Introduction. Given a differential operator 1 on Dk L2[0, 1] with compact resolvent, there are associated sequences of projections {Pm}j obtained by integrating the resolvent around circles whose radii tend to infinity. The problem of obtaining an eigenfunction expansion associated with LS is just the problem of showing that for each f E ekL2 [0, 1], limm 11f Pmf ll = 0. The usual technique for establishing such results involves an analysis of the kernel of the resolvent. If the operator is of high order, or one is dealing with a system of equations, this technique becomes unwieldy. We present an abstract approach to these problems which avoids the usual analysis of the resolvent kernel. Denote the orthogonal sum L2[0, 1] EDl. . . *D L2[0, 1] (k summands) by ?k L2[0, 1]. Let D = d/dx. A trivial but useful fact is that the operator e = iD acting componentwise on ek L2[0, 1] is self-adjoint if equipped with the domain of n-tuples (f1, . .. , f,,) of absolutely continuous functions whose derivatives are in L2[0, 1], and such that (fi, . . . , fn)(O) = (fl, fn)(l). Moreover the spectrum of e is {2iTnln E Z}, and each eigenvalue has multiplicity k. We begin by showing that the distribution of eigenvalues for this example is typical for a large class of unbounded operators. Information about the eigenvalue distribution for higher order differential operators is easily obtained by use of the spectral mapping theorem. Fixing notation, we let 69(e) be the domain of the operator X, and a(fD) its spectrum. Denote by C the complex numbers and by C* the Hilbert space adjoint of the operator C5. Finally we remark that many of the basic facts about ordinary differential operators can be found in Goldberg [5].