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

Mathematical Modelling of Natural Phenomena (2010)

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

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topBirman, 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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- M. Reed, B. Simon. Methods of modern mathematical physics. IV: Analysis of operators. Academic Press, New York, 1978. Zbl0401.47001
- 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.
- 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. Zbl0848.47032
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- 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.
- 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. Zbl0676.47032
- 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.
- M. Sh. Birman, M. Z. Solomyak. Spectral theory of selfadjoint operators in Hilbert space. D. Reidel Publishing Company, 1987, Dordrecht, Holland.
- 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. Zbl0344.47021
- 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. Zbl0518.47014
- 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.
- 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.
- 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. Zbl0377.47033
- 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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