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St. Petersburg Mathematical Journal

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Discrete spectrum of a periodic Schrödinger operator with variable metric perturbed by a nonnegative rapidly decaying potential

Author: V. A. Sloushch
Translated by: A. P. Kiselev
Original publication: Algebra i Analiz, tom 27 (2015), nomer 2.
Journal: St. Petersburg Math. J. 27 (2016), 317-326
MSC (2010): Primary 35P20
Published electronically: January 29, 2016
MathSciNet review: 3444465
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Abstract | References | Similar Articles | Additional Information

Abstract: The discrete spectrum is investigated that emerges in spectral gaps of the elliptic periodic operator $ A=-\mathrm {div} a(x)\mathrm {grad} +b(x)$, $ x\in \mathbb{R}^{d}$, perturbed by a nonnegative, ``rapidly'' decaying potential

$\displaystyle 0\le V(x)\sim v(x/\vert x\vert)\vert x\vert^{-\varrho }, \quad \vert x\vert\to +\infty ,\quad \varrho \ge d. $

The asymptotics of the number of eigenvalues for the perturbed operator $ B(t)=A+tV$, $ t>0$, that have crossed a fixed point of the gap, is established with respect to the large coupling constant $ t$.

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  • 1. 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
  • 2. Stanley Alama, Percy A. Deift, and Rainer Hempel, Eigenvalue branches of the Schrödinger operator 𝐻-𝜆𝑊 in a gap of 𝜎(𝐻), Comm. Math. Phys. 121 (1989), no. 2, 291–321. MR 985401
  • 3. M. Sh. Birman, Discrete spectrum of the periodic Schrödinger operator for non-negative perturbations, Mathematical results in quantum mechanics (Blossin, 1993) Oper. Theory Adv. Appl., vol. 70, Birkhäuser, Basel, 1994, pp. 3–7. MR 1308998,
  • 4. M. Sh. Birman and V. A. Sloushch, Discrete spectrum of the periodic Schrödinger operator with a variable metric perturbed by a nonnegative potential, Math. Model. Nat. Phenom. 5 (2010), no. 4, 32–53. MR 2662449,
  • 5. V. A. Sloushch, A Cwikel-type estimate as a consequence of some properties of the heat kernel, Algebra i Analiz 25 (2013), no. 5, 173–201 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 25 (2014), no. 5, 835–854. MR 3184610,
  • 6. -, Approximate commutation of a decaying potential and a function of elliptic operator, Algebra i Analiz 26 (2014), no. 5, 215-227; English transl., St. Petersburg Math. J. 26 (2015), no. 5.
  • 7. 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
  • 8. 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
  • 9. 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
  • 10. 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
  • 11. M. Sh. Birman and M. Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert operators, 2nd ed., revised and augmented, Lan', St. Petersburg, 2010. (Russian)
  • 12. M. Š. Birman and M. Z. Solomjak, Estimates for the singular numbers of integral operators, Uspehi Mat. Nauk 32 (1977), no. 1(193), 17–84, 271 (Russian). MR 0438186
  • 13. M. Sh. Birman and M. Z. Solomyak, Compact operators with power asymptotic behavior of the singular numbers, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 126 (1983), 21–30 (Russian, with English summary). Investigations on linear operators and the theory of functions, XII. MR 697420

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Additional Information

V. A. Sloushch
Affiliation: Department of Physics, Saint Petersburg State University, Ulyanovskaya st. 3, Petrodvorets, 198504 Saint Petersburg, Russia

Keywords: Periodic Schr\"odinger operator, discrete spectrum, spectral gap, asymptotics with respect to a large coupling constant, Cwikel-type estimate.
Received by editor(s): September 9, 2014
Published electronically: January 29, 2016
Additional Notes: The work was supported by SPbSU grant and RFBR grant 14-01-00760
Dedicated: To the memory of M. Sh. Birman
Article copyright: © Copyright 2016 American Mathematical Society

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