Kernel inspired factorizations of partial differential operators
As a practical tool to factor some constant coefficient linear partial differential operators, the following result is proved: Let \(\varphi(x_1,\dots,x_n)\) be a nonzero differentiable function, let \(q(z_1,\dots,z_n)\) be an irreducible polynomial. Let \(K^\varphi_q\) denote the operator \(\varphi(x) \circ q(\partial) \circ 1/\varphi(x)\). Then there exists a set \(F^\varphi_q \subset \ker K^\varphi_q\) such that a differential operator \(L\) factors through \(K^\varphi_q\) if and only if \(F^\varphi_q \subset \ker L\). The set \(F^\varphi_q\) constructed in the proof is parametrized by the algebraic variety \(q^{-1}(0)\). Notes on applications to Darboux transformations, quantum integrability, and bispectrality close the paper.
scientific article; zbMATH DE number 1334728
scientific article