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Mosc. Math. J., 2003, Volume 3, Number 1, Pages 45–61 (Mi mmj75)  

This article is cited in 5 scientific papers (total in 5 papers)

Periodic Schrödinger operators and Aharonov–Bohm Hamiltonians

B. Helffera, T. Hoffmann-Ostenhofb, N. S. Nadirashvilic

a Paris-Sud University 11
b International Erwin Schrödinger Institute for Mathematical Physics
c University of Chicago

Abstract: Let $H=-\Delta+V$ be a two-dimensional Schrödinger operator defined on a domain $\Omega\subset\mathbb R^2$ with Dirichlet boundary conditions. Suppose that $H$ and $\Omega$ are that $V(x_1,x_2)=V(-x_1,x_2)$ and that $(x_1,x_2)\in\Omega$ implies $(x_1+1,x_2)\in\Omega$ and $(-x_1,x_2)\in\Omega$. We investigate the associated Floquet operator $H_(q)$, $0\leq 1$. In particular, we show that the lowest eigenvalue $\lambda_q$ is simple for $q\neq 1/2$ and strictly increasing in $q$ for $0<q<1/2$ and that the associated complex-valued eigenfunction $u_q$ has empty zero set. For the Dirichlet realization of the Aharonov–Bohm Hamiltonian in an annulus-like domain with an axis of symmetry,
$$H_{A,V}=(i\partial_{x-1}+ A_1)^2+(i\partial x_2+A_2)^2+V$$
, we obtain similar results, where the parameter $q$ is replaced by the $\frac{1}{2\pi}$-flux through the hole, under the assumption that the magnetic field curl $A$ vanishes identically.

Key words and phrases: Schrödinger operator, magnetic field, eigenvalues.

DOI: https://doi.org/10.17323/1609-4514-2003-3-1-45-61

Full text: http://www.ams.org/.../abst3-1-2003.html
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MSC: 35B05
Received: May 7, 2002
Language:

Citation: B. Helffer, T. Hoffmann-Ostenhof, N. S. Nadirashvili, “Periodic Schrödinger operators and Aharonov–Bohm Hamiltonians”, Mosc. Math. J., 3:1 (2003), 45–61

Citation in format AMSBIB
\Bibitem{HelHofNad03}
\by B.~Helffer, T.~Hoffmann-Ostenhof, N.~S.~Nadirashvili
\paper Periodic Schr\"odinger operators and Aharonov--Bohm Hamiltonians
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 1
\pages 45--61
\mathnet{http://mi.mathnet.ru/mmj75}
\crossref{https://doi.org/10.17323/1609-4514-2003-3-1-45-61}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1996802}
\zmath{https://zbmath.org/?q=an:1043.35057}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208594100005}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Helffer B., Hoffmann-Ostenhof T., “Spectral theory for periodic Schrodinger operators with reflection symmetries”, Comm. Math. Phys., 242:3 (2003), 501–529  crossref  mathscinet  zmath  adsnasa  isi
    2. Geyler V.A., Šťovíček P., “Zero modes in a system of Aharonov-Bohm fluxes”, Rev. Math. Phys., 16:7 (2004), 851–907  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Helffer B., “Analysis of the bottom of the spectrum of Schrodinger operators with magnetic potentials and applications”, European Congress of Mathematics, 2005, 597–617  mathscinet  zmath  isi
    4. Pan Xing-Bin, “Nodal sets of solutions of equations involving magnetic Schrödinger operator in three dimensions”, J. Math. Phys., 48:5 (2007), 053521, 20 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Kachmar A., Pan X., “Superconductivity and the Aharonov-Bohm Effect”, C. R. Math., 357:2 (2019), 216–220  crossref  mathscinet  zmath  isi  scopus
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