Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T13:19:54.662Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverse problems for Aharonov–Bohm rings

Published online by Cambridge University Press:  26 November 2009

P. KURASOV
P. KURASOV*
Affiliation:
Department of Mathematics, LTH, Lund University, Box 118, 221 00 Lund, Sweden Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden Department of Physics, St Petersburg University, 198904 St. Peterhof, Russia. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The inverse problem for Schrödinger operators on metric graphs is investigated in the presence of magnetic field. Graphs without loops and with Euler characteristic zero are considered. It is shown that the knowledge of the Titchmarsh–Weyl matrix function (Dirichlet-to-Neumann map) for just two values of the magnetic field allows one to reconstruct the graph and potential on it provided a certain additional no-resonance condition is satisfied.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 2 , March 2010 , pp. 331 - 362
Copyright
Copyright © Cambridge Philosophical Society 2009

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

REFERENCES

[1]Albeverio, S. and Kurasov, P.Singular perturbations of differential operators. Solvable Schrödinger type operators. London Mathematical Society Lecture Note Series 271 (Cambridge University Press, 2000).CrossRefGoogle Scholar
[2]Avdonin, S. and Kurasov, P.Inverse problems for quantum trees. Inverse Probl. Imaging 2 (2008), 1–21.CrossRefGoogle Scholar
[3]Avdonin, S., Mikhaylov, V. and Rybkin, A.The boundary control approach to the Titchmarsh–Weyl m-function. I. The response operator and A-amplitude. Comm. Math. Phys. 275 (2005), 791–803.CrossRefGoogle Scholar
[4]Avron, J. E., Raveh, A. and Zur, B.Adiabatic quantum transport in multiply connected systems. Rev. Modern Phys. 60 (1988), 873–915.CrossRefGoogle Scholar
[5]Avron, J. E. and Sadun, L.Chern numbers and adiabatic transport in networks with leads. Phys. Rev. Lett. 62 (1989), 3082–3084.CrossRefGoogle ScholarPubMed
[6]Avron, J. E. and Sadun, L.Adiabatic quantum transport in networks with macroscopic components. Ann. Phys. 206 (1991), 440–493.CrossRefGoogle Scholar
[7]Belishev, M. I.Boundary spectral inverse problem on a class of graphs (trees) by the BC method. Inverse Problems 20 (2004), 647–672.CrossRefGoogle Scholar
[8]Belishev, M. I.On the boundary controllability of a dynamical system described by the wave equation on a class of graphs (on trees). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 308 (2004), Mat. Vopr. Teor. Rasprostr. Voln. 33, 23–47, 252 (English trans: J. Math. Sci. (N. Y.) 132 (2006), 11–25).Google Scholar
[9]Belishev, M. I.Recent progress in the boundary control method. Inverse Problems 23 (2007), R1–R67.CrossRefGoogle Scholar
[10]Belishev, M. I. and Vakulenko, A. F.Inverse problems on graphs: recovering the tree of strings by the BC-method. J. Inverse Ill-Posed Probl. 14 (2006), 29–46.CrossRefGoogle Scholar
[11]von Below, J. Can one hear the shape of a network? Partial differential equations on multistructures (Luminy, 1999), 19–36, Lecture Notes in Pure and Appl. Math. 219 (Dekker, 2001).Google Scholar
[12]Berkolaiko, G., Carlson, R., Fulling, S. A. and Kuchment, P. (Editors). Quantum graphs and their applications. Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held in Snowbird, UT, June 19–23, 2005. Contemporary Mathematics 415 (American Mathematical Society, 2006).CrossRefGoogle Scholar
