Kernel Inspired Factorizations of Partial Differential Operators

It is elementary to factor an ordinary differential operator given a nonzero function in its kernel. Here a higher dimensional generalization of this fact is developed and utilized to determine nontrivial factorizations of constant coefficient partial differential operators. Through Darboux transformations, this result is applied to the solution of linear partial differential equations, commutativity, quantum integrability and the ‘‘bispectral problem.’’