# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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, ${\alpha }_{1},...,{\alpha }_{r}$ are positive, $q\in \left(0,1\right)$ and $s=r/2$. Then the system

${L}_{j}={A}_{j}{A}_{j}^{+}-{\alpha }_{j}=q{A}_{j-1}^{+}{A}_{j-1},\phantom{\rule{2.em}{0ex}}{L}_{j+r}={T}^{-s}{L}_{j}{T}^{s},$

has an $r$-parametric family of solutions. The operator ${L}_{j}$ is bounded for each $j$ and its spectrum $\left\{{\lambda }_{j,0},{\lambda }_{j,1},...\right\}$ is discrete and is contained in the interval $\left[0,||{L}_{j}||\right)$. It can be found by using the Darboux scheme,

${\lambda }_{j,0}=0,\phantom{\rule{1.em}{0ex}}{\lambda }_{j+1,k+1}=q\left({\lambda }_{j,k}+{\alpha }_{j}\right),\phantom{\rule{1.em}{0ex}}{\lambda }_{j+r,k}={\lambda }_{j,k}·$

For each $j$, the eigenfunctions of the operator ${L}_{j}$ can also be obtained by using the Darboux scheme,

${A}_{j-1}{\psi }_{j,0}=0,\phantom{\rule{1.em}{0ex}}{\psi }_{j+1,k+1}={A}_{j}^{+}{\psi }_{j,k},$

and these eigenfunctions form a complete family in ${\text{L}}_{2}\left(ℤ\right)$

##### MSC:
 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 39A13 Difference equations, scaling ($q$-differences)
##### Keywords:
Darboux chain; dressing chain