Eigenfunction Expansions for Non-Symmetric Partial Differential Operators, I

Abstract

In two previous papers under the same title ([7], [8]), the writer has constructed a theory of eigenfunction expansions of a generalized PlancherelWeyl type for general classes of realizations of a partial differential operator L on an open subset G of El, the subnormal realizations in [7] and the decomposable realizations in [8]. These results were established without any restrictions upon G, the type of L, or on the smoothness or behaviour of the coefficients of L at the boundary of G or at infinity. Results of the same type were obtained for expansions in solutions of the equations (L CB) u = 0 for positive differential operators B. It is the purpose of the present paper to extend and generalize these results in two separate directions. In the first place, it is very difficult to determine in any concrete case whether a particular realization of a partial differential operator under specific boundary conditions falls within either of the two classes which we have studied, except in the relatively simple case in which L is symmetric. For this reason, we have begun the study elsewhere ([5], [61, [9] ) of the simplest of the basically non-symmetric cases, elliptic operators on unbounded domains under Dirichlet boundary conditions, but the results obtained at this level of generality do not as yet permit us to apply the theory as previously constructed. On the other hand, for the regular case on a bounded domain, the completeness of the eigenfunctions of the Dirichlet problem was proved by the writer in [4] by a method which extends to the whole class of regular variational boundary-value problems. It is of interest, therefore, to obtain some results on the completeness of the eigenfunctions of L for L singular but elliptic, without any assumption on the boundedness or smoothness in the large of its coefficients. In section 1, we show that if f is a distribution with compact support in G and if L satisfies certain simple local conditions at each point of G, then the orthogonality of f to each eigenfunction