Stein–Tomas theorem for a torus and the periodic Schrödinger operator with singular potential

Kachkovskiĭ, I.

doi:10.1090/S1061-0022-2013-01273-8

Stein–Tomas theorem for a torus and the periodic Schrödinger operator with singular potential
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by I. Kachkovskiĭ
Translated by: the author

St. Petersburg Math. J. 24 (2013), 939-948

DOI: https://doi.org/10.1090/S1061-0022-2013-01273-8

Published electronically: September 23, 2013

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Abstract:

A discrete version of the Stein–Tomas theorem for a torus is proved, except for the endpoint case. The result makes it possible to establish the absolute continuity of the spectrum of the periodic Schrödinger operator with a $\delta$-like potential concentrated on a hypersurface of nonzero curvature.

References

Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
M. Sh. Birman and T. A. Suslina, A periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity, Algebra i Analiz 11 (1999), no. 2, 1–40 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 11 (2000), no. 2, 203–232. MR 1702587
M. Sh. Birman, T. A. Suslina, and R. G. Shterenberg, Absolute continuity of the two-dimensional Schrödinger operator with delta potential concentrated on a periodic system of curves, Algebra i Analiz 12 (2000), no. 6, 140–177 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 6, 983–1012. MR 1816514
N. Burq, P. Gérard, and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), no. 3, 445–486 (English, with English and French summaries). MR 2322684, DOI 10.1215/S0012-7094-07-13834-1
L. I. Danilov, On absolute continuity of the spectrum of three- and four-dimensional periodic Schrödinger operators, J. Phys. A 43 (2010), no. 21, 215201, 13. MR 2644606, DOI 10.1088/1751-8113/43/21/215201
Edmund Hlawka, Über Integrale auf konvexen Körpern. I, Monatsh. Math. 54 (1950), 1–36 (German). MR 37003, DOI 10.1007/BF01304101
Alexander Iosevich and Eric Sawyer, Three problems motivated by the average decay of the Fourier transform, Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001) Contemp. Math., vol. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 205–215. MR 1979941, DOI 10.1090/conm/320/05608
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
I. Kachkovskiĭ and N. Filonov, Absolute continuity of the spectrum of the periodic Schrödinger operator in a layer and in a smooth cylinder, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 385 (2010), no. Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 41, 69–81, 235 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 178 (2011), no. 3, 274–281. MR 2749370, DOI 10.1007/s10958-011-0547-8
Alexander Ostrowski, On the Morse-Kuiper theorem, Aequationes Math. 1 (1968), 66–76. MR 242180, DOI 10.1007/BF01817558
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
Zhongwei Shen, On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Notices 1 (2001), 1–31. MR 1809495, DOI 10.1155/S1073792801000010
T. A. Suslina and R. G. Shterenberg, Absolute continuity of the spectrum of the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces, Algebra i Analiz 13 (2001), no. 5, 197–240 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 5, 859–891. MR 1882869
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Lawrence E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335–343. MR 334766
Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6
R. G. Shterenberg, Absolute continuity of the spectrum of the two-dimensional periodic Schrödinger operator with strongly subordinate magnetic potential, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 303 (2003), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 31, 279–320, 325–326 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 129 (2005), no. 4, 4087–4109. MR 2037543, DOI 10.1007/s10958-005-0344-3
Thomas H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and a preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254, DOI 10.1090/ulect/029