Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains

A chain of one-dimensional Schrödinger operators is called a “dressing chain” if they are connected by successive Darboux transformations. Particularly interesting are periodic dressing chains; they include finite-band operators and Painlevé equations as a special case. We investigate the Hamiltonian structure of these nonlinear lattices using V. Adler’s 2× 2 Lax pair. The Lax equation and the auxiliary linear problem of this Lax pair contain a shift, rather than a derivative, in the spectral parameter. Despite this unusual feature, we can construct a transition matrix around the periodic chain, an associated “spectral curve” and a set of Darboux coordinates (“spectral Darboux coordinates”). The dressing chain is thereby converted to a Hamiltonian system in these Darboux coordinates. Moreover, the Hamiltonian formalism is accompanied by an odd-dimensional Poisson structure. This induces a quadratic Poisson algebra of the matrix elements of the transition matrix. As a byproduct, we show that this Poisson structure is equivalent to another Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada. arXiv nlin.SI/0206049