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Semiclassical asymptotics and gaps in the spectra of periodic Schrödinger operators with magnetic wells


Authors: Bernard Helffer and Yuri A. Kordyukov
Journal: Trans. Amer. Math. Soc. 360 (2008), 1681-1694
MSC (2000): Primary 35P20, 35J10, 47F05, 81Q10
DOI: https://doi.org/10.1090/S0002-9947-07-04423-6
Published electronically: September 25, 2007
MathSciNet review: 2357710
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Abstract | References | Similar Articles | Additional Information

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 $ M$ such that $ H^1(M, \mathbb{R})=0$, 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.


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  • 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 $ K$-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: https://doi.org/10.1090/S0002-9947-07-04423-6
Received by editor(s): December 21, 2005
Received by editor(s) in revised form: September 12, 2006
Published electronically: 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).
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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