Answer by Robert Israel for Analytic continuation of instantaneous eigenstates of a time-dependent hamiltonian

An operator can have eigenvectors without being normal. Matters are complicated somewhat by having unbounded operators, but under some reasonable conditions you do have analyticity of eigenvectors for isolated eigenvalues. See Kato, "Perturbation Theory for Linear Operators", http://books.google.ca/books?id=8ji2kN_D3BwC