A Solvable Model for Scattering on a Junction and a Modified Analytic Perturbation Procedure

  • B. Pavlov
Part of the Operator Theory: Advances and Applications book series (OT, volume 197)


We consider a one-body spin-less electron spectral problem for a resonance scattering system constructed of a quantum well weakly connected to a noncompact exterior reservoir, where the electron is free. The simplest kind of the resonance scattering system is a quantum network, with the reservoir composed of few disjoint cylindrical quantum wires, and the Schrödinger equation on the network, with the real bounded potential on the wells and constant potential on the wires. We propose a Dirichlet-to-Neumann-based analysis to reveal the resonance nature of conductance across the star-shaped element of the network (a junction), derive an approximate formula for the scattering matrix of the junction, construct a fitted zero-range solvable model of the junction and interpret a phenomenological parameter arising in Datta- Das Sarma boundary condition, see [14], for T-junctions. We also propose using of the fitted zero-range solvable model as the first step in a modified analytic perturbation procedure of calculation of the corresponding scattering matrix.

Mathematics Subject Classification (2000)

Primary 47A40 47A48 47A55 Secondary 47N50 47N70 35Q40 


Junction Fitted zero-range model Dirichlet-to-Neumann map 


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  1. [1]
    V. Adamjan, D. Arov, On a class of scattering operators and characteristic operator-functions of contractions. (Russian) Dokl. Akad. Nauk SSSR 160 (1965) pp. 9–12.MathSciNetGoogle Scholar
  2. [2]
    V. Adamyan, B. Pavlov, Local Scattering problem and a Solvable model of Quantum Network. In: Operator Theory: Advances and Applications, Vol. 198, Birkhäuser Verlag, Basel/Switzerland, (2009) pp. 1–10.Google Scholar
  3. [3]
    V. Adamyan, B. Pavlov, A. Yafyasov, Modified Krein Formula and analytic perturbation procedure for scattering on arbitrary junction. International Newton Institute report series NI07016, Cambridge, 18 April 2007, 33p. To be published in the proceedings of M.G. Krein memorial conference, Odessa, April 2007.Google Scholar
  4. [4]
    N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space, (Frederick Ungar, Publ., New York, vol. 1, 1966) (translated from Russian by M. Nestel).Google Scholar
  5. [5]
    S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, London Math. Society Lecture Note Series 271. Cambridge University Press (2000).Google Scholar
  6. [6]
    S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable models in quantum mechanics. Springer-Verlag, New York, 1988.zbMATHGoogle Scholar
  7. [7]
    N. Bagraev, A. Mikhailova, B. Pavlov, L. Prokhorov, A. Yafyasov, Parameter regime of a resonance quantum switch. In: Phys. Rev. B, 71, 165308 (2005), pp. 1–16.Google Scholar
  8. [8]
    F.A. Berezin, L.D. Faddeev, A remark on Schrödinger equation with a singular potential Dokl. AN SSSR, 137 (1961) pp. 1011–1014.MathSciNetGoogle Scholar
  9. [9]
    M. Birman, A local test for the existence of wave operators. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 32 1968 914–942.zbMATHMathSciNetGoogle Scholar
  10. [10]
    V. Bogevolnov, A. Mikhailova, B. Pavlov, A. Yafyasov, About Scattering on the Ring In: “Operator Theory: Advances and Applications”, Vol. 124 (Israel Gohberg Anniversary Volume), Ed. A. Dijksma, A.M. Kaashoek, A.C.M. Ran, Birkhäuser, Basel (2001) pp. 155–187.Google Scholar
  11. [11]
    J. Bruening, B. Pavlov, On calculation of Kirchhoff constants of Helmholtz resonator. International Newton Institute report series NI07060-AGA, Cambridge, 04 September 2007, 40p.Google Scholar
  12. [12]
    R. Courant, D. Hilbert, Methods of mathematical physics. Vol. II. Partial differential equations. Reprint of the 1962 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York (1989). xxii+830 pp.Google Scholar
  13. [13]
    S. Datta, Electronic Transport in Mesoscopic systems. Cambridge University Press, Cambridge (1995).Google Scholar
  14. [14]
    S. Datta and B. Das Sarma, Electronic analog of the electro-optic modulator. Appl. Phys. Lett. 56, 7 (1990) pp. 665–667.CrossRefGoogle Scholar
