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A second order self-adjoint operator $\Delta=S\partial^2+U$ is uniquelydefined by its principal symbol $S$ and potential $U$ if it acts onhalf-densities. We analyse the potential $U$ as a compensating field (gaugefield) in the sense that it compensates the action of coordinatetransformations on the second derivatives in the same way as an affineconnection compensates the action of coordinate transformations on firstderivatives in the first order operator, a covariant derivative,$\nabla=\partial+\Gamma$. Usually a potential $U$ is derived from othergeometrical constructions such as a volume form, an affine connection, or aRiemannian structure, etc. The story is different if $\Delta$ is an oddoperator on a supermanifold. In this case the second order potential becomes aprimary object. For example, in the case of an odd symplectic supermanifold,the compensating field of the canonical odd Laplacian depends only on thissymplectic structure, and can be expressed by the formula obtained by K.Bering.We also study modular classes of odd Poisson manifolds via $\Delta$-operators,and consider an example of a non-trivial modular class which is related withthe Nijenhuis bracket.

Odd Laplacians: geometrical meaning of potential, and modular class