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Zbl 1047.37048
Smirnov, Sergey
Exactly solvable periodic darboux $q$-chains. (English)
[A] Mladenov, Iva\"ilo 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. 296-302 (2003). ISBN 954-90618-4-1/pbk

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 $\text{L}_2(\bbfZ)$
[Xianhua Tang (Changsha)]

MSC 2000:: *37K10 Completely integrable systems etc.
39A13 Difference equations, scaling ($q$-differences)

Keywords: Darboux chain; dressing chain

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