We give a characterisation of the spectral properties of linear differentialoperators with constant coefficients, acting on functions defined on a boundedinterval, and determined by general linear boundary conditions. The boundaryconditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with theproperties of the solution of a corresponding boundary value problem for thepartial differential equation $\partial_t q \pm iSq=0$. Namely, we are able toestablish an explicit correspondence between the properties of the family ofeigenfunctions of the operator, and in particular whether this family is abasis, and the existence and properties of the unique solution of theassociated boundary value problem. When such a unique solution exists, weconsider its representation as a complex contour integral that is obtainedusing a transform method recently proposed by Fokas and one of the authors. Theanalyticity properties of the integrand in this representation are crucial forstudying the spectral theory of the associated operator.
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