Luc Haine

Bispectral darboux transformations: An extension of the Krall polynomials

Orthogonal polynomials satisfying fourth order differential equations were classified by H. L. Krall [K2]. They can be obtained from very special instances of the (generalized) Laguerre and the Jacobi polynomials by the Darboux process applied to semi-infinite tridiagonal matrices [GH1]. In this paper, starting from any instance of the (generalized) Laguerre and the Jacobi polynomials, we construct a one parameter family of polynomials which are eigenfunctions of a fourth order differential operator. In general, these polynomials will satisfy a five term recursion relation. Some further examples involving higher order differential operators and recursion relations are also presented. We use a proper version of the idea of a bispectral involution first proposed by G. Wilson [W1] and then reinterpreted in the context of the Darboux transformation by other authors [KR], [BHY1]. The notion of a second order bispectral differential operator was introduced in [DG]. One says that such an operator L = −d2/dx2 + V(x) is bispectral if there exists a family of its eigenfunctions L(x, d/dx)ψ(x, k) = kψ(x, k)

The algebraic complete integrability of geodesic flow on SO(N)

AbstractWe study for which left invariant diagonal metrics λ onSO(N), the Euler-Arnold equations

The bispectral property of a q-deformation of the Schur polynomials and the q-KdV hierarchy

\dot X = [x,\lambda (X)], X = (x_{ij} ) \in so(N), \lambda _{ij} x_{ij} , \lambda _{ij} = \lambda _{ji}

Askey-Wilson type functions with bound states

can be linearized on an abelian variety, i.e. are solvable by quadratures. We show that, merely by requiring that the solutions of the differential equations be single-valued functions of complex timet∈ℂ, suffices to prove that (under a non-degeneracy assumption on the metric λ) the only such metrics are those which satisfy Manakovs conditions λij=(bi−bj) (ai−aj)−1. The case of degenerate metrics is also analyzed. ForN=4, this provides a new and simpler proof of a result of Adler and van Moerbeke [3].

The Jacobi polynomial ensemble and the Painlevé VI equation

Many hierarchies of the theory of solitons possess symmetries which do not belong to the hierarchy itself. These symmetries are known under the various names of additional, master or conformal symmetries. They were discovered by Fokas, Fuchssteiner and Oevel [9], [10], [25], Chen, Lee and Lin [4] and Orlov and Schulman [26]. They are intimately related to the bihamiltonian nature of the equations of the theory of solitons which was pioneered in the work of Magri [23] and Gel’fand and Dorfman [11].

The Bochner-Krall Problem: Some New Perspectives

We show that appropriate q-analogues of the Schur polynomials provide rational solutions of a q-deformation of the Nth KdV hierarchy. This allows us to construct explicit examples of bispectral commutative rings of q-difference operators.

A rational analogue of the Krall polynomials

Satos theory of infinite dimensional Grassmannians, has been applied to explain the geometry of the K-P equation ([S; DJKM]), it has been used as a tool to study blow up behaviors and to regularize the solutions near the blow up [A-vM2]. The point is that realizing the K-P flow as a holomorphic flow of planes, enables one to follow what happens to the limiting planes as the equation in the original bad coordinates blows up. The blow-up behaviors are characterized by the various strata the orbit of planes visits in the Grassmannian. In this paper such ideas are applied to the N-periodic Toda flow (on periodic Jacobi matrices) which translates into a flow on the space of N-periodic flags of planes in the Grassmannians. Indeed here the N-periodic Toda flow amounts to N coupled KP equations with special interactions between time flows [U-T]. How such matrices blow up has been studied in [FI; F1-Ha; A-vM1] for arbitrary Lie algebras and Kac-Moody Lie algebras, whereas this paper focusses on regularizing the flow near the blow up locus; that is, on finding the boundary of isospectral sets. If N-periodic Jacobi matrices

KP Trigonometric Solitons and an Adelic Flag Manifold

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.