  15. [15]
    Yu.N. Demkov, V.N. Ostrovskij, Zero-range potentials and their applications in Atomic Physics, Plenum Press, NY-London, (1988).Google Scholar
  16. [16]
    P. Exner, P. Seba, A new type of quantum interference transistor. Phys. Lett. A 129:8, 9, 477 (1988).CrossRefMathSciNetGoogle Scholar
  17. [17]
    P. Exner, O. Post, Convergence of graph-like thin manifolds J. Geom. Phys. 54,1, (2005) pp. 77–115.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    M. Faddeev, B. Pavlov, Scattering by resonator with the small opening. Proc. LOMI, v126 (1983). (English Translation J. of Sov. Math. v. 27, 2527 (1984).Google Scholar
  19. [19]
    E. Fermi, Sul motto dei neutroni nelle sostance idrogenate (in Italian) Richerca Scientifica 7 p. 13 (1936).Google Scholar
  20. [20]
    F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential. I. The case of an a.c. component in the spectrum. Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part II (Zürich, 1995). Helv. Phys. Acta 70, no. 1–2, (1997) pp 66–71.zbMATHMathSciNetGoogle Scholar
  21. [21]
    F. Gesztezy, Y. Latushkin, M. Mitrea and M. Zinchenko, Non-selfadjoint operators, infinite determinants and some applications, Russian Journal of Mathematical Physics, 12, 443–71 (2005).MathSciNetGoogle Scholar
  22. [22]
    F. Gesztezy, M. Mitrea and M. Zinchenko, On Dirichlet-to-Neumann maps and some applications to modified Fredholm determinants preprint, (2006).Google Scholar
  23. [23]
    V.I. Gorbachuk, M.L. Gorbachuk, Boundary value problems for operator differential equations. Translated and revised from the 1984 Russian original. Mathematics and its Applications (Soviet Series), 48. Kluwer Academic Publishers Group, Dordrecht, 1991. xii+347Google Scholar
  24. [24]
    D. Gramotnev, D. Pile, Double resonant extremely asymmetrical scattering of electromagnetic waves in non-uniform periodic arrays In: Opt. Quant. Electronics., 32, (2000) pp. 1097–1124.CrossRefGoogle Scholar
  25. [25]
    D. Grieser, Spectra of graph neighborhoods and scattering. Proc. Lond. Math. Soc. (3) 97, no. 3 (2008) pp. 718–752.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    M. Harmer, Hermitian symplectic geometry and extension theory. Journal of Physics A: Mathematical and General, 33 (2000) pp. 9193–9203.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    M. Harmer, Fitting parameters for a Solvable Model of a Quantum Network The University of Auckland, Department of Mathematics report series 514 (2004), 8 p.Google Scholar
  28. [28]
    M. Harmer, B. Pavlov, A. Yafyasov, Boundary condition at the junction, in: Journal of Computational Electronics, 6 (2007) pp. 153–157.CrossRefGoogle Scholar
  29. [29]
    T. Kato, Perturbation theory for linear operators Springer Verlag, Berlin-Heidelberg-NY, second edition (1976).zbMATHGoogle Scholar
  30. [30]
    J.P. Keating, J. Marlof, B. Winn, Value distribution of the eigenfunctions and spectral determinants of quantum star-graphs Communication of Mathematical Physics, 241, 2–3 (2003) pp. 421–452.zbMATHGoogle Scholar
  31. [31]
    J.P. Keating, B. Winn, No quantum ergodicity for star graphs Communication of Mathematical Physics, 250, 2 (2004) pp. 219–285.MathSciNetGoogle Scholar
  32. [32]
    G.R. Kirchhoff. Gesammelte Abhandlungen Publ. Leipzig: Barth, 1882, 641p.Google Scholar
  33. [33]
    J. Brüning, B. Pavlov, On calculation of Kirchhoff constants for Helmholtz resonator International Newton Institute, report series NI07060-AGA, Cambridge, 04 September, 2007, 38 p.Google Scholar
  34. [34]
    V. Kostrykin and R. Schrader, Kirchhoff’s rule for quantum wires. J. Phys. A: Math. Gen. 32, 595 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    M.A. Krasnosel’skij, On selfadjoint extensions of Hermitian Operators (in Russian) Ukrainskij Mat. Journal 1, 21 (1949).Google Scholar
  36. [36]
    M.G. Krein, Concerning the resolvents of an Hermitian operator with deficiency index (m,m), Doklady AN USSR, 52 (1946) pp. 651–654.MathSciNetGoogle Scholar
  37. [37]
