. Il se trouve que cette direction est celle du célèbre flot de Toda périodique.
LA - fre
KW - integrable systems; Toda lattices; Jacobi matrices; Lax equations; real Abelian varieties
UR - http://eudml.org/doc/75107
ER -

References

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  1. [1] M. ADLER, P. VANMOERBEKE, The Toda lattice, Dynkin diagrams, singularities and Abelian varieties, Invent. Math., 103 (1991), 223-278. Zbl0735.14031MR91m:14071
  2. [2] H. LANGE, Ch. BIRKENHAKE, Complex Abelian varieties, Grundlehren der math. Wissenschaften, Springer, 1992. Zbl0779.14012
  3. [3] P. VANMOERBEKE, D. MUMFORD, The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154. Zbl0502.58032MR80e:58028
  4. [4] A.G. REYMAN, M.A. SEMENOV-TIAN-SHANSKY, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II, Invent. Math., 63 (1981), 423-432. Zbl0442.58016MR82k:58049
  5. [5] P. VANHAECKE, Linearising two-dimensional integrable systems and the construction of action-angle variables, Math. Z., 211 (1992), 265-313. Zbl0758.58011MR93k:58125

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