[13]Boman, J. and Kurasov, P.Symmetries of quantum graphs and the inverse scattering problem. Adv. Appl. Math. 35 (2005), 58–70.CrossRefGoogle Scholar
[14]Borg, G.Eine Umlehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Acta Math. 78 (1946), 1–96.CrossRefGoogle Scholar
[15]Borg, G. Uniqueness theorems in the spectral theory of y” + (λ − q(x))y = 0. Den 11te Skandinaviske Matematikerkongress (Trondheim), 1949, 276–287.Google Scholar
[16]Brown, B. M. and Weikard, R.A Borg-Levinson theorem for trees. Proc. Royal Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), 3231–3243.Google Scholar
[17]Brüning, J., Chernozatonskii, L. A., Demidov, V. V. and Geyler, V. A.Transport properties of two-arc Aharonov-Bohm interferometers with scattering centers. Russ. J. Math. Phys. 14 (2007), 417–422.CrossRefGoogle Scholar
[18]Bulgakov, E. N., Pichugin, K. N., Sadreev, A. F. and Rotter, I.Bound states in the continuum in open Aharonov-Bohm rings. JETP Letters, 84 (2006), 430–435.CrossRefGoogle Scholar
[19]Büttiker, M., Imry, Y. and Azbel, M. Ya.Quantum oscillations in one-dimensional normal-metal rings. Phys. Rev A 30 (1984), 1982–1989.CrossRefGoogle Scholar
[20]Büttiker, M., Imry, Y., Landauer, R. and Pinhas, S.Generalized many-channel conductance formula with application to small rings. Phys. Rev. B 31 (1985), 6207–6215.CrossRefGoogle ScholarPubMed
[21]Carlson, R.Inverse eigenvalue problems on directed graphs. Trans. Amer. Math. Soc. 351 (1999), 4069–4088.CrossRefGoogle Scholar
[22]Colin de Verdière, Y. Spectres de variétés riemanniennes et spectres de Graphes (French) [Spectra of Riemannian manifolds and spectra of graphs], Proc. ICM 1986, Vol. 1, 2, 522–530 (Amer. Math. Soc., 1987).Google Scholar
[23]Colin de Verdière, Y.Spectres de Graphes (Société Mathématiques de France, 1998).Google Scholar
[24]Exner, P. and Šeba, P.Free quantum motion on a branching graph. Rep. Math. Phys. 28 (1989), 7–26.CrossRefGoogle Scholar
[25]Friedlander, L.Genericity of simple eigenvalues for a metric graph. Israel J. Math. 146 (2005), 149–156.CrossRefGoogle Scholar
[26]Fulling, S. A., Kuchment, P. and Wilson, J. H.Index theorems for quantum graphs. J. Phys. A: Math. Theor. 40 (2007), 14165–14180.CrossRefGoogle Scholar
[27]Garnett, J. and Trubowitz, E.Gaps and bands of one-dimensional periodic Schrödinger operators. Comment. Math. Helv. 59 (1984), 258–312.CrossRefGoogle Scholar
[28]Garnett, J. and Trubowitz, E.Gaps and bands of one-dimensional periodic Schrödinger operators. II. Comment. Math. Helv. 62 (1987), 18–37.CrossRefGoogle Scholar
[29]Gerasimenko, N. I. and Pavlov, B. S.Scattering problems on noncompact graphs. Teoret. Mat. Fiz. 74 (1988), 345–359 (English trans: Theoret. and Math. Phys. 74 (1988), 230–240).Google Scholar
[30]Gerasimenko, N. I.Inverse scattering problem on a noncompact graph. Teoret. Mat. Fiz. 75 (1988), 187–200 (Englsih trans: Theoret. and Math. Phys. 75 (1988), 460–470).Google Scholar
[31]Gesztesy, F. and Simon, B.A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure. Ann. of Math. (2) 152 (2000), 593–643.CrossRefGoogle Scholar
[32]Gesztesy, F. and Tsekanovskii, E.On matrix-valued Herglotz functions. Math. Nachr. 218 (2000), 61–138.3.0.CO;2-D>CrossRefGoogle Scholar
[33]Gutkin, B. and Smilansky, U.Can one hear the shape of a graph? J. Phys. A 34 (2001), 6061–6068.CrossRefGoogle Scholar
