Bispectrality

In its simplest form, the bispectral problem can be stated as follows: Find the differential operators $L$ in $x$ for which there exists a differential operator $B$ in $k$ and a common eigenfunction, and such that

$$\begin{cases}L\psi(x,k)=f(k)\psi(x,k),\\B\psi(x,k)=\theta(x)\psi(x,k),\end{cases}$$

for certain functions $f$, $\theta$, typically polynomial functions.

It was first brought out by F.A. Grünbaum [a2] in the context of medical imaging. It was solved in [a1] for second-order $L$; there are essentially two families of solutions, and the remarkable observation that one of these are rational KdV-solutions gave rise to the related question of finding bispectral commutative algebras of differential operators, and their relationship with KP-flows, [a4], [a5] (cf. also Korteweg–de Vries equation; KP-equation). This line of research produced further examples, links with representation theory, and further open questions such as the characterization of bispectral algebras of rank higher than one.

The problem of bispectrality is however richer, in that it can be (and was originally) posed for a pair of operators, one integral and one differential; moreover, the variables in question can be continuous, discrete or mixed. Grünbaum and others produced large classes of examples, including banded matrices and generalizations of classical orthogonal polynomials, and suggested that bispectrality has its deeper roots in symmetry groups.

For generalizations to partial differential operators, as well as open directions, cf. [a3]

References

[a1]	J.J. Duistermaat, F.A. Grünbaum, "Differential equations in the spectral parameter" Comm. Math. Phys. , 103 (1986) pp. 177–240
[a2]	F.A. Grünbaum, "Some nonlinear evolution equations and related topics arising in medical imaging" Phys. D , 18 (1986) pp. 308–311
[a3]	"The bispectral problem (Montreal, PQ, 1997)" J. Harnad (ed.) A. Kasman (ed.) , CRM Proc. Lecture Notes , Amer. Math. Soc. (1998)
[a4]	J.P. Zubelli, F. Magri, "Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV" Comm. Math. Phys. , 141 : 2 (1991) pp. 329–351
[a5]	G. Wilson, "Bispectral commutative ordinary differential operators" J. Reine Angew. Math. , 442 (1993) pp. 177–204