General methods for constructing bispectral operators

Abstract

We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by new classes of bispectral operators.q-alg/9605011 IntroductionThe bispectral problem of J. J. Duistermaat and F. A. Grünbaum [9] consists of finding all bispectral ordinary differential operators, i.e. operators L(x, ∂ x ) having a family of eigenfunctions ψ(x, z), which are also eigenfunctions for another differential operator Λ(z, ∂ z ) in the spectral parameter:In [3] we constructed large families of solutions to this problem, generalizing all previously known results (cf. [9,19,16,17,13], etc.). We obtained them as special Darboux transformations (called "polynomial") from the most obvious solutionsthe (generalized) Bessel and Airy ones. We recall that the Bessel (respectivelyAiry) operators have the formFor fixed ψ, all operators L(x, ∂ x ) for which ψ is an eigenfunction form a commutative algebra A ψ , called the spectral algebra. Then SpecA ψ is an algebraic curve [6] -the spectral curve. The dimension of the space of eigenfunctions ψ (= g.c.d. ordL, L ∈ A ψ ) is called a rank of A ψ . Following G. Wilson [17] we call the spectral algebra A ψ bispectral iff there exists an operator Λ(z, ∂ z ) satisfying (0.2).The purpose of this paper is to present general methods for constructing bispectral operators (respectively -algebras). Some of our statements are only abstract *

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

“…The following introductory material is mainly borrowed from [6]. Here we present it in a form suitable for the continuous-discrete version of the bispectral problem.…”

“…A nice feature of their construction is that a connection to representation theory is underlined and heavily used. Our construction uses only automorphism of algebras as defined in [6] and seems to be simpler and easier to apply elsewhere.…”

Askey-Wilson type functions with bound states

“…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].…”

“…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 3 explains the method of bispectral Darboux transformations. Roughly, the method consists in finding the most general "rational" factorization of some constant coefficient polynomial in 4) for appropriate choices of the parameters a, b, c, d and of the bound states x i to be added. The factorization involves m free parameters defining a new tridiagonal operatorL, by exchanging the two factors…”