Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-16T14:48:00.142Z Has data issue: false hasContentIssue false
  • English
  • Français

Decay rates at infinity for solutions to periodic Schrödinger equations

Part of: Partial differential equations Elliptic equations and systems

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)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bardos, C. and Merigot, M.. Asymptotic decay of the solution of a second-order elliptic equation in an unbounded domain. Applications to the spectral properties of a Hamiltonian. Proc. Roy. Soc. Edinburgh Sect. A76 (1977), 323–344.CrossRefGoogle Scholar
2Cruz-Sampedro, J.. Unique continuation at infinity of solutions to Schrödinger equations with complex-valued potentials. Proc. Edinb. Math. Soc. 42 (1999), 143–153.CrossRefGoogle Scholar
3Froese, R., Herbst, I., Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. L 2-Exponential lower bounds to solutions of the Schrödinger equation. Comm. Math. Phys. 87 (1982a), 265–286.CrossRefGoogle Scholar
4Froese, R., Herbst, I., Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. On the absence of positive eigenvalues for one-body Schrödinger operators. J. d'Analyse 41 (1982b), 272–284.CrossRefGoogle Scholar
5Kenig, C., Silvestre, L. and Wang, J.-N.. On Landis' conjecture in the plane. Comm. Partial Differ. Equ. 40 (2015), 766–789.CrossRefGoogle Scholar
6Kondratiev, V. A. and Landis, E. M.. Qualitative theory of second order linear partial differential equations. Itogi Nauki i Tekhniki: Sovremennye Problemy Mat.: Fundamental'nye Napravleniya,vol. 32, VINITI, Moscow, 1988,pp. 99–218, (English transl. in Partial Differential Equations III, Encyclopedia of Mathematical Sciences vol. 32 (Berlin: Springer, 1991).Google Scholar
7Kuchment, P. A.. Floquet theory for partial differential equations. Operator theory: advances and applications, vol. 60 (Basel: Birkhäuser, 1993).CrossRefGoogle Scholar
8Meshkov, V. Z.. Weighted differential inequalities and their application for estimating the rate of decrease at infinity of solutions of second-order elliptic equations. Trudy Mat. Inst. Steklov 190 (1989), 139–158 (English transl. Proc. Steklov Inst. Math. 190 No. 1 (1992) 145–166).Google Scholar
9Meshkov, V. Z.. On the possible rate of decay at infinity of solutions of second order partial differential equations. Mat. Sbornik 182 (1991) 364–383 (English transl. Math. USSR Sbornik 72 No. 2 (1992) 343–361).Google Scholar
10Pall, G.. The distribution of integers represented by binary quadratic forms. Bull. Amer. Math. Soc. 49 (1943), 447–449.CrossRefGoogle Scholar
11Reed, M. and Simon, B.. Methods of modern mathematical physics IV: analysis of operators (San Diego: Academic Press, 1979).Google Scholar
12Simon, B.. Schrödinger semigroups. Bull. Amer. Math. Soc. 7 (1982), 447–526.CrossRefGoogle Scholar
13Thomas, L. E.. Time dependent approach to scattering from impurities in a crystal. Comm. Math. Phys. 33 (1973), 335–343.CrossRefGoogle Scholar