On the commutativity of self-adjoint extensions of partial differential operators

Pairs of linear partial differential operators of order 2R, R=1,2,... with the constant coefficients before the higher order derivatives, defined on a certain subspace of the set C infinity ( antiΩ), (antiΩ is a bounded domain in Rsup(n)) are considered. Necessary and sufficient conditions for the commutativity of their self-adjoit extensions are proved, some assumptions are made about the coefficients, about the boundary of Ω and about domains of the extensions. Applications of these results to the problem of defining compatible observables as commuting self-adjoint operators in quantum theory are considered