Grünbaum, F. Alberto 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]. Reviewer: Angela Pasquale (Clausthal-Zellerfeld) Cited in 24 Documents 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.) Keywords:bispectral problem; hypergeometric functions; matrix valued spherical functions; eigenvalue problems; ordinary differential operators Citations:Zbl 0625.34007 PDFBibTeX XMLCite \textit{F. A. Grünbaum}, NATO Sci. Ser. II, Math. Phys. Chem. 30, 129--140 (2001; Zbl 0999.47018)