On the spectrum of a limit-periodic Schrödinger operator

Skriganov, M.; Sobolev, A.

doi:10.1090/S1061-0022-06-00931-9

On the spectrum of a limit-periodic Schrödinger operator
HTML articles powered by AMS MathViewer

by M. M. Skriganov and A. V. Sobolev
Translated by: the authors

St. Petersburg Math. J. 17 (2006), 815-833

DOI: https://doi.org/10.1090/S1061-0022-06-00931-9

Published electronically: July 27, 2006

PDF | Request permission

Abstract:

The spectrum of the perturbed polyharmonic operator $H = (-\Delta )^l + V$ in $\textsf {L}^2(\mathbb R^d)$ with a limit-periodic potential $V$ is studied. It is shown that if $V$ is periodic in one direction in $\mathbb R^d$ and $8l > d+3$, $d \not = 1 ({mod}4)$, then the spectrum of $H$ contains a semiaxis. The proof is based on the properties of periodic operators.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
Joseph Avron and Barry Simon, Almost periodic Schrödinger operators. I. Limit periodic potentials, Comm. Math. Phys. 82 (1981/82), no. 1, 101–120. MR 638515
B. E. J. Dahlberg and E. Trubowitz, A remark on two-dimensional periodic potentials, Comment. Math. Helv. 57 (1982), no. 1, 130–134. MR 672849, DOI 10.1007/BF02565850
Kathryn E. Hare and Toby C. O’Neil, $N$-fold sums of Cantor sets, Mathematika 47 (2000), no. 1-2, 243–250 (2002). MR 1924501, DOI 10.1112/S0025579300015850
Bernard Helffer and Abderemane Mohamed, Asymptotic of the density of states for the Schrödinger operator with periodic electric potential, Duke Math. J. 92 (1998), no. 1, 1–60. MR 1609321, DOI 10.1215/S0012-7094-98-09201-8
Yulia E. Karpeshina, On the density of states for the periodic Schrödinger operator, Ark. Mat. 38 (2000), no. 1, 111–137. MR 1749362, DOI 10.1007/BF02384494
—, 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.
Sergei V. Konyagin, Maxim M. Skriganov, and Alexander V. Sobolev, On a lattice point problem arising in the spectral analysis of periodic operators, Mathematika 50 (2003), no. 1-2, 87–98 (2005). MR 2136353, DOI 10.1112/S0025579300014819
Pedro Mendes and Fernando Oliveira, On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity 7 (1994), no. 2, 329–343. MR 1267692
Leonid Parnovski and Alexander V. Sobolev, Lattice points, perturbation theory and the periodic polyharmonic operator, Ann. Henri Poincaré 2 (2001), no. 3, 573–581. MR 1846857, DOI 10.1007/PL00001046
M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov. 171 (1985), 122 (Russian). MR 798454
M. M. Skriganov, The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential, Invent. Math. 80 (1985), no. 1, 107–121. MR 784531, DOI 10.1007/BF01388550
M. M. Skriganov and A. V. Sobolev, Asymptotic estimates for spectral bands of periodic Schrödinger operators, Algebra i Analiz 17 (2005), no. 1, 276–288 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 1, 207–216. MR 2140682, DOI 10.1090/S1061-0022-06-00900-9
M. A. Šubin, Spectral theory and the index of elliptic operators with almost-periodic coefficients, Uspekhi Mat. Nauk 34 (1979), no. 2(206), 95–135 (Russian). MR 535710