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Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics

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Abstract

Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph Γ . We calculate the Lagrangian gluing conditions at vertices \({v\in \Gamma }\) for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network.

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References

  1. Birman M.S. (1962). Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions (Russian, English summary). Vestnik Leningrad. Univ 17(1): 22–55

    MathSciNet  Google Scholar 

  2. Blekher P.M. (1972). On operators that depend meromorphically on a parameter. Moscow Univ. Math. Bull. 24(5–6): 21–26

    MATH  Google Scholar 

  3. Dell’Antonio G. and Tenuta L. (2006). Quantum graphs as holonomic constraints. J. Math. Phys. 47(072102): 1–21

    MathSciNet  Google Scholar 

  4. Duclos P. and Exner P. (1995). Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7: 73–102

    Article  MATH  MathSciNet  Google Scholar 

  5. Freidlin M. and Wentzel A. (1993). Diffusion processes on graphs and averaging principle. Ann. Probab. 21(4): 2215–2245

    MATH  MathSciNet  Google Scholar 

  6. Exner, P., Seba, P.: Electrons in semiconductor microstructures: a challengee to operator theorists. In: Schrödinger Operators, Standard and Nonstandard (Dubna 1988), Singapure: World Scientific, 1989 pp. 79–100

  7. Exner P. and Post O. (2005). Convergence of spectra of graph-like thin manifolds. J. Geom. Phys. 54: 77–115

    Article  MATH  MathSciNet  Google Scholar 

  8. Kostrykin V. and Schrader R. (1999). Kirchhoff’s rule for quantum waves. J. Phys. A: Mathematical and General 32: 595–630

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Kuchment P. (2002). Graph models of wave propagation in thin structures. Waves in Random Media 12: 1–24

    Article  ADS  MathSciNet  Google Scholar 

  10. Kuchment P. (2004). Quantum graphs. I. Some basic structures. Waves in Random Media 14(1): 107–128

    Article  ADS  MathSciNet  Google Scholar 

  11. Kuchment P. (2005). Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A: Mathematical and General 38(22): 4887–4900

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Kuchment P. and Zeng H. (2001). Convergence of spectra of mesoscopic systems collapsing onto a graph. J. Math. Anal. Appl. 258: 671–700

    Article  MATH  MathSciNet  Google Scholar 

  13. Kuchment, P., Zeng, H.: Asymptotics of spectra of Neumann Laplacians in thin domains. In: Advances in Differential Equations and mathematical Physics, Yu. Karpeshina etc. (editors), Contemporary Mathematics 387, Providence, RI: Amer. Math. Sec., 2003 pp. 199–213

  14. Molchanov, S., Vainberg, B.: Transition from a network of thin fibers to quantum graph: an explicitly solvable model. Contemporary Mathematics 115, Providence, RI: Amer. Math. Sec., 2006 pp. 227–239

  15. Novikov, S.: Schrödinger operators on graphs and symplectic geometry. The Arnold fest., Fields Inst. Commun. 24, Providence, RI: Amer. Math. Sec., 1999 pp. 397–413

  16. Post O. (2005). Branched quantum wave guides with Dirichle BC: the decoupling case. J. Phys. A: Mathematical and General 38(22): 4917–4932

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Reed, M., Simon, B.: Methods of modern mathematical Physics, IV: Analysis of operators, New York: Acadamic Press, A Subsidiary of Harcourt Brace Jovnnovich, Publishers, 1978

  18. Rubinstein J. and Schatzman M. (2001). Variational problems on myltiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum. Arch. Ration. Mech. Anal. 160(4): 293–306

    Google Scholar 

  19. Vainberg B. (1968). On analytic properties of the resolvent for a class of sheaves of operators. Math. USSR-Sb 6: 241–273

    Article  Google Scholar 

  20. Vainberg B. (1975). On short wave asymptotic behavior of solutions of stationary problems and the asymptotic behavior as t → ∞ of solutions of non-stationary problems. Russ. Math. Surv. 30(2): 1–58

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dept. of Mathematics, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    S. Molchanov & B. Vainberg

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  1. S. Molchanov
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  2. B. Vainberg
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Correspondence to B. Vainberg.

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Communicated by B. Simon

The authors were supported partially by the NSF grant DMS-0405927.

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Molchanov, S., Vainberg, B. Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics. Commun. Math. Phys. 273, 533–559 (2007). https://doi.org/10.1007/s00220-007-0220-8

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  • DOI: https://doi.org/10.1007/s00220-007-0220-8

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