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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.
|
[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. |
[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. |
[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. |
[5] |
M. I. Belishev, Recent progress in the boundary control method,, Inverse Problems, 23 (2007).
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[13] |
N. I. Gerasimenko and B. Pavlov, Scattering problems on noncompact graphs,, Teoret. Mat. Fiz., 74 (1988), 345.
|
[14] |
N. I. Gerasimenko, Inverse scattering problem on a noncompact graph,, Teoret. Mat. Fiz., 75 (1988), 187.
|
[15] |
M. Harmer, Hermitian symplectic geometry and extension theory,, J. Phys. A, 33 (2000), 9193.
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.
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.
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.
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).
|
[20] |
P. Kuchment, Quantum graphs. I. Some basic structures,, Waves in Random Media, 14 (2004).
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. 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. |
[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. |
[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. |
[25] |
V. Yurko, Inverse problems for differential operators of arbitrary orders on trees (Russian),, Mat. Zametki, 83 (2008), 139.
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.
|
[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. |
[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. |
[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. |
[5] |
M. I. Belishev, Recent progress in the boundary control method,, Inverse Problems, 23 (2007).
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[13] |
N. I. Gerasimenko and B. Pavlov, Scattering problems on noncompact graphs,, Teoret. Mat. Fiz., 74 (1988), 345.
|
[14] |
N. I. Gerasimenko, Inverse scattering problem on a noncompact graph,, Teoret. Mat. Fiz., 75 (1988), 187.
|
[15] |
M. Harmer, Hermitian symplectic geometry and extension theory,, J. Phys. A, 33 (2000), 9193.
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.
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.
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.
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).
|
[20] |
P. Kuchment, Quantum graphs. I. Some basic structures,, Waves in Random Media, 14 (2004).
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. 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. |
[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. |
[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. |
[25] |
V. Yurko, Inverse problems for differential operators of arbitrary orders on trees (Russian),, Mat. Zametki, 83 (2008), 139.
doi: doi:10.1134/S000143460801015X. |
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