  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.