    P. Kuchment, H. Zeng, Convergence of Spectra of mesoscopic Systems Collapsing onto Graph Journal of Mathematical Analysis and Applications, 258 (2001) pp. 671–700.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    P. Kuchment, Graph models for waves in thin structures Waves in Periodic and Random Media, 12, 1 (2002) R 1–R 24.MathSciNetGoogle Scholar
  39. [39]
    S. Lall, P. Krysl, J. Marsden, Structure-preserving model reduction for mechanical systems In: Complexity and nonlinearity in physical systems (Tucson, AZ, 2001), Phys. D 184, 1–4 (2003) pp. 304–318.zbMATHMathSciNetGoogle Scholar
  40. [40]
    Lax, Peter D., Phillips, Ralph, S. Scattering theory. Second edition. With appendices by Cathleen S. Morawetz and Georg Schmidt. Pure and Applied Mathematics, 26. Academic Press, Inc., Boston, MA, (1989) xii+309 pp.Google Scholar
  41. [41]
    M.S. Livshits, Method of nonselfadjoint operators in the theory of wave guides In: Radio Engineering and Electronic Physics. Publ. by American Institute of Electrical Engineers, 1 (1962) pp. 260–275.Google Scholar
  42. [42]
    O. Madelung, Introduction to solid-state theory. Translated from German by B.C. Taylor. Springer Series in Solid-State Sciences, 2. Springer-Verlag, Berlin, New York (1978).Google Scholar
  43. [43]
    A. Mikhailova, B. Pavlov, L. Prokhorov, Modeling of quantum networks’ arXiv mathph: 031238, 2004, 69 p.Google Scholar
  44. [44]
    A. Mikhailova, B. Pavlov, L. Prokhorov, Intermediate Hamiltonian via Glazman splitting and analytic perturbation for meromorphic matrix-functions. In: Mathematische Nachrichten, 280, 12, (2007) pp. 1376–1416.zbMATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    A. Mikhailova, B. Pavlov, Remark on the compensation of singularities in Krein’s formula In: Operator Theory: Advances and Applications Vol. 186, Proceedings of OTAMP06, Lund. Editors: S. Naboko, P. Kurasov. pp. 325–337.Google Scholar
  46. [46]
    R. Mittra, S. Lee, Analytical techniques in the theory of guided waves The Macmillan Company, NY, Collier-Macmillan Limited, London, 1971.zbMATHGoogle Scholar
  47. [47]
    L. Ko, R. Mittra, A new approach based on a combination of integral equation and asymptotic techniques for solving electromagnetic scattering problems IEEE Trans. Antennas and Propagation AP-25, no. 2, (1977) pp. 187–197.CrossRefMathSciNetGoogle Scholar
  48. [48]
    J. von Neumann, Mathematical foundations of quantum mechanics Twelfth printing. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, (1996).zbMATHGoogle Scholar
  49. [49]
    R.G. Newton, Scattering theory of waves and particlesNewton, Reprint of the 1982 second edition [Springer, New York; MR0666397 (84f:81001)], with list of errata prepared for this edition by the author. Dover Publications, Inc., Mineola, NY, 2002.Google Scholar
  50. [50]
    N. Nikol’skii, S. Khrushchev, A functional model and some problems of the spectral theory of functions (Russian) Translated in Proc. Steklov Inst. Math. 1988, no. 3, 101–214. Mathematical physics and complex analysis (Russian). Trudy Mat. Inst. Steklov. 176 (1987), 97–210, 327.Google Scholar
  51. [51]
    B. Sz.-Nagy, C., Foias, Harmonic analysis of operators on Hilbert space. Translated from the French and revised North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest 1970 xiii+389 pp.Google Scholar
  52. [52]
    S. Novikov, Schrödinger operators on graphs and symplectic geometry In: The Arnoldfest (Toronto, ON, 1997), Ed.: Fields Inst. Commun., 24, Amer. Math. Soc., Providence, RI, (1999) pp. 397–413.Google Scholar
  53. [53]
    B. Pavlov, On one-dimensional scattering of plane waves on an arbitrary potential, Teor. i Mat. Fiz., v. 16, N1, 1973, pp. 105–119.zbMATHGoogle Scholar
  54. [54]
    B. Pavlov, The theory of extensions and explicitly solvable models (in Russian) Uspekhi Mat. Nauk, 42, (1987) pp. 99–131.zbMATHMathSciNetGoogle Scholar
  55. [55]
    B. Pavlov, S-Matrix and Dirichlet-to-Neumann Operators In: Encyclopedia of Scattering, ed. R. Pike, P. Sabatier, Academic Press, Harcourt Science and Tech. Company (2001) pp. 1678–1688.Google Scholar
  56. [56]
    B. Pavlov, I. Antoniou, Jump-start in analytic perturbation procedure for Friedrichs model. In J. Phys. A: Math. Gen. 38 (2005) pp. 4811–4823.zbMATHCrossRefMathSciNetGoogle Scholar
