Non Self Adjoint Operators and Completeness: Eigenfunction Expansions for Generalized Differential Operators

Physical observables are often taken to correspond to Hermitian operators on an appropriate space (usually a Hilbert or nuclear space). The eigenvalues of such operators are real, and the corresponding eigenfunctions can be shown to be complete under appropriate circumstances, say, in the sense of Parseval. Boundary conditions, i.e. the domain of definition, play a large role in determining whether an operator is self adjoint or not, and consequentially whether the theory of self adjoint operators can be leveraged in the mathematical analysis. In this talk we explore an alternative formulation of generalized linear differential operators on appropriate subdomains of the space of tempered distributions. The operators in question are not self adjoint in general, and indeed can have complex eigenvalues. We will have a system and adjoint system, with corresponding eigenvectors and adjoint eigenvectors, which together retain their completeness under appropriate assumptions. Particular boundary conditions will correspond to sub-manifolds of the eigenspaces, recovering the classical theory but allowing for more general boundary conditions.