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Asymptotic bounds for spectral bands of periodic
Schrödinger operators
Author(s):
M.
M.
Skriganov;
A.
V.
Sobolev
Translated by:
the authors
Original publication:
Algebra i Analiz,
tom 17
(2005),
vypusk 1.
Journal:
St. Petersburg Math. J.
17
(2006),
207-216.
MSC (2000):
Primary 35P15, 11H06
Posted:
January 19, 2006
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References |
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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:
-
- 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.
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- 8.
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- 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)
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- 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:
10.1090/S1061-0022-06-00900-9
PII:
S 1061-0022(06)00900-9
Keywords:
Periodic operators,
lattices
Received by editor(s):
8/APR/2005
Posted:
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
Copyright of article:
Copyright
2006,
American Mathematical Society
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