American Institute of Mathematical Sciences

Advanced
November  2010, 4(4): 579-598. doi: 10.3934/ipi.2010.4.579

Inverse problems for quantum trees II: Recovering matching conditions for star graphs

Sergei Avdonin 1, , Pavel Kurasov 2, and  Marlena Nowaczyk 3,

1. 

Department of Mathematics and Statistics, University of Alaska, Fairbanks, AK 99775-6660
2. 

Dept. of Mathematics, LTH, Lund Univ., Box 118, 221 00 Lund
3. 

Institute of Mathematics, PAN, ul. Św.Tomasza 30, 31-027 Kraków, Poland

Received  November 2009 Revised  May 2010 Published  September 2010

The inverse problem for the Schrödinger operator on a star graph is investigated. It is proven that such Schrödinger operator, i.e. the graph, the real potential on it and the matching conditions at the central vertex, can be reconstructed from the Titchmarsh-Weyl matrix function associated with the graph boundary. The reconstruction is also unique if the spectral data include not the whole Titchmarsh-Weyl function but its principal block (the matrix reduced by one dimension). The same result holds true if instead of the Titchmarsh-Weyl function the dynamical response operator or just its principal block is known.
Keywords: matching conditions., quantum graphs, inverse problems.
Mathematics Subject Classification: Primary: 81C05, 35R30, 35L05, 93B05, 49E1.
Citation: Sergei Avdonin, Pavel Kurasov, Marlena Nowaczyk. Inverse problems for quantum trees II: Recovering matching conditions for star graphs. Inverse Problems & Imaging, 2010, 4 (4) : 579-598. doi: 10.3934/ipi.2010.4.579
References:
[1]

S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Problems and Imaging, 2 (2008), 1-21.  Google Scholar

[2]

S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings, Zeit. Angew. Math. Mech., 90 (2010), 136-150. doi: doi:10.1002/zamm.200900295.  Google Scholar

[3]

S. Avdonin, V. Mikhaylov and A. Rybkin, The boundary control approach to the Titchmarsh-Weyl $m$-function. I. The response operator and the $A$-amplitude, Comm. Math. Phys., 275 (2007), 791-803. doi: doi:10.1007/s00220-007-0315-2.  Google Scholar

[4]

M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672. doi: doi:10.1088/0266-5611/20/3/002.  Google Scholar

[5]

M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1-R67. doi: doi:10.1088/0266-5611/23/5/R01.  Google Scholar

[6]

M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inv. Ill-Posed Problems, 14 (2006), 29-46. doi: doi:10.1515/156939406776237474.  Google Scholar

[7]

B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3231-3243.  Google Scholar

[8]

B. M. Brown and R. Weikard, On inverse problems for finite trees, in "Methods of Spectral Analysis in Mathematical Physics, Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2006," 31-48. Google Scholar

[9]

R. Carlson, Inverse eigenvalue problems on directed graphs, Trans. Amer. Math. Soc., 351 (1999), 4069-4088. doi: doi:10.1090/S0002-9947-99-02175-3.  Google Scholar

[10]

P. Exner and P. Šeba, Free quantum motion on a branching graph, Rep. Math. Phys., 28 (1989), 7-26. doi: doi:10.1016/0034-4877(89)90023-2.  Google Scholar

[11]

G. Freiling and V. Yurko, Inverse problems for Sturm-Liouville operators on noncompact trees, Results Math., 50 (2007), 195-212. doi: doi:10.1007/s00025-007-0246-4.  Google Scholar

[12]

G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions, Applicable Analysis, 86 (2007), 653-667. doi: doi:10.1080/00036810701303976.  Google Scholar

[13]

N. I. Gerasimenko and B. Pavlov, Scattering problems on noncompact graphs, Teoret. Mat. Fiz., 74 (1988), 345-359; Eng. transl. in Theoret. and Math. Phys., 74 (1988), 230-240.  Google Scholar

[14]

N. I. Gerasimenko, Inverse scattering problem on a noncompact graph, Teoret. Mat. Fiz., 75 (1988), 187-200; Eng. transl. in Theoret. and Math. Phys., 75 (1988), 460-470.  Google Scholar

[15]

M. Harmer, Hermitian symplectic geometry and extension theory, J. Phys. A, 33 (2000), 9193-9203. doi: doi:10.1088/0305-4470/33/50/305.  Google Scholar

[16]

