Differential operators commuting with invariant functions

Differential Operators Commuting with Invariant Functions

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 ...

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.

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...

Invariant differential operators

on the complex upper half-plane H, provably invariant under the linear fractional action of SL2(R), but it is oppressive to verify this directly. Worse, the goal is not merely to verify an expression presented as a deus ex machina, but, rather to systematically generate suitable expressions. An important part of this intention is understanding reasons for the existence of invariant operators, a...