Discrete Spectrum of the Periodic Schrödinger Operator with a Variable Metric Perturbed by a Nonnegative Potential

M. Sh. Birman; V. A. Sloushch

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 4, page 32-53
  • ISSN: 0973-5348

Abstract

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We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L2(ℝd) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics is power-like and depends on the observation point.

How to cite

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Birman, M. Sh., and Sloushch, V. A.. "Discrete Spectrum of the Periodic Schrödinger Operator with a Variable Metric Perturbed by a Nonnegative Potential." Mathematical Modelling of Natural Phenomena 5.4 (2010): 32-53. <http://eudml.org/doc/197667>.

@article{Birman2010,
abstract = {We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L2(ℝd) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics is power-like and depends on the observation point. },
author = {Birman, M. Sh., Sloushch, V. A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {periodic Schrödinger operator; discrete spectrum; spectral gaps; asymptotics in the large coupling constant limit.; asymptotics in the large coupling constant limit},
language = {eng},
month = {5},
number = {4},
pages = {32-53},
publisher = {EDP Sciences},
title = {Discrete Spectrum of the Periodic Schrödinger Operator with a Variable Metric Perturbed by a Nonnegative Potential},
url = {http://eudml.org/doc/197667},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Birman, M. Sh.
AU - Sloushch, V. A.
TI - Discrete Spectrum of the Periodic Schrödinger Operator with a Variable Metric Perturbed by a Nonnegative Potential
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 32
EP - 53
AB - We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L2(ℝd) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics is power-like and depends on the observation point.
LA - eng
KW - periodic Schrödinger operator; discrete spectrum; spectral gaps; asymptotics in the large coupling constant limit.; asymptotics in the large coupling constant limit
UR - http://eudml.org/doc/197667
ER -

References

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  1. M. Sh. Birman. The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential. (Russian) Algebra i Analiz8 (1996), no. 1, 3–20; English transl., St. Petersburg Math. J. 8 (1997), no. 1, 1–14.  
  2. M. Reed, B. Simon. Methods of modern mathematical physics. IV: Analysis of operators. Academic Press, New York, 1978.  
  3. M. M. Skriganov. Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators. (Russian) Trudy Mat. Inst. Steklov, vol. 171, 1985, 171 pp. English transl., Proc. Steklov Inst. Math., 1987, no. 2, 121 pp.  
  4. M. Sh. Birman. The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. I. Regular perturbations. Boundary value problems, Schrödinger operators, deformation quantization, pp. 334–352, Math. Top., 8, Akademie Verlag, Berlin, 1995.  
  5. M. Sh. Birman, G. E. Karadzhov, M. Z. Solomyak. Boundedness conditions and spectrum estimates for the operators b(X)a(D) and their analogs. Estimates and asymptotics for discrete spectra of integral and differential equations. Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 85–106.  
  6. M. Sh. Birman. Discrete spectrum in the gaps of a continuous one for perturbation with large coupling constant. Estimates and asymptotics for discrete spectra of integral and differential equations. Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 57–73.  
  7. S. Alama, P. A. Deift, R. Hempel. Eigenvalue branches of the Schrödinger operator H − λW in a gap of σ(H). Commun. Math. Phys.121 (1989), 291-321.  
  8. M. Sh. Birman. Discrete spectrum of the periodic Schrödinger operator for non–negative perturbations. Mathematical results in quantum mechanics (Blossin, 1993), 3–7. Oper. Theory Adv. Appl., Vol. 70, Birkhäuser, Basel, 1994.  
  9. M. Sh. Birman, M. Z. Solomyak. Spectral theory of selfadjoint operators in Hilbert space. D. Reidel Publishing Company, 1987, Dordrecht, Holland.  
  10. M. Sh. Birman, M. Z. Solomyak. Estimates for the singular numbers of integral operators. (Russian) Uspekhi Mat. Nauk32 (1977), no. 1 (193), 17–84. English transl., Russian Math. Surveys 32, no. 1 (1977), 15–89.  
  11. M. Sh. Birman, M. Z. Solomyak. Compact operators with power-like asymptotics of singular numbers. (Russian) Investigations on linear operators and the theory of functions, 12. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI)126 (1983), 21–30. English transl., J. Soviet Math. 27 (1984), 2442–2447.  
  12. V. A. Sloushch. Generalizations of the Cwikel estimate for integral operators. (Russian) Trudy Sankt-Peterburgskogo mat. obshchestva, vol. 14 (2008), 169-196. English transl., Proc. St. Petersburg Math. Soc., vol. XIV, Amer. Math. Soc. Transl. (2), vol. 228, 2009.  
  13. M. Sh. Birman, M. Z. Solomyak. Negative discrete spectrum of the Schrödinger operator with large coupling constant: a qualitative discussion. Order, disorder, and chaos in quantum systems (Dubna, 1989). Oper. Theory Adv. Appl., vol. 46, Birkhäuser, Basel, 1990, pp. 3-16.  
  14. M. Sh. Birman, M. Z. Solomyak. Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. (Russian) Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 13, no. 3 (1977), 13–21. English transl., Vestnik Leningrad Univ. Math. 10 (1982), 237–247.  
  15. M. Sh. Birman, M. Z. Solomyak. Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. II. (Russian) Vestnik Leningrad. Univ. Mat. Mekh. Astronom.13, no. 3 (1979), 5–10. English transl., Vestnik Leningrad Univ. Math. 12 (1980), 155–161.  

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