[34]Habib, B., Tatic, E. and Shayegan, M. Strong Aharonov-Bohm oscillations in GaAs two dimensional holes. arXiv:cond-mat/0612638v2, 7 Mar 2007.CrossRefGoogle Scholar
[35]Harmer, M.Hermitian symplectic geometry and the factorization of the scattering matrix on graphs. J. Phys. A 33 (2000), 9015–9032.CrossRefGoogle Scholar
[36]Harmer, M.Inverse scattering for the matrix Schrödinger operator and Schrödinger operator on graphs with general self-adjoint boundary conditions. Kruskal, 2000 (Adelaide). ANZIAM J. 44 (2002), 161–168.CrossRefGoogle Scholar
[37]Harmer, M.Inverse scattering on matrices with boundary conditions. J. Phys. A 38 (2005), 4875–4885.CrossRefGoogle Scholar
[38]Its, A. R. and Matveev, V. B.Hill operators with a finite number of lacunae (Russian). Funksional. Anal. i Priložen. 9 (1975), 69–70.Google Scholar
[39]Its, A. R. and Matveev, V. B.Schrödinger operators with the finite-band spectrum and the N-soliton solutions of the Korteweg-de Vries equation (Russian). Teoret. Mat. Fiz. 23 (1975), 51–68.Google Scholar
[40]Kostrykin, V. and Schrader, R.Kirchoff's rule for quantum wires. J. Phys. A 32 (1999), 595–630.CrossRefGoogle Scholar
[41]Kostrykin, V. and Schrader, R.Kirchhoff's rule for quantum wires. II. The inverse problem with possible applications to quantum computers. Fortschr. Phys. 48 (2000), 703–716.3.0.CO;2-O>CrossRefGoogle Scholar
[42]Kostrykin, V. and Schrader, R.Quantum wires with magnetic fluxes. Comm. Math. Phys. 237 (2003), 161–179.CrossRefGoogle Scholar
[43]Kostrykin, V. and Schrader, R. The inverse scattering problem for metric graphs and the travelling salesman problem. arXiv:math-ph/0603010.Google Scholar
[44]Kottos, T. and Smilansky, U.Periodic orbit theory and spectral statistics for quantum graphs. Ann. Physics 274 (1999), 76–124.CrossRefGoogle Scholar
[45]Kuchment, P.Quantum graphs. I. Some basic structures. Special section on quantum graphs. Waves Random Media 14 (2004), S107–S128.CrossRefGoogle Scholar
[46]Kuchment, P.Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A 38 (2005), 4887–4900.CrossRefGoogle Scholar
[47]Kurasov, P.Graph Laplacians and topology. Arkiv för Matematik 46 (2008), 95–111.CrossRefGoogle Scholar
[48]Kurasov, P.Schrödinger operators on graphs and geometry. I. Essentially bounded potentials. J. Funct. Anal. 254 (2008), 934–953.CrossRefGoogle Scholar
[49]Kurasov, P. and Nowaczyk, M.Inverse spectral problem for quantum graphs. J. Phys. A 38 (2005), 4901–4915.CrossRefGoogle Scholar
[50]Kurasov, P. and Stenberg, F.On the inverse scattering problem on branching graphs. J. Phys. A 35 (2002) 101–121.CrossRefGoogle Scholar
[51]Levinson, N.On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase. Danske Vid. Selsk. Mat.-Fys. Medd. 25 (1949), no. 9, 29 pp.Google Scholar
[52]Levitan, B. M. and Gasymov, M. G.Determination of a differential equation by two spectra (Russian). Uspehi Mat. Nauk 19 (1964), 3–63.Google Scholar
[53]Lyubarskiĭ, Yu. I. and Marchenko, V. A.Direct and inverse problems of multichannel scattering (Russian). Funktsional. Anal. i Prilozhen. 41 (2007), 58–77.CrossRefGoogle Scholar
[54]Marchenko, V. A.Some questions of the theory of one-dimensional linear differential operators of the second order. I (Russian). Trudy Moskov. Mat. Obšč. 1 (1952), 327–420.Google Scholar
[55]Marchenko, V. A.Sturm–Liouville operators and applications. Oper. Theory Adv. Appl. 22 (1986).Google Scholar
