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Semiclassical asymptotics and gaps in the spectra of periodic Schrödinger operators with magnetic wells
Author(s):
Bernard
Helffer;
Yuri
A.
Kordyukov
Journal:
Trans. Amer. Math. Soc.
360
(2008),
1681-1694.
MSC (2000):
Primary 35P20, 35J10, 47F05, 81Q10
Posted:
September 25, 2007
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Abstract:
We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrödinger operator with magnetic wells on a noncompact Riemannian manifold  such that  , equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries, has an arbitrarily large number of spectral gaps in the semi-classical limit.
References:
-
- 1.
- S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations. Mathematical Notes 29, Princeton University Press, Princeton, 1982. MR 745286 (85f:35019)
- 2.
- J. Brüning, S. Yu. Dobrokhotov, K. V. Pankrashkin. The spectral asymptotics of the two-dimensional Schrödinger operator with a strong magnetic field. I. Russ. J. Math. Phys. 9 (2002), no. 1, 14-49; II. Russ. J. Math. Phys. 9 (2002), no. 4, 400-416 (see also e-print version math-ph/0411012). MR 1965505 (2005e:81052)
- 3.
- U. Carlsson, An infinite number of wells in the semi-classical limit. Asymptotic Anal. 3 (1990), no. 3, 189-214. MR 1076447 (91i:35134)
- 4.
- H. D. Cornean, G. Nenciu, Two dimensional magnetic Schrödinger operators: width of mini bands in the tight binding approximation, Ann. Henri Poincaré 1 (2000), 203-222. MR 1770797 (2001g:81059)
- 5.
- M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit. London Mathematical Society Lecture Notes Series, 268, Cambridge University Press, Cambridge, 1999. MR 1735654 (2001b:35237)
- 6.
- R. L. Frank, On the tunneling effect for magnetic Schrödinger operators in antidot lattices. Asymptot. Anal. 48 (2006), no. 1-2, 91-120. MR 2233380 (2007b:81074)
- 7.
- B. Helffer, A. Mohamed, Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells. J. Funct. Anal. 138 (1996), 40-81. MR 1391630 (97h:35177)
- 8.
- B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit. I. Comm. Partial Differential Equations 9 (1984), 337-408. MR 740094 (86c:35113)
- 9.
- B. Helffer, J. Sjöstrand, Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation. Ann. Inst. H. Poincaré Phys. Théor. 42, no. 2 (1985), 127-212. MR 798695 (87a:35142)
- 10.
- B. Helffer, J. Sjöstrand, Effet tunnel pour l'équation de Schrödinger avec champ magnétique. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 625-657. MR 963493 (91c:35043)
- 11.
- B. Helffer, J. Sjöstrand, Analyse semi-classique pour l'équation de Harper (avec application à l'équation de Schrödinger avec champ magnétique). Mém. Soc. Math. France (N.S.) 34 (1988). MR 1003937 (91d:81024)
- 12.
- B. Helffer, J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, In: Schrödinger operators (Sønderborg, 1988), Lecture Notes in Phys., 345, Springer, Berlin, 1989, pp. 118-197. MR 1037319 (91g:35078)
- 13.
- R. Hempel, I. Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps. Commun. Math. Phys. 169 (1995), 237-259. MR 1329195 (96a:81026)
- 14.
- I. Herbst, S. Nakamura. Schrödinger operators with strong magnetic fields: quasi-periodicity of spectral orbits and topology. In: Differential operators and spectral theory, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999, pp. 105-123. MR 1730507 (2001g:81066)
- 15.
- Yu. A. Kordyukov, V. Mathai, M. Shubin, Equivalence of projections in semiclassical limit and a vanishing theorem for higher traces in
 -theory. J. Reine Angew. Math. 581 (2005), 193-236. MR 2132676 (2007b:58040) - 16.
- Yu. A. Kordyukov, Spectral gaps for periodic Schrödinger operators with strong magnetic fields. Commun. Math. Phys. 253 (2005), 371-384. MR 2140253 (2006h:81079)
- 17.
- V. Mathai, M. Shubin, Semiclassical asymptotics and gaps in the spectra of magnetic Schrödinger operators. Geometriae Dedicata 91 (2002), 155-173. MR 1919898 (2004f:58040)
- 18.
- S. Nakamura, Band spectrum for Schrödinger operators with strong periodic magnetic fields. In: Partial differential operators and mathematical physics (Holzhau, 1994), Operator Theory: Advances and Applications. vol. 78, Birkhäuser, Basel, 1995, pp. 261-270. MR 1365340 (97a:81038)
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Additional Information:
Bernard
Helffer
Affiliation:
Département de Mathématiques, Bâtiment 425, F91405 Orsay Cédex, France
Email:
Bernard.Helffer@math.u-psud.fr
Yuri
A.
Kordyukov
Affiliation:
Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky str., 450077 Ufa, Russia
Email:
yurikor@matem.anrb.ru
DOI:
10.1090/S0002-9947-07-04423-6
PII:
S 0002-9947(07)04423-6
Received by editor(s):
December 21, 2005
Received by editor(s) in revised form:
September 12, 2006
Posted:
September 25, 2007
Additional Notes:
The first author acknowledges support from the SPECT programme of the ESF and from the European Research Network `Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems' with contract number HPRN-CT-2002-00277.
The second author acknowledges support from the Russian Foundation of Basic Research (grant no. 04-01-00190).
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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