  We give several generalizations of Rellich's classical uniqueness theorem to
unbounded domains. We give a natural half-space generalization for
super-exponentially decaying inhomogeneities using real variable techniques. We
also prove under super-exponential decay a discrete generalization where the
inhomogeneity only needs to vanish in a suitable cone.
  The more traditional complex variable techniques are used to prove the
half-space result again, but with less exponential decay, and a variant with
polynomial decay, but with supports exponentially thin at infinity. As an
application, we prove the discreteness of non-scattering energies for
non-compactly supported potentials with suitable asymptotic behaviours and
supports.
