A second order self-adjoint operator = S@2 + U is uniquely dened by its
principal symbol S and potential U if it acts on half-densities. We analyse the potential U as a compensating eld (gauge eld) in the sense that it compensates the action of coordinate transformations on the second derivatives in the same way as an ane connection compensates the action of coordinate transformations on rst derivatives in the rst order operator, a covariant derivative, r = @ + 􀀀. Usually a potential U is derived from other geometrical constructions such as a volume form, an ane connection, or a Riemannian structure, etc. The story is dierent if is an odd operator on a supermanifold. In this case the second order potential becomes a primary object. For example, in the case of an odd symplectic supermanifold, the compensating eld of the canonical odd Laplacian depends only on this symplectic structure, and can be expressed by the formula obtained by K.Bering. We also study modular classes of odd Poisson manifolds via -operators, and consider an example of a non-trivial modular class which is related with the Nijenhuis bracket.