Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
Published electronically: September 25, 2007
MathSciNet review: 2357710
Full-text PDF Free Access

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.


References [Enhancements On Off] (What's this?)

  • 1. Shmuel Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of 𝑁-body Schrödinger operators, Mathematical Notes, vol. 29, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982. MR 745286
  • 2. J. Brüning, S. Yu. Dobrokhotov, and 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. MR 1965505
  • 3. Ulf Carlsson, An infinite number of wells in the semi-classical limit, Asymptotic Anal. 3 (1990), no. 3, 189–214. MR 1076447
  • 4. H. D. Cornean and G. Nenciu, Two-dimensional magnetic Schrödinger operators: width of mini bands in the tight binding approximation, Ann. Henri Poincaré 1 (2000), no. 2, 203–222 (English, with English and French summaries). MR 1770797, 10.1007/PL00001003
  • 5. Mouez Dimassi and Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999. MR 1735654
  • 6. Rupert L. Frank, On the tunneling effect for magnetic Schrödinger operators in antidot lattices, Asymptot. Anal. 48 (2006), no. 1-2, 91–120. MR 2233380
  • 7. Bernard Helffer and Abderemane Mohamed, Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal. 138 (1996), no. 1, 40–81. MR 1391630, 10.1006/jfan.1996.0056
  • 8. B. Helffer and J. Sjöstrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations 9 (1984), no. 4, 337–408. MR 740094, 10.1080/03605308408820335
  • 9. B. Helffer and J. Sjöstrand, Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation, Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), no. 2, 127–212 (French, with English summary). MR 798695
  • 10. B. Helffer and J. Sjöstrand, Effet tunnel pour l’équation de Schrödinger avec champ magnétique, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 4, 625–657 (1988) (French). MR 963493
  • 11. B. Helffer and 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), 113 pp. (1989) (French, with English summary). MR 1003937
  • 12. B. Helffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988) Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 118–197 (French). MR 1037319, 10.1007/3-540-51783-9_19
  • 13. Rainer Hempel and Ira Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps, Comm. Math. Phys. 169 (1995), no. 2, 237–259. MR 1329195
  • 14. Ira Herbst and Shu Nakamura, Schrödinger operators with strong magnetic fields: quasi-periodicity of spectral orbits and topology, Differential operators and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 189, Amer. Math. Soc., Providence, RI, 1999, pp. 105–123. MR 1730507
  • 15. Y. Kordyukov, V. Mathai, and M. Shubin, Equivalence of spectral projections in semiclassical limit and a vanishing theorem for higher traces in 𝐾-theory, J. Reine Angew. Math. 581 (2005), 193–236. MR 2132676, 10.1515/crll.2005.2005.581.193
  • 16. Yuri A. Kordyukov, Spectral gaps for periodic Schrödinger operators with strong magnetic fields, Comm. Math. Phys. 253 (2005), no. 2, 371–384. MR 2140253, 10.1007/s00220-004-1134-3
  • 17. V. Mathai and M. Shubin, Semiclassical asymptotics and gaps in the spectra of magnetic Schrödinger operators, Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), 2002, pp. 155–173. MR 1919898, 10.1023/A:1016245930716
  • 18. S. Nakamura, Band spectrum for Schrödinger operators with strong periodic magnetic fields, Partial differential operators and mathematical physics (Holzhau, 1994), Oper. Theory Adv. Appl., vol. 78, Birkhäuser, Basel, 1995, pp. 261–270. MR 1365340

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35P20, 35J10, 47F05, 81Q10

Retrieve articles in all journals with MSC (2000): 35P20, 35J10, 47F05, 81Q10


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.