# 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 & 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. 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. 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. 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. 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). 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. 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. Google Scholar [8] B. M. Brown and R. Weikard, On inverse problems for finite trees,, in, (2006), 31. Google Scholar [9] R. Carlson, Inverse eigenvalue problems on directed graphs,, Trans. Amer. Math. Soc., 351 (1999), 4069. 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. 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. 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. 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. Google Scholar [14] N. I. Gerasimenko, Inverse scattering problem on a noncompact graph,, Teoret. Mat. Fiz., 75 (1988), 187. Google Scholar [15] M. Harmer, Hermitian symplectic geometry and extension theory,, J. Phys. A, 33 (2000), 9193. 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. 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. 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. 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). Google Scholar [20] P. Kuchment, Quantum graphs. I. Some basic structures,, Waves in Random Media, 14 (2004). 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. Google Scholar [22] P. Kurasov and F. Stenberg, On the inverse scattering problem on branching graphs,, J. Phys. A, 35 (2002), 101. 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. 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. 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. doi: doi:10.1134/S000143460801015X. Google Scholar

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##### References:
 [1] S. Avdonin and P. Kurasov, Inverse problems for quantum trees,, Inverse Problems and Imaging, 2 (2008), 1. 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. 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. 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. 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). 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. 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. Google Scholar [8] B. M. Brown and R. Weikard, On inverse problems for finite trees,, in, (2006), 31. Google Scholar [9] R. Carlson, Inverse eigenvalue problems on directed graphs,, Trans. Amer. Math. Soc., 351 (1999), 4069. 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. 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. 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. 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. Google Scholar [14] N. I. Gerasimenko, Inverse scattering problem on a noncompact graph,, Teoret. Mat. Fiz., 75 (1988), 187. Google Scholar [15] M. Harmer, Hermitian symplectic geometry and extension theory,, J. Phys. A, 33 (2000), 9193. 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. 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. 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. 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). Google Scholar [20] P. Kuchment, Quantum graphs. I. Some basic structures,, Waves in Random Media, 14 (2004). 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. Google Scholar [22] P. Kurasov and F. Stenberg, On the inverse scattering problem on branching graphs,, J. Phys. A, 35 (2002), 101. 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. 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. 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. doi: doi:10.1134/S000143460801015X. Google Scholar
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