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


Author: I. Kachkovskiĭ
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 6.
Journal: St. Petersburg Math. J. 24 (2013), 939-948
MSC (2010): Primary 35J10
Published electronically: September 23, 2013
MathSciNet review: 3097555
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Abstract | References | Similar Articles | Additional Information

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.


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  • 1. Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • 2. 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
  • 3. 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
  • 4. 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, 10.1215/S0012-7094-07-13834-1
  • 5. 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, 10.1088/1751-8113/43/21/215201
  • 6. Edmund Hlawka, Über Integrale auf konvexen Körpern. I, Monatsh. Math. 54 (1950), 1–36 (German). MR 0037003
  • 7. 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, 10.1090/conm/320/05608
  • 8. Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
  • 9. 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 Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 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, 10.1007/s10958-011-0547-8
  • 10. Alexander Ostrowski, On the Morse-Kuiper theorem, Aequationes Math. 1 (1968), 66–76. MR 0242180
  • 11. 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
  • 12. Zhongwei Shen, On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Notices 1 (2001), 1–31. MR 1809495, 10.1155/S1073792801000010
  • 13. 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
  • 14. 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
  • 15. Lawrence E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335–343. MR 0334766
  • 16. Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 0358216, 10.1090/S0002-9904-1975-13790-6
  • 17. 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 Linein. 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, 10.1007/s10958-005-0344-3
  • 18. 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 preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254

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Additional Information

I. Kachkovskiĭ
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskay ul. 3, Petrodvorets, St. Petersburg 198054, Russia
Email: ilya.kachkovskiy@gmail.com

DOI: https://doi.org/10.1090/S1061-0022-2013-01273-8
Keywords: Stein--Tomas theorem, Schr\"odinger operator, periodic coefficients, absolutely continuous spectrum
Received by editor(s): June 15, 2012
Published electronically: September 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society