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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].
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)