Bispectral Algebras of Commuting Ordinary Differential Operators

Bakalov, Bojko; Horozov, Emil; Yakimov, Milen

doi:10.1007/s002200050244

Cited by 48 publications

(126 citation statements)

References 29 publications

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“…The next theorem summarizes the technology of bispectral Darboux transformations, initiated in [3], [4] and [23]. It was adapted and applied to the case of difference operators in [14].…”

Section: Bispectral Darboux Transformationsmentioning

confidence: 99%

“…The basic technique to establish Theorem 6.2 is the so-called method of bispectral Darboux transformations which was developed by Bakalov, Horozov and Yakimov [3], [4], and by Kasman and Rothstein [23], in relation with a program aiming at describing all bispectral commutative rings of differential operators. In [14], the method was adapted to attack more systematically Krall's original problem.…”

Section: Introductionmentioning

confidence: 99%

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Askey-Wilson type functions with bound states

Haine

Илиев

2006

Ramanujan J

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Abstract. The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of q-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Grünbaum.

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“…The next theorem summarizes the technology of bispectral Darboux transformations, initiated in [3], [4] and [23]. It was adapted and applied to the case of difference operators in [14].…”

Section: Bispectral Darboux Transformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Askey-Wilson type functions with bound states

Haine

Илиев

2006

Ramanujan J

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“…For instance, by replacing t → c(x) + t in the Schur polynomials, one finds q-Schur polynomials. The latter were obtained by Haine and Iliev [9] by using the q-Darboux transforms; the latter had been studied by Horozov and coworkers in [5,6]. The n-soliton solution to the KdV (for N = 2) (for this formulation, see [4]), τ (t) = det δ i,j − a j y i + y j e Moreover the vertex operator for the 1-Toda lattice is a reduction of the 2-Toda lattice vertex operator (see [2]), given by X(t, y, z) = −χ * (z)X(−t, z)X(t, y)χ(y) = z y − z e for q-KdV, having the typical vertex operator properties.…”

Section: Examples and Vertex Operatorsmentioning

confidence: 99%

The solution to the q-KdV equation

1998

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Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this paper is to show that any KdV solution leads effectively to a solution of the q-approximation of KdV. Two different q-KdV approximations were proposed, first one by E. Frenkel [7] and a variation by Khesin, Lyubashenko and Roger [10]. We show there is a dictionary between the solutions of q-KP and the 1-Toda lattice equations, obeying some special requirement; this is based on an algebra isomorphism between difference operators and D-operators, where Df (x) = f (qx). Therefore every notion about the 1-Toda lattice can be transcribed into q-language.Consider the q-difference operators D and D q , defined by Df (y) = f (qy) and D q f (y) := f (qy) − f (y) (q − 1)y , and the q-pseudo-differential operators

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“…Thus, we obtain a geometric approach to some of the results of [BHY2,BHY3] relating W 1+∞ , bispectrality and the adèlic Grassmannian. (For a discussion of bispectrality for solutions of KP and its multicomponent versions in terms of Dbundles see [BN2].)…”

Section: W 1+∞ -Symmetry and Integrable Systemsmentioning

confidence: 99%

$${\mathcal{W}}$$ -Symmetry of the Adèlic Grassmannian

Ben‐Zvi

Nevins

2009

Commun. Math. Phys.

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Abstract. We give a geometric construction of the W 1+∞ vertex algebra as the infinitesimal form of a factorization structure on an adèlic Grassmannian. This gives a concise interpretation of the higher symmetries and Bäcklund-Darboux transformations for the KP hierarchy and its multicomponent extensions in terms of a version of "W 1+∞ -geometry": the geometry of D-bundles on smooth curves, or equivalently torsion-free sheaves on cuspidal curves.

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Bispectral Algebras of Commuting Ordinary Differential Operators

Cited by 48 publications

References 29 publications

Askey-Wilson type functions with bound states

Askey-Wilson type functions with bound states

The solution to the q-KdV equation

$${\mathcal{W}}$$ -Symmetry of the Adèlic Grassmannian

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