Decay rates at infinity for solutions to periodic Schrödinger equations

Elton, Daniel Mark (2020) Decay rates at infinity for solutions to periodic Schrödinger equations. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 150 (3). pp. 1113-1126. ISSN 0308-2105

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Abstract

We consider the equation ∆u = Vu in the half-space Rd+ , d ≥ 2 where V has certain periodicity properties. In particular we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation ∆u = Vu is studied as part of a broader class of elliptic evolution equations.