ar X iv : m at h - ph / 0 20 70 22 v 1 1 8 Ju l 2 00 2 Exactly solvable periodic Darboux q - chains ∗
نویسنده
چکیده
where Aj = −d/dx + fj(x) are first order differential operators. A Darboux chain is called periodic if Lj+r = Lj for some r and for all j = 1, 2, . . . . Number r is called the period of a Darboux chain. The operator L+ α2 appears to be the harmonic oscillator in the particular case r = 1. Periodic Darboux chains lead to integrable systems of differential equations for functions fj, which are examined in [1]. The cases α = 0 and α 6= 0, where α = ∑rj=1 αj , are cardinally different. The operators of a periodic Darboux chain are finite-gap if α = 0. If α 6= 0, then the equations for fj lead to the Painlevé equations or their higher analogues; relations (1) define the discrete spectrum of the chain operators Lj: the spectrum of each of these operators consists of r arithmetic sequences (see [1]). Consider the following “q-analogue” of a Darboux chain:
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