M. Harmer, Inverse scattering on matrices with boundary conditions, J. Phys. A, 38 (2005), 4875-4885. doi: doi:10.1088/0305-4470/38/22/012.  Google Scholar

[17]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630. doi: doi:10.1088/0305-4470/32/4/006.  Google Scholar

[18]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires. II. The inverse problem with possible applications to quantum computers, Fortschr. Phys., 48 (2000), 703-716. doi: doi:10.1002/1521-3978(200008)48:8<703::AID-PROP703>3.0.CO;2-O.  Google Scholar

[19]

P. Kuchment, "Waves in Periodic and Random Media. Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference Held at Mount Holyoke College, South Hadley, MA, June 22-28, 2002,'' Contemporary Mathematics, 339 (2003), (Providence, RI: American Mathematical Society).  Google Scholar

[20]

P. Kuchment, Quantum graphs. I. Some basic structures, Waves in Random Media, 14 (2004), S107-S128. doi: doi:10.1088/0959-7174/14/1/014.  Google Scholar

[21]

P. Kurasov and M. Nowaczyk, Geometric properties of quantum graphs and vertex scattering matrices, Opuscula Math., 30 (2010), 295-309. Google Scholar

[22]

P. Kurasov and F. Stenberg, On the inverse scattering problem on branching graphs, J. Phys. A, 35 (2002), 101-121. doi: doi:10.1088/0305-4470/35/1/309.  Google Scholar

[23]

V. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems, 21 (2005), 1075-1086. doi: doi:10.1088/0266-5611/21/3/017.  Google Scholar

[24]

V. Yurko, On the reconstruction of Sturm-Liouville operators on graphs (Russian), Mat. Zametki, 79 (2006), 619-630; translation in Math. Notes, 79 (2006), 572-582. doi: doi:10.1007/s11006-006-0064-0.  Google Scholar

[25]

V. Yurko, Inverse problems for differential operators of arbitrary orders on trees (Russian), Mat. Zametki, 83 (2008), 139-152; translation in Math. Notes, 83 (2008), 125-137. doi: doi:10.1134/S000143460801015X.  Google Scholar

show all references

References:
[1]

S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Problems and Imaging, 2 (2008), 1-21.  Google Scholar

[2]

S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings, Zeit. Angew. Math. Mech., 90 (2010), 136-150. doi: doi:10.1002/zamm.200900295.  Google Scholar

[3]

S. Avdonin, V. Mikhaylov and A. Rybkin, The boundary control approach to the Titchmarsh-Weyl $m$-function. I. The response operator and the $A$-amplitude, Comm. Math. Phys., 275 (2007), 791-803. doi: doi:10.1007/s00220-007-0315-2.  Google Scholar

[4]

M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672. doi: doi:10.1088/0266-5611/20/3/002.  Google Scholar

[5]

M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1-R67. doi: doi:10.1088/0266-5611/23/5/R01.  Google Scholar

[6]

M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inv. Ill-Posed Problems, 14 (2006), 29-46. doi: doi:10.1515/156939406776237474.  Google Scholar

[7]

B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3231-3243.  Google Scholar

[8]

B. M. Brown and R. Weikard, On inverse problems for finite trees, in "Methods of Spectral Analysis in Mathematical Physics, Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2006," 31-48. Google Scholar

[9]

R. Carlson, Inverse eigenvalue problems on directed graphs, Trans. Amer. Math. Soc., 351 (1999), 4069-4088. doi: doi:10.1090/S0002-9947-99-02175-3.  Google Scholar

[10]

P. Exner and P. Šeba, Free quantum motion on a branching graph, Rep. Math. Phys., 28 (1989), 7-26. doi: doi:10.1016/0034-4877(89)90023-2.  Google Scholar

[11]

G. Freiling and V. Yurko, Inverse problems for Sturm-Liouville operators on noncompact trees, Results Math., 50 (2007), 195-212. doi: doi:10.1007/s00025-007-0246-4.  Google Scholar

[12]

G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions, Applicable Analysis, 86 (2007), 653-667. doi: doi:10.1080/00036810701303976.  Google Scholar

[13]

N. I. Gerasimenko and B. Pavlov, Scattering problems on noncompact graphs, Teoret. Mat. Fiz., 74 (1988), 345-359; Eng. transl. in Theoret. and Math. Phys., 74 (1988), 230-240.  Google Scholar

[14]

N. I. Gerasimenko, Inverse scattering problem on a noncompact graph, Teoret. Mat. Fiz., 75 (1988), 187-200; Eng. transl. in Theoret. and Math. Phys., 75 (1988), 460-470.  Google Scholar

