Rellich type theorems for unbounded domains

Esa V. Vesalainen

doi:10.3934/ipi.2014.8.865

Article Contents

2014, Volume 8, Issue 3: 865-883. Doi: 10.3934/ipi.2014.8.865

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Rellich type theorems for unbounded domains

Esa V. Vesalainen^1,

1.
Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hallstromin katu 2b), FI-00014 University of Helsinki

Received: December 31, 2013

Revised: April 30, 2014

Published: July 2014

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Abstract

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.

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

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