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Rellich type theorems for unbounded domains

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  • 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.
    Mathematics Subject Classification: Primary: 35P25, 35J05; Secondary: 35R30, 81U40.


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