Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T00:19:58.275Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  12 May 2010

M. Sh. Birman  and
V. A. Sloushch
Get access

Abstract

We study discrete spectrum in spectral gaps of an elliptic periodic second orderdifferential operator in L 2(ℝd )perturbed by a decaying potential. It is assumed that a perturbation is nonnegative andhas a power-like behavior at infinity. We find asymptotics in the large coupling constantlimit for the number of eigenvalues of the perturbed operator that have crossed a givenpoint inside the gap or the edge of the gap. The corresponding asymptotics is power-likeand depends on the observation point.

Keywords

periodic Schrödinger operatordiscrete spectrumspectral gapsasymptotics in the large coupling constant limit.
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 32 - 53
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Birman, M. Sh.. The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential . (Russian) Algebra i Analiz 8 (1996), no. 1, 3–20; English transl., St. Petersburg Math. J. 8 (1997), no. 1, 1–14. Google Scholar
M. Reed, B. Simon. Methods of modern mathematical physics. IV: Analysis of operators. Academic Press, New York, 1978.
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.
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.
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.
Alama, S., Deift, P. A., Hempel, R.. Eigenvalue branches of the Schrödinger operator H − λW in a gap of σ(H). Commun. Math. Phys. 121 (1989), 291-321. CrossRefGoogle Scholar
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.
Birman, M. Sh., Solomyak, M. Z.. Estimates for the singular numbers of integral operators . (Russian) Uspekhi Mat. Nauk 32 (1977), no. 1 (193), 17–84. English transl., Russian Math. Surveys 32, no. 1 (1977), 15–89. Google Scholar
Birman, M. Sh., Solomyak, M. Z.. 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. Google Scholar
Sloushch, V. A.. 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. Google Scholar
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.
Birman, M. Sh., Solomyak, M. Z.. 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. Google Scholar
Birman, M. Sh., Solomyak, M. Z.. 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. Google Scholar