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Russian Math. Surveys 57:4 (2002), to appear

Exactly solvable periodic Darboux q -chains
Ivan A. Dynnikov and Sergey V. Smirnov

Let L1 , L2 , . . . be selfadjoint differential operators acting on R. They form a Darboux chain if they satisfy the relation Lj = Aj A+ - j = A+ 1 Aj -1 , (1) j j- 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 + appears to be the harmonic oscillator in the particular case r = 1. 2 Periodic Darboux chains lead to integrable systems of differential equations for functions fj , which r are examined in [1]. The cases = 0 and = 0, where = j =1 j , are cardinally different. The operators of a periodic Darboux chain are finite-gap if = 0. If = 0, then the equations for fj lead to the Painlevґ equations or their higher analogues; relations (1) define the discrete spectrum of the chain e 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: Lj = Aj A+ - j = q A j
+ j -1

A

j -1

,

(2)

where Aj = aj + bj T are difference operators on one-dimensional lattice Z and T is the shift operator: (Aj f )(n) = aj (n)f (n) + bj (n)f (n + 1), aj (n), bj (n) R\{0}. Without loss of generality we may assume that bj (n) > 0 for all j, n. One of the important features of difference operators in comparison to differential ones is the possibility to define a periodic chain in various ways. We say that a chain (2) is periodic with period r and shift s if the relation L
j +r

=T

-s

Lj T

s

(3)

holds for all j 1. In the literature the case s = 0 has mostly been considered. However the possibility of factorization in "the reverse order" was mentioned in [2, 4]: operators Aj = aj + bj T can be replaced by operators of the form Aj = aj T + + bj ; such factorization with s = 0 is, in fact, equivalent to the factorization "in the right order" for s = r (the operator Lj is replaced by T -j Lj T j ). Apparently, the general formulation with an arbitrary s has not been considered before, while the following special cases have been studied in the literature. 1. = 0, q = 1, r is arbitrary [2]. Operators Lj are finite-gap. 2. > 0, q = 1, r = 1 (difference analogue of the harmonic oscillator) [2]. In this case, symmetric operators Lj acting on the space of functions on the lattice Z do not exist. Nevertheless, there exists a solution on the "half-line" Z>0 . The spectrum of the operator L + is exactly the same as that of the 2 harmonic oscillator: k = (k + 1 ); the eigenfunctions are expressed in terms of the Charlier polynomials, 2 and, therefore, form a complete family in L2 (Z>0 ). 3. r = 1, > 0, 0 < q < 1 (or < 0, 1 < q ) [2, 5] (q -oscillator). The spectrum of the operator L lies in the interval [0, 1qq ) (or in (0, qq1 if q > 1); it forms a "q -arithmetic sequence". It is mentioned in [2] - - that, in this case, the operator L is unbounded, and co jectured that L has continuous spectrum in the interval ( 1qq , ). - 4. Another version of the q -oscillator is considered in [6]. It is presented by a difference operator on the whole "line" Z. In our settings, this version of the q -oscillator can be interpreted as the case s = 1, r = 2, 1 = 2 , ( and q are the same as in 3). A particular solution that is symmetric about the origin

This work was partially supp orted by Russian Foundation for Fundamental Research (grants 02-01-00659, 00-15-96011)

1

was found in [6]. This case is specific because here the operator L is bounded and has no continuous spectrum. We claim here that the same property holds for a q -chain of an arbitrary even period r with the shift s = r/2: the chain operators are bounded and have no continuous spectrum. We also provide explicitly the general solution of the problem in the case s = 1, r = 2. Theorem. Suppose r is even, 1 , . . . , r are positive, q satisfies the inequality 0 < q < 1, and we have s = r/2. Then the system (2),(3) has an r-parametric family of solutions. The operator Lj is bounded for each j ; its spectrum {j,0 , j,1 , . . . } is discrete and is contained in the interval [0, Lj ). It can be found by using the Darboux scheme:
j,0

= 0,

j +1,k+1

= q (

j,k

+ j ),

j +r,k

=

j,k

.

