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Sufficient conditions are given on the coefficients of a second order, semibounded, formally selfadjoint differential operator on R', not necessarily elliptic, so that the closure of the operator restricted to CO' (Rn) is selfadjoint. The results are based on A. E. Nussbaum's notion of quasi- analytic vectors.

Essential selfadjointness of certain partial differential operators on $R\sp{n}$