Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator

TL;DR

This paper investigates the factorization of differential operators on the algebra of densities, focusing on the generalized Sturm-Liouville operator on the line, and identifies conditions and obstructions for such factorizations.

Contribution

It introduces a criterion for factorization of generalized Sturm-Liouville operators on densities, extending the understanding of operator factorizability beyond traditional function spaces.

Findings

Obstructions to factorization on the algebra of densities are analyzed.

A criterion for factorizability based on classical Sturm-Liouville solutions is established.

The possibility of incomplete factorization is demonstrated.

Abstract

We consider factorization problem for differential operators on the commutative algebra of densities (defined either algebraically or in terms of an auxiliary extended manifold) introduced in 2004 by Khudaverdian and Voronov in connection with Batalin-Vilkovisky geometry. We consider the case of the line, where unlike the familiar setting (where operators act on functions) there are obstructions for factorization. We analyze these obstructions. In particular, we study the "generalized Sturm-Liouville" operators acting on the algebra of densities on the line. This in a certain sense is in between the 1D and 2D cases. We establish a criterion of factorizabily for the generalized Sturm-Liouville operator in terms of solution of the classical Sturm-Liouville equation. We also establish the possibility of an incomplete factorization.