  57. [57]
    B. Pavlov, V. Kruglov, Operator Extension technique for resonance scattering of neutrons by nuclei. In: Hadronic Journal 28 (2005) pp. 259–268.zbMATHGoogle Scholar
  58. [58]
    B. Pavlov, V. Kruglov, Symplectic operator-extension technique and zero-range quantum models In: New Zealand mathematical Journal 34,2 (2005) pp. 125–142.zbMATHMathSciNetGoogle Scholar
  59. [59]
    B. Pavlov, A. Yafyasov, Standing waves and resonance transport mechanism in quantum networks With A. Yafyasov, Surface Science 601 (2007), pp. 2712–2716.CrossRefGoogle Scholar
  60. [60]
    B. Pavlov, A star-graph model via operator extension Mathematical Proceedings of the Cambridge Philosophical Society, Volume 142, Issue 02, March 2007, pp. 365–384.zbMATHCrossRefMathSciNetGoogle Scholar
  61. [61]
    B. Pavlov, T. Rudakova, V. Ryzhii, I. Semenikhin, Plasma waves in two-dimensional electron channels: propagation and trapped modes. Russian Journal of mathematical Physics, 14, 4 (2004) pp. 465–487.CrossRefMathSciNetGoogle Scholar
  62. [62]
    L. Petrova, B. Pavlov, Tectonic plate under a localized boundary stress: fitting of a zero-range solvable model. Journal of Physics A, 41 (2008) 085206 (15 pp.).CrossRefMathSciNetGoogle Scholar
  63. [63]
    H. Poincaré, Methodes nouvelles de la mécanique celeste Vol. 1 (1892), Second edition: Dover, New York (1957).Google Scholar
  64. [64]
    C. Presilla, J. Sjostrand, Transport properties in resonance tunnelling heterostructures In: J. Math. Phys. 37, 10 (1996), pp. 4816–4844.zbMATHCrossRefMathSciNetGoogle Scholar
  65. [65]
    I. Prigogine, Irreversibility as a Symmetry-breaking Process In: Nature, 246, 9 (1973).CrossRefGoogle Scholar
  66. [66]
    Lord Rayleigh, The theory of Helmholtz resonator Proc. Royal Soc. London 92 (1916) pp. 265–275.CrossRefGoogle Scholar
  67. [67]
    J. Rubinstein, M. Shatzman, Variational approach on multiply connected thin strips I: Basic estimates and convergence of the Laplacian spectrum Arch. Ration. Mech. Analysis 160, 4, 271 (2001).zbMATHCrossRefGoogle Scholar
  68. [68]
    M. Schatzman, On the eigenvalues of the Laplace operator on a thin set with Neumann boundary conditions Applicable Anal. 61, 293 (1996).CrossRefMathSciNetGoogle Scholar
  69. [69]
    I.A. Shelykh, N.G. Galkin, and N.T. Bagraev, Quantum splitter controlled by Rashba spin-orbit coupling. Phys. Rev.B 72, 235316 (2005).CrossRefGoogle Scholar
  70. [70]
    J.H. Schenker, M. Aizenman, The creation of spectral gaps by graph decoration Letters of Mathematical Physics, 53, 3, (2000) pp. 253–262.zbMATHCrossRefMathSciNetGoogle Scholar
  71. [71]
    J. Shirokov, Strongly singular potentials in three-dimensional Quantum Mechanics (In Russian) Teor. Mat. Fiz. 42 1 (1980) pp. 45–49.MathSciNetGoogle Scholar
  72. [72]
    J. Splettstoesser, M. Governale, and U. Zülicke, Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling. Phys. Rev. B, 68:165341, (2003).CrossRefGoogle Scholar
  73. [73]
    P. Streda, P. Seba, Antisymmetric spin filtering in one-dimensional electron systems via uniform spin-orbit coupling Phys. Rev. Letters 90, 256601 (2003).CrossRefGoogle Scholar
  74. [74]
    J. Sylvester, G. Uhlmann, The Dirichlet to Neumann map and applications. In: Proceedings of the Conference “Inverse problems in partial differential equations (Arcata, 1989)”, SIAM, Philadelphia, 101 (1990).Google Scholar
  75. [75]
    A. Wentzel, M. Freidlin, Reaction-diffusion equations with randomly perturbed boundary conditions Annals of Probability 20, 2 (1992) pp. 963–986.CrossRefMathSciNetGoogle Scholar
  76. [76]
    E.P. Wigner, On a class of analytic functions from the quantum theory of collisions Annals of mathematics, 2, N. 53, 36 (1951).CrossRefMathSciNetGoogle Scholar
  77. [77]
    H.Q. Xu, Diode and transistor behaviour of three-terminal ballistic junctions Applied Phys. Letters 80, 853 (2002).CrossRefGoogle Scholar

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© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • B. Pavlov
    • 1
  1. 1.V.A. Fock Institute for Physics of St.-Petersburg UniversityPetrodvoretsRussia

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