Journal of Mathematical Sciences

, Volume 176, Issue 3, pp 458–474

Spectral estimates for Schrödinger operators with sparse potentials on graphs

Article

A construction of “sparse potentials,” suggested by the authors for the lattice \( {\mathbb{Z}^d} \), d > 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D > 2. For the Schrödinger operator − Δ − αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(−Δ − αV) of negative eigenvalues of − Δ − αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(−Δ − αV) under very mild regularity assumptions. A similar construction works also for the lattice \( {\mathbb{Z}^2} \), where D = 2. Bibliography: 13 titles.

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References

  1. 1.
    G. Rozenblum and M. Solomyak, “Counting Schrödinger boundstates: semiclassics and beyond,” In: Sobolev Spaces in Mathematics. II. Applications in Analysis and Partial Differential Equations, pp. 329–354, International Mathematical Series 9, Springer and Tamara Rozhkovskaya Publisher (2009).Google Scholar
  2. 2.
    G. Rozenblum and M. Solomyak, “On the spectral estimates for the Schrödinger type operators: the case of small local dimension” [in Russian], Funk. Anal. Pril. 44, No. 4, 21–33 (2010); English transl.: Funct. Anal. Appl. 44, No. 4, 259–269 (2010).CrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Rozenblum and M. Solomyak, “On the spectral estimates for the Schrödinger operator on \( {\mathbb{Z}^d} \), d ⩾ 3” [in Russian], Probl. Mat. Anal. 41, 107–126 (2009); English transl.: J. Math. Sci., New York, 159, No. 2, 241–263 (2009).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D. Pearson, “Singular continuous measures in scattering theory,” Commun. Math. Phys. 60, 13–36 (1976).CrossRefGoogle Scholar
  5. 5.
    S. Molchanov and B. Vainberg, “Spectrum of multidimensional Schrödinger operators with sparse potentials,” In: Analytical and Computational Methods in Scattering and Applied Mathematics (Newark, DE, 1998), pp. 231–254, Chapman and Hall/CRC Res. Notes Math. 417, Boca Raton, FL (2000).Google Scholar
  6. 6.
    S. Molchanov and B. Vainberg, “Scattering on the system of the sparse bumps: multidimensional case,” Appl. Anal. 71, No. 1–4, 167–185 (1999).MATHMathSciNetGoogle Scholar
  7. 7.
    N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry on Groups, Cambridge Univ. Press, Cambridge (1992).Google Scholar
  8. 8.
    M. Sh. Birman and M. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space [in Russian], 2nd ed. Lan’, S. Petersburg etc. (2010); English transl.: D. Reidel Publishing Co., Dordrecht (1987).Google Scholar
  9. 9.
    I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Non-Selfadjoint Operators in Hilbert Space [in Russian], Nauka, Moscow (1965); English transl.: Am. Math. Soc., Providence, RI (1969).Google Scholar
  10. 10.
    W. H. McCrea and F. J. W. Whipple, “Random paths in two and three dimensions,” Proc. Royal Soc. Edinburgh 60, 281–298 (1940).MathSciNetGoogle Scholar
  11. 11.
    F. Spitzer, Principles of Random Walk, Springer (2001).Google Scholar
  12. 12.
    F. Calogero, “Upper and lower limits for the number of bound states in a given central potential,” Commun. Math. Phys 1, 80–88 (1965).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press (1978).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Chalmers University of Technology, University of GothenburgGothenburgSweden
  2. 2.The Weizmann Institute of ScienceRehovotIsrael

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