Arkiv för Matematik

Discrete spectrum of the perturbed Dirac operator

Mikhail Sh. Birman and Ari Laptev

Full-text: Open access

Abstract

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operator D(α)=D0−αV1 acting in L2(R3; C4) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.

Article information

Source
Ark. Mat., Volume 32, Number 1 (1994), 13-32.

Dates
Received: 17 June 1993
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898268

Digital Object Identifier
doi:10.1007/BF02559521

Mathematical Reviews number (MathSciNet)
MR1277918

Zentralblatt MATH identifier
0858.47004

Rights
1994 © Institut Mittag-Leffler

Citation

Birman, Mikhail Sh.; Laptev, Ari. Discrete spectrum of the perturbed Dirac operator. Ark. Mat. 32 (1994), no. 1, 13--32. doi:10.1007/BF02559521. https://projecteuclid.org/euclid.afm/1485898268


Export citation

References

  • [BL] Bergh, J. and Löfström, J., Interpolation Spaces. An Introduction., Grundlehren der mathematischen Wissenschaften 223, Springer-Verlag Berlin-Heidelberg-New York, 1976.
  • [B1] Birman, M. Sh., Discrete spectrum in the gaps of a continuous one for perturbations with large coupling constant, Adv. Soviet Math. 7 (1991), 57–73.
  • [B2] Birman, M. Sh., Discrete spectrum in a gaps of a perturbed periodic Schrödinger operator. I. Regular perturbations, to appear in Adv. Soviet Math.
  • [BKS] Birman, M. Sh., Karadzhov, G. E. and Solomyak, M. Z., Boundedness conditions and spectrum estimates for the operators b(X) a (D) and their analogs, Adv. Soviet Math. 7 (1991), 85–106.
  • [BS1] Birman, M. Sh. and Solomyak, M. Z., Asymptotic behavior of the spectrum of weakly polar integral operators, Izv. Akad. Nauk. SSSR Ser. Mat. 34 (1970), 1142–1158 (Russian). English transl: Math. USSR-Izv. 4 (1970), 1151–1168.
  • [BS2] Birman, M. Sh. and Solomyak, M. Z., Spectral asymptotics of pseudodifferential operators with anisotropic homogeneous symbols. I, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 13, (1977), 13–21. II, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 13 (1979), 5–10 (Russian).
  • [BS3] Birman, M. Sh. and Solomyak, M. Z., Spectral Theory of Selfadjoint Operators in Hilbert Space, D. Reidel Publ. Comp., Dordrecht-Boston, 1987.
  • [BS4] Birman, M. Sh. and Solomyak, M. Z., Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, Adv. Soviet Math. 7 (1991), 1–55.
  • [BS5] Birman, M. Sh. and Solomyak, M. Z., Schrödinger Operator. Estimates for number of bound states as function-theoretical problem, Amer. Math. Soc. Transl. (2)150 (1992), 1–54.
  • [C] Cwikel, M., Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. 106 (1977), 93–100.
  • [K] Klaus, M., On the point spectrum of Dirac operators, Helv. Phys. Acta 53 (1980), 453–462.
  • [L] Laptev, A., Asymptotics of the negative discrete spectrum of a class of Schrödinger operators with large coupling constant, Proc. Amer. Math. Soc. (2)119 (1993), 481–488.
  • [T] Thaller, B.The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, Berlin-Heidelberg-New York, 1992.
  • [Y] Yafaev, D., Mathematical Scattering Theory, Transl. Math. Monographs, 105, Amer. Math. Soc., Providence, R.I., 1992.