  In our previous works, we introduced, for each (super)manifold, a commutative
algebra of densities. It is endowed with a natural invariant scalar product. In
this paper, we study geometry of differential operators of second order on this
algebra. In the more conventional language they correspond to certain operator
pencils. We consider the self-adjoint operators and analyze the operator
pencils that pass through a given operator acting on densities of a particular
weight. There are `singular values' for pencil parameters. They are related
with interesting geometric picture. In particular, we obtain operators that
depend on certain equivalence classes of connections (instead of connections as
such). We study the corresponding groupoids. From this point of view we analyze
two examples: the canonical Laplacian on an odd symplectic supermanifold
appearing in Batalin--Vilkovisky geometry and the Sturm--Liouville operator on
the line, related with classical constructions of projective geometry. We also
consider the canonical second order semi-density arising on odd symplectic
supermanifolds, which has some similarity with mean curvature of surfaces in
Riemannian geometry.
