The Schrödinger operator H = −Δ + V is considered in a layer or in a d-dimensional cylinder. The potential V is assumed to be periodic with respect to a lattice. The absolute continuity of H is established, provided that V ∈ L p,loc, where p is a real number greater than d/2 in the case of a layer and p > max(d/2, d − 2) for a cylinder. Bibliography: 14 titles.
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References
M. Sh. Birman and T. A. Suslina, “Absolute continuity of a two-dimensional periodic magnetic Hamiltonian with discontinuous vector potential,” Algebra Analiz, 10, No. 4, 1–36 (1998).
M. Sh. Birman and T. A. Suslina, “Periodic magnetic Hamiltonian with variable metrics. Problem of absolute continuity,” Algebra Aualiz, 11. No. 2, 1–40 (1999).
L. I. Danilov, “On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator,” J. Phys. A: Math. Theor., 42, 275204 (2009).
N. Filonov and I. Kachkovskii, “Absolute continuity of the spectrum of a periodic Schrödinger operator in a miiltidimensional cylinder,” Algebra Analiz, 21, No. 1, 133–152 (2009).
T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin-Heidelberg-New York (1966).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4: Analysis of Operators, Academic Press, New—York (1978).
E. Shargorodsky and A. V. Sobolev, “Quasiconformal mappings and periodic spectral problems in dimension two,” J. Anal. Math., 91, 67–103 (2003).
Z. Shen, “On absolute continuity of the periodic Schrödinger operators,” Intern. Math. Res. Notes, No. 1, 1–31 (2001).
R. G. Shterenberg and T. A. Suslina, “Absolute continuity of the spectrum of the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces,” Algebra Analiz, 13, No. 5, 197–240 (2001).
R. G. Shterenberg and T. A. Suslina, “Absolute continuity of the spectrum of the magnetic Schrödinger operator with a metric in a two-dimensional periodic waveguide,” Algebra Analiz, 14, No. 2, 159–206 (2002).
H. F. Smith and C. D. Sogge, “On the L p norm of spectral clusters for compact manifolds with boundary,” Acta Mathematica, 198, No. 1, 107–153 (2007).
C. D. Sogge, “Concerning the L p norm of spectral clusters for second—order elliptic operators on compact manifolds,” J. Funct. Anal., 77, No. 1, 123–138 (1988).
T. A. Suslina, “On the absence of eigenvalues of a periodic matrix Schrödinger operator in a layer,” Russ. J. Math. Phys., 8, No. 4, 463–486 (2001).
L. Thomas, “Time dependent approach to scattering from impurities in a crystal,” Commun. Math. Phys., 33, 335–343 (1973).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 385, 2010, pp. 69–82.
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Kachkovskiy, I., Filonov, N. Absolute continuity of the spectrum of the periodic Schrödinger operator in a layer and in a smooth cylinder. J Math Sci 178, 274–281 (2011). https://doi.org/10.1007/s10958-011-0547-8
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DOI: https://doi.org/10.1007/s10958-011-0547-8