For each j , the eigenfunctions of the operator Lj also can be obtained by using the Darboux scheme: A
j -1

j,0

= 0,

j +1,k+1

= A+ j

j,k

;

these eigenfunctions form a complete family in L2 (Z). A similar assertion holds for 1 , . . . , r < 0, 1 < q (in this case, the point 0 is not included in the spectrum of Lj ). Now we present an explicit form of the operators A1 , A2 for r = 2. Prop osition. or r = 2 = 2s, 1 , 2 > 0, 0 < q < 1, the general solution of the problem (2),(3) has the F form a1 (n) = 2n , b1 (n) = 2n+1 , a2 (n) = 2n-1 , b2 (n) = 2n , where = ±1, n = 1 cn - 2 q -n-- 2 + cn+1 q -2n-2-1 , 2 (1 - q -2(n+) )(1 - q -2(n++1) ) cn =
1

n =

1 cn+1 - 2 q n++ 2 + cn q 2n+2+1 , 2 (1 - q 2(n+) )(1 - q 2(n++1) )

1

(4)

1 + 2 1 - 2 + (-1)n , 1-q 1+q
+1 - -1 -1 - +1

R is arbitrary, the parameter satisfies the restrictions c[] q
-
1

+ c[

]-1

q < 2 < min(c[] q
1

+ c[

]-1

q

, c[] q

+ c[

]-1

q

)

if Z, /

and 2 = c q 2 + c-1 q - 2 if Z (in this case, in fractions (4), one has to cancel the factor (1 - q 2(n+) ) 1 for n (mod 2) and the factor (1 - q 2(n++1) ) for n + 1(mod 2)). Here, we set = - [] - 2 and [] stands for the integral part of . Observation. In the case r = 2, converge to the harmonic oscillator d x = nh, T = exp(h dx ) and assume operator on R. Then for any f C L
1,2

1 = 2 , operators Lj + 2j constructed from the above solutions as q 1 in the following sense: take = 0, = -1, q = exp(- 1 h2 ), 4 that n is real in formulae (4) and that the operator Lj is a difference 2 (R) we have 1 d2 2 f (x) = - 2 + 1 x2 f (x) + o(h). 2 dx 4

+

If 1 = 2 , then the operator Lj converges in the same sense to - (j + j d2 + 2 dx 16
+1

)2

x2 -

j (j - j +1 )(j + 3j - 2 4(j + j +1 )2 x2

+1

)

,

where j +2 = j . In the cases 2,3 mentioned above there is no link of that kind between the discrete and the continuous models. Thus the considered case s = r/2 gives, in a certain sense, a proper discretization of a Darboux chain. We hope to prove that a q -chain converges in the same way to an ordinary Darboux chain for an arbitrary even r. The numerical experiment confirms that a q -chain of the period 6 converges to an ordinary Darboux chain of the period 3 if j = j +3 . 2

References
[1] A. P. Veselov, A. B. Shabat. Dressing chains and the spectral theory of SchrЁ odinger operators. Functional Anal. Appl., 27 (1993), no. 2, 121. [2] S. P. Novikov, I. A. Taimanov. Difference analogs of the harmonic oscillator. Appendix II in [3]. odinger operators and Laplace [3] S. P. Novikov, A. P. Veselov. Exactly solvable two-dimensional SchrЁ transformations. Solitons, Geometry, and Topology: on the Crossroad, ed. V. M. Buchstaber, S. P. Novikov. AMS Trans. Ser. 2, 179 (1997), 109132. [4] S. P. Novikov, I. A. Dynnikov. Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds. Russian Math. Surveys, 52 (1997), no. 5(317), 175234. [5] V. Spiridonov, L. Vinet, A. Zhedanov. Difference SchrЁ odinger operators with linear and exponential discrete spectra. Lett. Math. Phys, 129 (1993), 6773. [6] N. Atakishiyev, A. Frank, K. Wolf. A simple difference realization of the Heisenberg q -algebra. J. Math. Phys, 35(7), 1994, 32533260.

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