RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mosc. Math. J.: Year: Volume: Issue: Page: Find

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

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
References: PDF file   HTML file

Bibliographic databases:

MSC: 35B05
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} 

• http://mi.mathnet.ru/eng/mmj75
• http://mi.mathnet.ru/eng/mmj/v3/i1/p45

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Geyler V.A., Šťovíček P., “Zero modes in a system of Aharonov-Bohm fluxes”, Rev. Math. Phys., 16:7 (2004), 851–907
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
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.
5. Kachmar A., Pan X., “Superconductivity and the Aharonov-Bohm Effect”, C. R. Math., 357:2 (2019), 216–220