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Discrete Spectrum of a Periodic Schrödinger Operator Perturbed by a Rapidly Decaying Potential

Annales Henri Poincaré volume 23, pages 1883–1907 (2022)Cite this article

Abstract

Let \([\lambda ,\mu ]\) be an interval contained in a spectral gap of a periodic Schrödinger operator H. Consider \(H(\alpha )=H-\alpha V\) where V is a fast decaying positive function. We study the asymptotic behavior of the number of eigenvalues of \(H(\alpha )\) in \([\lambda ,\mu ]\) as \(\alpha \rightarrow \infty \).

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References

  1. Alama, S., Avellaneda, M., Deift, P., Hempel, R.: On the existence of eigenvalues of a divergence-form operator \(A+{\lambda }B\) in a gap of \(\sigma (A)\). Asympt. Anal. 8(4), 311–344 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Alama, S., Deift, P., Hempel, R.: Eigenvalue branches of the Schrödinger operator \(H-{\lambda }W\) in a gap of \(\sigma (H)\). Commun. Math. Phys. 121(2), 291–321 (1989)

    ADS  Article  Google Scholar 

  3. Birman, M.: Discrete spectrum in gaps of a continuous one for perturbations with large coupling constants. Adv. Sov. Math. 7, 57–73 (1991)

    MathSciNet  MATH  Google Scholar 

  4. Birman, M., Laptev, A.: Discrete spectrum of the perturbed Dirac operator. Ark. Matematik 32(1), 13–32 (1994)

    ADS  MathSciNet  Article  Google Scholar 

  5. Birman, M., Sloushch, V.: Discrete spectrum of the periodic Schrödinger operator with a variable metric perturbed by a nonnegative potential. Math. Model. Nat. Phenom. 5(4), 32–53 (2010)

    MathSciNet  Article  Google Scholar 

  6. Birman, M., Solomyak, M.: Spectral Theory of Self-adjoint Operators in Hilbert Space, 2nd edn. Izdatelstvo Lan, St. Petersburg (2010)

    Google Scholar 

  7. Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrodinger operators. Ann. Math. (2) 106(1), 93–100 (1977)

    MathSciNet  Article  Google Scholar 

  8. Deift, P., Hempel, R.: On the existence of eigenvalues of the Schrödinger operator \(H-{\lambda }W\) in a gap of \(\sigma (H)\). Commun. Math. Phys. 103, 461–490 (1986)

    ADS  Article  Google Scholar 

  9. Gesztesy, F., Gurarie, D., Holden, H., Klaus, M., Sadun, L., Simon, B., Vogl, P.: Trapping and cascading of eigenvalues in the large coupling constant limit. Commun. Math. Phys. 118, 597–634 (1988)

    ADS  Article  Google Scholar 

  10. Gesztesy, F., Simon, B.: On a theorem of Deift and Hempel. Commun. Math. Phys. 116, 503–505 (1988)

    ADS  MathSciNet  Article  Google Scholar 

  11. Hempel, R.: On the asymptotic distribution of the eigenvalue branches of the Schrödinger operator \( H\pm {\lambda }W\) in a spectral gap of \(H\). J. Reine Angew. Math. 399, 38–59 (1989)

    MathSciNet  MATH  Google Scholar 

  12. Hempel, R.: Eigenvalues in gaps and decoupling by Neumann boundary conditions. J. Math. Anal. Appl. 169(1), 229–259 (1992)

    MathSciNet  Article  Google Scholar 

  13. Hempel, R.: Eigenvalues of Schrödinger operators in gaps of the essential spectrum: an overview. In: Contemporary Mathematics, vol. 458. AMS, Providence, RI (2008)

  14. Klaus, M.: On the point spectrum of Dirac operators. Helv. Phys. Acta 53, 453–462 (1980)

    MathSciNet  Google Scholar 

  15. Klaus, M.: Some applications of the Birman–Schwinger principle. Helv. Phys. Acta 55, 49–68 (1980)

    MathSciNet  Google Scholar 

  16. Lieb, E.: Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Am. Math. Soc. 82, 751–753 (1976)

    Article  Google Scholar 

  17. Lieb, E.: The number of bound states of one-body Schrödinger operators and the Weyl problem. In: Geometry of the Laplace Operator (Proceedings of Symposia in Pure Mathematics, 1979), pp. 241–252

  18. Pushnitski, A.: Operator theoretic methods for the eigenvalue counting function in spectral gaps. Ann. Henri Poincare 10, 793–822 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  19. Rotfeld, SYu.: Remarks on singular numbers of the sum of totally continuous operators. Funct. Anal. Appl. 1(3), 95–96 (1967)

    MathSciNet  Google Scholar 

  20. Rozenbljum, G.: The disctribution of discrete spectrum for singular differential operators. Dokl. Akad. Nauk SSSR 202:1012–1015. Soviet Math. Dokl. 13, 245–249 (1972)

  21. Safronov, O.: The discrete spectrum of selfadjoint operators under perturbations of variable sign. Commun. PDE 26(3–4), 629–649 (2001)

    MathSciNet  Article  Google Scholar 

  22. Safronov, O.: The discrete spectrum of the perturbed periodic Schrödinger operator in the large coupling constant limit. Commun. Math. Phys. 218(1), 217–232 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  23. Safronov, O.: The amount of discrete spectrum of a perturbed periodic Schrödinger operator inside a fixed interval \(({\lambda }_1, {\lambda }_2)\). Int. Math. Not. 9, 411–423 (2004)

    Article  Google Scholar 

  24. Seiler, E., Simon, B.: Bounds in the Yukawa2 quantum field theory: upper bound on pressure, Hamiltonian bound and linear lower bound. Commun. Math. Phys. 45, 99–114 (1975)

    ADS  MathSciNet  Article  Google Scholar 

  25. Sobolev, A.V.: Weyl asymptotics for the discrete spectrum of the perturbed Hill operator. Adv. Sov. Math. 7, 159–178 (1991)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, UNCC, Charlotte, USA

    Oleg Safronov

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  1. Oleg Safronov
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Correspondence to Oleg Safronov.

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Safronov, O. Discrete Spectrum of a Periodic Schrödinger Operator Perturbed by a Rapidly Decaying Potential. Ann. Henri Poincaré 23, 1883–1907 (2022). https://doi.org/10.1007/s00023-021-01141-1

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  • DOI: https://doi.org/10.1007/s00023-021-01141-1

Mathematics Subject Classification

  • 81Q10 (35P20)