[15]

M. Harmer, Hermitian symplectic geometry and extension theory, J. Phys. A, 33 (2000), 9193-9203. doi: doi:10.1088/0305-4470/33/50/305.  Google Scholar

[16]

M. Harmer, Inverse scattering on matrices with boundary conditions, J. Phys. A, 38 (2005), 4875-4885. doi: doi:10.1088/0305-4470/38/22/012.  Google Scholar

[17]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630. doi: doi:10.1088/0305-4470/32/4/006.  Google Scholar

[18]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires. II. The inverse problem with possible applications to quantum computers, Fortschr. Phys., 48 (2000), 703-716. doi: doi:10.1002/1521-3978(200008)48:8<703::AID-PROP703>3.0.CO;2-O.  Google Scholar

[19]

P. Kuchment, "Waves in Periodic and Random Media. Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference Held at Mount Holyoke College, South Hadley, MA, June 22-28, 2002,'' Contemporary Mathematics, 339 (2003), (Providence, RI: American Mathematical Society).  Google Scholar

[20]

P. Kuchment, Quantum graphs. I. Some basic structures, Waves in Random Media, 14 (2004), S107-S128. doi: doi:10.1088/0959-7174/14/1/014.  Google Scholar

[21]

P. Kurasov and M. Nowaczyk, Geometric properties of quantum graphs and vertex scattering matrices, Opuscula Math., 30 (2010), 295-309. Google Scholar

[22]

P. Kurasov and F. Stenberg, On the inverse scattering problem on branching graphs, J. Phys. A, 35 (2002), 101-121. doi: doi:10.1088/0305-4470/35/1/309.  Google Scholar

[23]

V. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems, 21 (2005), 1075-1086. doi: doi:10.1088/0266-5611/21/3/017.  Google Scholar

[24]

V. Yurko, On the reconstruction of Sturm-Liouville operators on graphs (Russian), Mat. Zametki, 79 (2006), 619-630; translation in Math. Notes, 79 (2006), 572-582. doi: doi:10.1007/s11006-006-0064-0.  Google Scholar

[25]

V. Yurko, Inverse problems for differential operators of arbitrary orders on trees (Russian), Mat. Zametki, 83 (2008), 139-152; translation in Math. Notes, 83 (2008), 125-137. doi: doi:10.1134/S000143460801015X.  Google Scholar

[1]

Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1
[2]

James B. Kennedy, Jonathan Rohleder. On the hot spots of quantum graphs. Communications on Pure &amp; Applied Analysis, 2021, 20 (9) : 3029-3063. doi: 10.3934/cpaa.2021095
[3]

Travis G. Draper, Fernando Guevara Vasquez, Justin Cheuk-Lum Tse, Toren E. Wallengren, Kenneth Zheng. Matrix valued inverse problems on graphs with application to mass-spring-damper systems. Networks & Heterogeneous Media, 2020, 15 (1) : 1-28. doi: 10.3934/nhm.2020001
[4]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19
[5]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
[6]

De-Han Chen, Daijun Jiang, Irwin Yousept, Jun Zou. Variational source conditions for inverse Robin and flux problems by partial measurements. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021050
[7]

Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks & Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483
[8]

Constanza Riera, Matthew G. Parker, Pantelimon Stǎnicǎ. Quantum states associated to mixed graphs and their algebraic characterization. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021015
[9]

Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231
[10]

José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
[11]

J. M. Mazón, Julio D. Rossi, J. Toledo. Optimal matching problems with costs given by Finsler distances. Communications on Pure &amp; Applied Analysis, 2015, 14 (1) : 229-244. doi: 10.3934/cpaa.2015.14.229
[12]

Sergio Grillo, Leandro Salomone, Marcela Zuccalli. Explicit solutions of the kinetic and potential matching conditions of the energy shaping method. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021022
[13]

Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008
[14]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
[15]

Klaus-Jochen Engel, Marjeta Kramar Fijavž. Waves and diffusion on metric graphs with general vertex conditions. Evolution Equations & Control Theory, 2019, 8 (3) : 633-661. doi: 10.3934/eect.2019030
[16]

Pengwen Chen, Changfeng Gui. Alpha divergences based mass transport models for image matching problems. Inverse Problems & Imaging, 2011, 5 (3) : 551-590. doi: 10.3934/ipi.2011.5.551
[17]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[18]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[19]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1
[20]

Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences