In this case, we can understand the implications of the domain issues for the spectral theorem. If we use the first choice of domain (with no boundary conditions), all functions formula_78 for formula_79 are eigenvectors, with eigenvalues formula_80, and so the spectrum is the whole complex plane. If we use the second choice of domain (with Dirichlet boundary conditions), "A" has no eigenvectors at all. If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for "A", the functions formula_81. Thus, in this case finding a domain such that "A" is self-adjoint is a compromise: the domain has to be small enough so that "A" is symmetric, but large enough so that formula_82.