Differential operators commuting with convolution integral operators

On Commuting Differential Operators

The theory of commuting linear differential expressions has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P,L). Gesztesy and the present author have established a relationship of this circle of ideas with the property that all solutions of the differential equations Ly = zy, z ∈ C, are meromorphic. In this paper this relationship is explored ...

Differential Operators Commuting with Invariant Functions

On Commuting Matrix Differential Operators

If the differential expressions P and L are polynomials (over C) of another differential expression they will obviously commute. To have a P which does not arise in this way but satisfies [P,L] = 0 is rare. Yet the question of when it happens has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P,L). In this paper the question is answered in the...

Commuting differential operators with regular singularities

We study a system of partial differential equations defined by commuting family of differential operators with regular singularities. We construct ideally analytic solutions depending on a holomorphic parameter. We give some explicit examples of differential operators related to SL(n, R) and completely integrable quantum systems.

Bispectral algebras of commuting ordinary differential operators

We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank N . It combines and unifies the ideas of Duistermaat–Grünbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectra...