# American Institute of Mathematical Sciences

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

 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.
Citation: Sergei Avdonin, Pavel Kurasov, Marlena Nowaczyk. Inverse problems for quantum trees II: Recovering matching conditions for star graphs. Inverse Problems and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [21] P. Kurasov and M. Nowaczyk, Geometric properties of quantum graphs and vertex scattering matrices, Opuscula Math., 30 (2010), 295-309. [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. [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. [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. [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.

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##### References:
 [1] S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Problems and Imaging, 2 (2008), 1-21. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [21] P. Kurasov and M. Nowaczyk, Geometric properties of quantum graphs and vertex scattering matrices, Opuscula Math., 30 (2010), 295-309. [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. [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. [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. [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.
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