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Exactly solvable periodic darboux $$q$$-chains. (English) Zbl 1047.37048
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 296-302 (2003).
The authors consider a difference $$q$$-analogue of the dressing chain and prove the following theorem: Suppose $$r$$ is even, $$\alpha_1, \ldots, \alpha_r$$ are positive, $$q\in (0, 1)$$ and $$s=r/2$$. Then the system $L_{j}=A_jA^+_j-\alpha_j=qA^+_{j-1}A_{j-1},\qquad L_{j+r}=T^{-s}L_jT^s,$ has an $$r$$-parametric family of solutions. The operator $$L_j$$ is bounded for each $$j$$ and its spectrum $$\{\lambda_{j,0}, \lambda_{j,1}, \ldots\}$$ is discrete and is contained in the interval $$[0, ||L_j||)$$. It can be found by using the Darboux scheme, $\lambda_{j,0}=0,\quad \lambda_{j+1,k+1}=q(\lambda_{j,k}+\alpha_j), \quad \lambda_{j+r,k}=\lambda_{j,k}.$ For each $$j$$, the eigenfunctions of the operator $$L_j$$ can also be obtained by using the Darboux scheme, $A_{j-1}\psi_{j,0}=0,\quad \psi_{j+1,k+1}=A^+_j\psi_{j,k} ,$ and these eigenfunctions form a complete family in $$\text{L}_2(\mathbb{Z})$$
For the entire collection see [Zbl 1008.00022].
##### MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 39A13 Difference equations, scaling ($$q$$-differences)
##### Keywords:
Darboux chain; dressing chain