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Communications in Mathematical Physics

, Volume 241, Issue 1, pp 111–142 | Cite as

Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains

  • Kanehisa TakasakiEmail author
Article

Abstract

A chain of one-dimensional Schrödinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlevé equations . The $N$-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated ``spectral curve'' and a set of Darboux coordinates called ``spectral Darboux coordinates''. The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.

Keywords

Control Parameter Periodic Lattice Hamiltonian System Transition Matrix Poisson Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of Fundamental Sciences, Faculty of Integrated Human StudiesKyoto UniversitySakyo-kuJapan

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