Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

Discrete spectrum of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. II. Internal gaps


Author: T. A. Suslina
Translated by: the author
Original publication: Algebra i Analiz, tom 15 (2003), nomer 2.
Journal: St. Petersburg Math. J. 15 (2004), 249-287
MSC (2000): Primary 35P20
DOI: https://doi.org/10.1090/S1061-0022-04-00810-6
Published electronically: January 29, 2004
MathSciNet review: 2052132
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The discrete spectrum in the spectral gaps is studied in the case of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. The main goal is to find asymptotics (for the large coupling constant) of the number of eigenvalues that have been ``born'' (or have ``died'') at the edges of the gap. The high-energy (Weyl) asymptotics and the threshold asymptotics are distinguished. At the right edge of the gap, a competition between the Weyl contribution and the threshold contribution may occur. The case of a semiinfinite gap was studied in part I of the paper.


References [Enhancements On Off] (What's this?)

  • [B1] M. Š. Birman, On the spectrum of singular boundary-value problems, Mat. Sb. (N.S.) 55 (97) (1961), 125–174 (Russian). MR 0142896
  • [B2] M. Sh. Birman, Discrete spectrum in the gaps of a continuous one for perturbations with large coupling constant, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 57–73. MR 1306508
  • [B3] M. Sh. Birman, The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. I. Regular perturbations, Boundary value problems, Schrödinger operators, deformation quantization, Math. Top., vol. 8, Akademie Verlag, Berlin, 1995, pp. 334–352. MR 1389015
  • [B4] M. Sh. Birman, The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential, Algebra i Analiz 8 (1996), no. 1, 3–20 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 1, 1–14. MR 1392011
  • [B5] M. Sh. Birman, The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. II. Nonregular perturbations, Algebra i Analiz 9 (1997), no. 6, 62–89 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1073–1095. MR 1610239
  • [BL] M. Sh. Birman and A. Laptev, The negative discrete spectrum of a two-dimensional Schrödinger operator, Comm. Pure Appl. Math. 49 (1996), no. 9, 967–997. MR 1399202, https://doi.org/10.1002/(SICI)1097-0312(199609)49:9<967::AID-CPA3>3.3.CO;2-O
  • [BLSu] M. Sh. Birman, A. Laptev, and T. A. Suslina, The discrete spectrum of a two-dimensional second-order periodic elliptic operator perturbed by a decreasing potential. I. A semi-infinite gap, Algebra i Analiz 12 (2000), no. 4, 36–78 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 4, 535–567. MR 1793617
  • [BS1] M. Š. Birman and M. Z. Solomjak, \cyr Spektral′naya teoriya samosopryazhennykh operatorov v gil′bertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
    M. Sh. Birman and M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. MR 1192782
  • [BS2] M. Sh. Birman and M. Z. Solomyak, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 1–55. MR 1306507
  • [BS3] M. Sh. Birman and M. Solomyak, On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential, J. Anal. Math. 83 (2001), 337–391. MR 1828497, https://doi.org/10.1007/BF02790267
  • [GoKr] I. C. Gohberg and M. G. Kreĭn, \cyr Vvedenie v teoriyu lineĭnykh nesamosopryazhennykh operatorov v gil′bertovom prostranstve, Izdat. “Nauka”, Moscow, 1965 (Russian). MR 0220070
    I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142
  • [He] B. Helffer, Around Floquet eigenvalues, Preprint, Mittag-Leffler Inst., 2002.
  • [Iv] Victor Ivrii, Accurate spectral asymptotics for periodic operators, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1999) Univ. Nantes, Nantes, 1999, pp. Exp. No. V, 11. MR 1718966
  • [S] M. Solomyak, Piecewise-polynomial approximation of functions from 𝐻^{𝑙}((0,1)^{𝑑}), 2𝑙=𝑑, and applications to the spectral theory of the Schrödinger operator, Israel J. Math. 86 (1994), no. 1-3, 253–275. MR 1276138, https://doi.org/10.1007/BF02773681
  • [Su] T. A. Suslina, On discrete spectrum in the gaps of a two-dimensional periodic elliptic operator perturbed by a decaying potential, Preprint, Mittag-Leffler Inst., 2002; Contemp. Math. (to appear).

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 35P20

Retrieve articles in all journals with MSC (2000): 35P20


Additional Information

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 1, Petrodvorets, St. Petersburg 198904, Russia
Email: tanya@petrov.stoic.spb.su

DOI: https://doi.org/10.1090/S1061-0022-04-00810-6
Keywords: Periodic operator, perturbation, discrete spectrum, spectral gap, threshold effect
Received by editor(s): January 14, 2003
Published electronically: January 29, 2004
Additional Notes: Supported by RFBR (grant no. 02-01-00798)
Dedicated: Dedicated to my dear teacher Mikhail Shlemovich Birman on the occasion of his anniversary
Article copyright: © Copyright 2004 American Mathematical Society