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A chain of one-dimensional Schr\"odinger operators connected by successiveDarboux 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 ageneralization of the fourth or fifth (depending on the parity of $N$)Painlev\'e equations . The $N$-periodic dressing chain has two different Laxrepresentations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Laxpair can be used to construct a transition matrix around the periodic lattice.One can thereby define an associated ``spectral curve'' and a set of Darbouxcoordinates called ``spectral Darboux coordinates''. The equations of motion ofthe dressing chain can be converted to a Hamiltonian system in these Darbouxcoordinates. The symplectic structure of this Hamiltonian formalism turns outto be consistent with a Poisson structure previously studied by Veselov,Shabat, Noumi and Yamada.Comment: latex2e, 41 pages, no figure; (v2) some minor errors are corrected;  (v3) fully revised and shortend, and some results are improve

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