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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Asymptotic bounds for spectral bands of periodic Schrödinger operators


Authors: M. M. Skriganov and A. V. Sobolev
Translated by: the authors
Original publication: Algebra i Analiz, tom 17 (2005), nomer 1.
Journal: St. Petersburg Math. J. 17 (2006), 207-216
MSC (2000): Primary 35P15, 11H06
DOI: https://doi.org/10.1090/S1061-0022-06-00900-9
Published electronically: January 19, 2006
MathSciNet review: 2140682
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Abstract | References | Similar Articles | Additional Information

Abstract: The precise upper and lower bounds for the multiplicity of the spectrum band overlapping are given for the multidimensional periodic Schrödinger operators with rational period lattices. These bounds are based on very recent results on the lattice point problem.


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

  • 1. F. Götze, Lattice point problems and values of quadratic forms, Invent. Math. 157 (2004), 195-226. MR 2135188
  • 2. B. Helffer and A. Mohamed, Asymptotic of the density of states for the Schrödinger operator with periodic electric potential, Duke Math. J. 92 (1998), 1-60. MR 1609321 (99e:35166)
  • 3. E. Krätzel, Lattice points, Math. Appl. (East European Ser.), vol. 33, Kluwer Acad. Publ. Group, Dordrecht, 1988. MR 0998378 (90e:11144)
  • 4. E. Landau, Zur analytischen Zahlentheorie der definiten quadratischen Formen. (Über die Gitterpunkte in einem mehrdimensionalen Ellipsoid), Berichte Math.-Natur. Kl. (Berlin) 31 (1915), 458-476.
  • 5. -, Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Math. Z. 21 (1924), 126-132.
  • 6. L. Parnovski and A. V. Sobolev, On the Bethe-Sommerfeld conjecture for the polyharmonic operator, Duke Math. J. 107 (2001), 209-238. MR 1823047 (2002d:35050)
  • 7. -, Lattice points, perturbation theory and the periodic polyharmonic operator, Ann. Henri Poincaré 2 (2001), 573-581. MR 1846857 (2002j:35235)
  • 8. M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Acad. Press, New York-London, 1978. MR 0493421 (58:12429c)
  • 9. M. M. Srkiganov, Geometric and arithmetic methods in the spectral theory of multi-dimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 171 pp.; English transl., Proc. Steklov Inst. Math. 1987, no. 2 (171), 121 pp. MR 0798454 (87h:47110); MR 0905202 (88g:47038)
  • 10. -, The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential, Invent. Math. 80 (1985), 107-121. MR 0784531 (86i:35107)
  • 11. M. M. Skriganov and A. V. Sobolev, Variation of the number of lattice points in large balls, Acta Arith. (2005) (to appear).

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Additional Information

M. M. Skriganov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: skrig@pdmi.ras.ru

A. V. Sobolev
Affiliation: School of Mathematics, University of Birmingham, Edgbaston Birmingham, B152TT, United Kingdom
Email: asobolev@bham.ac.uk

DOI: https://doi.org/10.1090/S1061-0022-06-00900-9
Keywords: Periodic operators, lattices
Received by editor(s): April 8, 2005
Published electronically: January 19, 2006
Additional Notes: The first author was supported by RFBR (grant no. 02-01-00086) and by INTAS (grant no. 00-429).
Dedicated: Dedicated to L. D. Faddeev on his 70th birthday
Article copyright: © Copyright 2006 American Mathematical Society

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