On Second Order Differential Operators

Abstract

Nevertheless, (1.2) is meaningful only under differentiability conditions which are unnatural for (1.1). An adjoint to (1.1) exists always, but it cannot be written in terms of derivatives with respect to x. The of A appears to be (1) that it is of local character, (2) that whenever f has a local minimum at xo and f(xo) = 0, then Af(xo) > 0 (provided Af is meaningful). The purpose of this paper is to derive the most general operator with this property. (For a rigorous formulation see section 2.) It will be seen that we are led not only to a useful generalization, but also to an appreciable simplification of the classical theory of second order differential operators.2 The impetus to this investigation came from a reconsideration of the now classical problem to derive the most general diffusion operator.3 It turns out that the two problems are almost identical. Note that our characteristic property remains unaffected by an arbitrary topological transformation of the x-axis. In principle our operators can therefore not be expressed in terms of differentiations with respect to one fixed scale.