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Decay rates at infinity for solutions to periodic Schrödinger equations

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  30 January 2019

Daniel M. Elton
Daniel M. Elton*
Affiliation:
Department of Mathematics and Statistics, Fylde College, Lancaster University, LancasterLA1 4YF, UK (d.m.elton@lancaster.ac.uk)
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Abstract

We consider the equation Δu = Vu in the half-space ${\open R}_ + ^d $, 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 Δu = Vu is studied as part of a broader class of elliptic evolution equations.

Keywords

Periodic Schrödinger operatorElliptic evolution equationCarleman estimateUnique continuation

MSC classification

Primary: 35J10: Schrödinger operator
Secondary: 35J15: Second-order elliptic equations 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1113 - 1126
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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