[56]Marchenko, V. A. and Ostrovsky, I. V.A characterization of the spectrum of the Hill operator (Russian). Mat. Sb. (N.S.) 97 (139) (1975), 540–606.Google Scholar
[57]Marchenko, V. A. and Ostrovsky, I. V.Approximations of Periodic by Finite-zone potentials. Selecta Matematica Sovietica 6 (1987), 101–136 (Originally published: Vestnik Khar'kovsk. Univ., no. 205; Prikl. Mat. i Mekh. 45 (1980), 4–40).Google Scholar
[58]Novikov, S.Discrete Schrödinger operators and topology. Asian J. Math. 2 (1998), 921–934.CrossRefGoogle Scholar
[59]Nowaczyk, M.Inverse spectral problems for quantum graphs with rationally dependent edges. Oper. Theory Adv. Appl. 174 (2007), 105–116.Google Scholar
[60]Nowaczyk, M. Inverse Spectral Problems for Quantum Graphs, Licentiate Thesis (Lund University, 2005).Google Scholar
[61]Nowaczyk, M. Inverse Problems for Graph Laplacians, PhD Thesis (Lund University, 2008).Google Scholar
[62]Pivovarchik, V.Scattering in a loop-shaped waveguide. Oper. Theory Adv. Appl. 124 (2001), 527–543.Google Scholar
[63]Pöschel, J. and Trubowitz, E.Inverse Spectral Theory (Academic Press, 1987).Google Scholar
[64]Pokornyi, Yu. V., Penkin, O. M., Pryadiev, V. L., Borovskikh, A.V., Lazarev, K. P. and Shabrov, S. A.Differential equations on geometric graphs (in Russian) (Fiziko-Matematicheskaya Literatura, Moscow, 2005).Google Scholar
[65]Ramm, A. and Simon, B.A new approach to inverse spectral theory. III. Short-range potentials. J. Anal. Math. 80 (2000), 319–334.CrossRefGoogle Scholar
[66]Reed, M. and Simon, B.Methods of Modern Mathematical Physics. I-IV (Academic Press, 1972–1979).Google Scholar
[67]Roth, J.-P. Le spectre du laplacien sur un graphe (French) [The spectrum of the Laplacian on a graph]. Théorie du potentiel (Orsay, 1983), 521–539, Lecture Notes in Math. 1096 (Springer, 1984).CrossRefGoogle Scholar
[68]Russo, S., Oostinga, J. B., Wehenkel, D., Heersche, H. B., Shams Sobhani, Samira, Vandersypen, L. M. K. and Morpurgo, A. F.Observation of Aharonov–Bohm conductance oscillations in a graphene ring. Phys. Rev B 77 (2008), 085413.CrossRefGoogle Scholar
[69]Shelykh, I. A., Galkin, N. G. and Bagraev, N. T.Conductance of a gated Aharonov-Bohm ring touching a quantum wire. Phys. Rev. B 74 (2006), 165331.CrossRefGoogle Scholar
[70]Simon, B.A new approach to inverse spectral theory. I Fundamental formalism. Ann. of Math. 150 (1999), 1029–1057.CrossRefGoogle Scholar
[71]Texier, Ch. and Büttiker, M.Local Friedel sum rule on graphs. Phys. Rev. B 67 (2003), 245410.CrossRefGoogle Scholar
[72]Tokuno, A., Ashikawa, M. and Demler, E. Dynamics of of one-dimensional Bose liquids: Andreev-like reflection at Y-junctions and absence of the Aharonov–Bohm effect. ArXiv:cond.mat/0703610v3, 14 Apr 2008.Google Scholar
[73]Yeyati, A. L. and Büttiker, M.Aharonov-Bohm oscillations in a mesoscopic ring with a quantum dot. Phys. Rev. B 52 (1995), 14360–14363.CrossRefGoogle Scholar
[74]Yurko, V.On the reconstruction of Sturm-Liouville operators on graphs (Russian). Mat. Zametki 79 (2006), 619–630 (English trans: Math. Notes 79 (2006), 572–582).Google Scholar
[75]Yurko, V.Inverse spectral problems for Sturm-Liouville operators on graphs. Inverse Problems 21 (2005), 1075–1086.CrossRefGoogle Scholar
[76]Yurko, V.Recovering differential pencils on compact graphs. J. Differential Equations 244 (2008), 431–443.CrossRefGoogle Scholar