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Mathematical reminiscences
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 |
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. |
show all references
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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