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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, α 1 ,...,α r are positive, q(0,1) and s=r/2. Then the system

L j =A j A j + -α j =qA j-1 + A j-1 ,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 {λ j,0 ,λ j,1 ,...} is discrete and is contained in the interval [0,||L j ||). It can be found by using the Darboux scheme,

λ j,0 =0,λ j+1,k+1 =q(λ j,k +α j ),λ j+r,k =λ j,k ·

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

A j-1 ψ j,0 =0,ψ j+1,k+1 =A j + ψ j,k ,

and these eigenfunctions form a complete family in L 2 ()

37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
39A13Difference equations, scaling (q-differences)