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Localized Solutions of the Schrödinger Equation on Hybrid Spaces. Relation to the Behavior of Geodesics and to Analytic Number Theory

  • Andrei Shafarevich
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 273)

Abstract

In this chapter, we review our results concerning localized asymptotic solutions of time-dependent Schrödinger equation on hybrid spaces. We describe the connections of this problem to the problem of global behavior of geodesics on Riemannian manifolds and to certain problems of the analytic number theory.

Keywords

Schrödinger equation Hybrid spaces Localised asymptotic solutions of time-dependent Schrödinger equation 

Notes

Acknowledgments

The work was supported by the Russian Scientific Foundation (grant 16-11-10069).

References

  1. 1.
    Pavlov, B.S.: Model of the zero-range potential with internal structure. Theor. Math. Phys. 59, 345–353 (1984)CrossRefGoogle Scholar
  2. 2.
    Exner, P., Seba, P.: Quantum motion in a halfline connected to a plane. J. Math. Phys. 28, 386–391 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bruning, J., Geyler, V.A.: Scattering on compact manifolds with infinitely thin horns. J. Math. Phys. 44, 371 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Tolchennikov A.A.: The kernel of Laplace-Beltrami operators with zero-radius potential or on decorated graphs. Sbornik Math. 199(7), 1071 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chernyshev V.L., Shafarevich A.I.: Statistics of Gaussian packets on metric and decorated graphs. Philos. Trans. R. Soc. A. 372(2007), Article number: 20130145 (2013)Google Scholar
  6. 6.
    Chernyshev V.L., Tolchennikov A.A., shafarevich A.I.: Behaviour of quasi-particles on hybrid spaces. Relations to the geometry of geodesics and to the problems of analytic number theory. Regul. Chaotic Dyn. 21(5), 531–537 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chernyshev, V.L., Shafarevich, A.I.: Semiclassical asymptotics and statistical properties of Gaussian packets for the nonstationary Schrodinger equation on a geometric graph. Russ. J. Math. Phys. 15, 2534 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chernyshev, V.L.: Time-dependent Schrodinger equation: statistics of the distribution of Gaussian packets on a metric graph. Proc. Steklov Inst. Math. 270, 246262 (2010)CrossRefGoogle Scholar
  9. 9.
    Chernyshev V.L., Tolchennikov A.A.: Asymptotic estimate for the counting problems corresponding to the dynamical system on some decorated graphs. Ergod. Theory Dyn. Syst. (To appear)Google Scholar
  10. 10.
    Maslov, V.P.: Perturbation Theory and Asymptotic Methods. Dunod, Paris (1972)Google Scholar
  11. 11.
    Skriganov, M.M.: Ergodic theory on SL(n), Diophantine approximations and anomalies in the lattice point problem. Invent. Math. 132, 172 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Paternain, G.P.: Geodesic Flows. Birkhauser, Boston (1999)CrossRefGoogle Scholar
  13. 13.
    Ma R.: On the topological entropy of geodesic flows. J. Differ. Geom 45, 74–93 (1997)Google Scholar
  14. 14.
    Knopfmacher, J.: Abstract Analytic Number Theory, 2nd edn. Dover Publishing, New York (1990)zbMATHGoogle Scholar
  15. 15.
    Nazaikinskii V.E.: On the entropy of the Bose-Maslov gas. In: Doklady Mathematics, vol. 448, no. 3, pp. 266–268 (2013)Google Scholar
  16. 16.
    Chernyshev V.L., Minenkov D.S., Nazaikinskii V.E.: The asymptotic behavior of the number of elements in an additive arithmetical semigroup in the case of an exponential function of counting of the generators. Funct. Anal. Appl. 50(2) (2016) (In press)Google Scholar
  17. 17.
    Chernyshev V.L., Minenkov D.S., Nazaikinskii V.E.: About Bose-Maslov in the case of an infinite number of degrees of freedom. In: Doklady Mathematics, vol. 468, no. 6, pp. 618–621 (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.M.V. Lomonosov Moscow State UniversityMoscowRussia

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