The bispectral problem: An overview (Q2760153)

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 \textit{J. J. Duistermaat} and \textit{F. A. Grünbaum} [Commun. Math. Phys. 103, 177-240 (1986; Zbl 0625.34007)]. NEWLINENEWLINENEWLINEIn this survey article the author discusses the original problem and its extensions to multivariable and matrix-valued settings.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00053].