St. Petersburg Mathematical Society

Author(s): M. M. Skriganov; A. V. Sobolev
Translated by: the authors
Original publication: Algebra i Analiz, tom 17 (2005), vypusk 5.
Journal: St. Petersburg Math. J. 17 (2006), 815-833.
MSC (2000): Primary 35P99.
Posted: July 27, 2006
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Abstract: The spectrum of the perturbed polyharmonic operator $H = (-\Delta)^l + V$ in $\textsf{L}^2(\mathbb{R}^d)$ with a limit-periodic potential

is studied. It is shown that if

is periodic in one direction in $\mathbb{R}^d$ and

, $d \not = 1 ({\mathrm{{mod}}}4)$ , then the spectrum of

contains a semiaxis. The proof is based on the properties of periodic operators.

1.: M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Nat. Bureau of Standards Appl. Math. Ser., vol. 55, U. S. Government Printing Office, Washington, DC, 1964. MR 0167642 (29:4914)
2.: J. Avron and B. Simon, Almost periodic Schrödinger operators. I. Limit periodic potentials, Comm. Math. Phys. 82 (1981/82), 101-120. MR 0638515 (84i:34023)
3.: B. E. J. Dahlberg and E. Trubowitz, A remark on two-dimensional periodic potentials, Comment. Math. Helv. 57 (1982), 130-134. MR 0672849 (84h:35119)
4.: K. E. Hare and T. C. O'Neil, N-fold sums of Cantor sets, Mathematika 47 (2000), no. 1-2, 243-250 (2002). MR 1924501 (2003h:28014)
5.: 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)
6.: Yu. Karpeshina, On the density of states for the periodic Schrödinger operator, Ark. Mat. 38 (2000), 111-137. MR 1749362 (2001g:47088)
7.: -, Spectral properties of a polyharmonic operator with limit-periodic potential in two dimensions, talk at the Workshop on Spectral Theory of Schrödinger Operators, Montreal, July 2004.
8.: S. V. Konyagin, M. M. Skriganov, and A. V. Sobolev, On a lattice point problem arising in the spectral analysis of periodic operators, Mathematika 50 (2003), 87-98 (2005). MR 2136353 (2005k:11207)
9.: P. Mendes and F. Oliveira, On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity 7 (1994), 329-343. MR 1267692 (95j:58123)
10.: L. Parnovski and A. V. Sobolev, Lattice points, perturbation theory and the periodic polyharmonic operator, Ann. H. Poincaré 2 (2001), 573-581. MR 1846857 (2002j:35235)
11.: M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 172 pp.; English transl., Proc. Steklov Inst. Math. 1987, no. 2. MR 0798454 (87h:47110)
12.: -, The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential, Invent. Math. 80 (1985), 107-121. MR 0784531 (86i:35107)
13.: M. M. Skriganov and A. V. Sobolev, Asymptotic estimates for special bands of periodic Schrödinger operators, Algebra i Analiz 17 (2005), no. 1, 276-288; English transl., St. Petersburg Math. J. 17 (2006), no. 1. MR 2140682 (2005m:35205)
14.: M. A. Shubin, Spectral theory and the index of elliptic operators with almost-periodic coefficients, Uspekhi Mat. Nauk 34 (1979), no. 2, 95-135; English transl., Russian Math. Surveys 34 (1979), no. 2, 109-158. MR 0535710 (81f:35090)

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 35P99.

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: Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, United Kingdom
Email: A.V.Sobolev@sussex.ac.uk

DOI: 10.1090/S1061-0022-06-00931-9
PII: S 1061-0022(06)00931-9
Keywords: Perturbed polyharmonic operator, limit-periodic potentials, Bethe--Sommerfeld conjecture.
Received by editor(s): 6/APR/2005
Posted: July 27, 2006
Additional Notes: The first author acknowledges the support of RFBR (grant no. 05-01-00935).
Dedicated: In memory of O. A. Ladyzhenskaya
Copyright of article: Copyright 2006, American Mathematical Society

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ISSN: 1547-7371(e) ISSN: 1061-0022(p)

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