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The bispectral problem: An overview. (English) Zbl 0999.47018

Bustoz, Joaquin (ed.) et al., Special functions 2000: current perspective and future directions. Proceedings of the NATO Advanced Study Institute, Tempe, AZ, USA, May 29-June 9, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 30, 129-140 (2001).
The bispectral problem consists in finding and classifying all possible situations in which the eigenvalue problems \(L(x,d/dx)\varphi(x,k)=k\varphi(x,k)\) and \(B(k,d/dk)=\Theta(x)\varphi(x,k)\) corresponding to two ordinary differential operators of the form \(L(x,d/dx):=-(d/dx)^2+V(x)\) and \(B(k,d/dk):=\sum_{j=0}^M b_j(k) (d/dk)^j\) admit a simultaneous solution \(\varphi(x,k)\). This problem has been stated and solved by J. J. Duistermaat and F. A. Grünbaum [Commun. Math. Phys. 103, 177-240 (1986; Zbl 0625.34007)].
In this survey article the author discusses the original problem and its extensions to multivariable and matrix-valued settings.
For the entire collection see [Zbl 0969.00053].

MSC:

47A75 Eigenvalue problems for linear operators
47E05 General theory of ordinary differential operators
47A10 Spectrum, resolvent
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 0625.34007
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