Inverse Problems for Quantum Graphs: Recent Developments and Perspectives

• Authors: P. Kurasov
• Languages of publication: EN

Abstracts

EN

An introduction into the area of inverse problems for the Schrödinger operators on metric graphs is given. The case of metric finite trees is treated in detail with the focus on matching conditions. For graphs with loops we show that for almost all matching conditions the potential on the loop is not determined uniquely by the Titchmarsh-Weyl function. The class of all admissible potentials is characterized.

Keywords

EN

03.65.Nk; 73.63.-b; 85.35.-p

Discipline

• 73.63.-b: Electronic transport in nanoscale materials and structures(see also 73.23.-b Electronic transport in mesoscopic systems)
• 03.65.Nk: Scattering theory
• 85.35.-p: Nanoelectronic devices

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2011
• Volume: 120
• Issue: 6A
• Pages: A-132-A-141

Dates

published

2011-12

Contributors

author

P. Kurasov

• Dept. of Mathematics, LTH, Lund Univ., Box 118, 221 00 Lund, Sweden
• Dept. of Mathematics, Stockholm Univ., 106 91, Stockholm, Sweden
• Dept. of Physics, St. Petersburg Univ., 198904 St. Peterhof, Russia

References

• [1] S. Avdonin, P. Kurasov, Inverse Probl. Imaging 2, 1 (2008)
• [2] S. Avdonin, P. Kurasov, M. Nowaczyk, Inverse Probl. Imaging 4, 579 (2010)
• [3] S. Avdonin, P. Kurasov, Inverse problems for quantum trees III: recovering metric graph and complete solution of the problem
• [4] S. Avdonin, V. Mikhaylov, A. Rybkin, Commun. Math. Phys. 275, 791 (2005)
• [5] M.I. Belishev, Inverse Problems 20, 647 (2004)
• [6] M.I. Belishev, J. Math. Sci. (New York) 132, 11 (2006)
• [7] M.I. Belishev, A.F. Vakulenko, J. Inverse Ill-Posed Probl. 14, 29 (2006)
• [8] M.I. Belishev, Inverse Problems 23, R1 (2007)
• [9] J. Boman, P. Kurasov, Adv. Appl. Math. 35, 58 (2005)
• [10] B.M. Brown, R. Weikard, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461, 3231 (2005)
• [11] L.D. Faddeev, Usp. Mat. Nauk 14, 57119 (1959)
• [12] G. Freiling, V. Yurko, Appl. Anal. 86, 653 (2007)
• [13] J. Garnett, E. Trubowitz, Comment. Math. Helv. 59, 258 (1984)
• [14] J. Garnett, E. Trubowitz, Comment. Math. Helv. 62, 18 (1987)
• [15] I.M. Gelfand, B.M. Levitan, Izv. Akad. Nauk SSSR, Ser. Mat. 15, 309360 (1951)
• [16] N.I. Gerasimenko, B.S. Pavlov, Theoret. Math. Phys. 74, 230 (1988)
• [17] N.I. Gerasimenko, Theoret. Math. Phys. 75, 460 (1988)
• [18] B. Gutkin, U. Smilansky, J. Phys. A 34, 6061 (2001)
• [19] P. Kurasov, M. Nowaczyk, J. Phys. A 38, 4901 (2005)
• [20] P. Kurasov, F. Stenberg, J. Phys. A 35, 101 (2002)
• [21] P. Kurasov, Ark. Matemat. 46, 95 (2008)
• [22] P. Kurasov, J. Funct. Anal. 254, 934 (2008)
• [23] P. Kurasov, Math. Proc. Cambridge Philos. Soc. 148, 331 (2010)
• [24] P. Kurasov, Acta Phys. Pol. A 116, 765 (2009)
• [25] P. Kurasov, 'Can one distinguish quantum trees from the boundary?', accepted for publication in Proc. AMS
• [26] P. Kurasov, 'How does magnetic field held to solve the inverse problem for quantum graphs?', in preparation
• [27] P. Kurasov, M. Enerbäck, 'Aharonov-Bohm ring touching a quantum wire: how to model it and to solve the inverse problem', accepted for publication in Rep. Math. Phys
• [28] B.M. Levitan, M.G. Gasymov, Usp. Mat. Nauk 19, 3 (1964)
• [29] Yu.I. Lyubarskiĭ, V.A. Marchenko, Funktsional. Anal. Prilozhen. 41, 58 (2007)
• [30] H.P. McKean, E. Trubowitz, Commun. Pure Appl. Math. 29, 143 (1976)
• [31] V.A. Marchenko, Usp. Mat. Nauk (N.S.) 12, 143145 (1957)
• [32] V.A. Marchenko, I.V. Ostrovsky, Mat. Sb. (N.S.) 97, 540 (1975)
• [33] V.A. Marchenko, I.V. Ostrovsky, Selecta Matemat. Soviet. 6, 101 (1987)
• [34] V. Pivovarchik, Oper. Theory Adv. Appl. 124, 527 (2001)
• [35] R. Rueckriemen, 'Recovering quantum graphs from their Bloch spectrum', preprint arXiv:1101.6002v2, 10 Feb 2011
• [36] A. Rybkin, Inverse Probl. Imaging 3, 139 (2009)
• [37] V. Yurko, Math. Notes 79, 572 (2006)
• [38] V. Yurko, Inverse Problems 21, 1075 (2005)
• [39] V. Yurko, Operators Matrices 2, 543 (2008)
• [40] V. Yurko, Matem. Sbornik 200, 147 (2009); translation in Sbornik: Mathematics 200, 1403 (2009)
• [41] V. Yurko, Inverse Problems 25, 105008 (2009)
• [42] V. Yurko, Applied Math. Lett. 23, 875 (2010)
• [43] V. Yurko, J. Inverse Ill-Posed Problems 18, 245 (2010)

Identifiers

DOI

10.12693/APhysPolA.120.A-132