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On the quotient quantum graph with respect to the regular representation
Gazi University, Faculty of Science, Department of Mathematics, Ankara-Turkey |
Given a quantum graph $ \Gamma $, a finite symmetry group $ G $ acting on it and a representation $ R $ of $ G $, the quotient quantum graph $ \Gamma /R $ is described and constructed in the literature [
References:
[1] |
R. Band, G. Berkolaiko, C. H. Joyner and W. Liu, Quotients of finite-dimensional operators by symmetry representations, arXiv: 1711.00918v3. |
[2] |
R. Band, O. Parzanchevski and G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A: Math. Theor., 42 (2009), 175202.
doi: 10.1088/1751-8113/42/17/175202. |
[3] |
R. Band, T. Shapira and U. Smilansky,
Nodal domains on isospectral quantum graphs: the resolution of isospectrality?, J. Phys. A: Math. Gen., 39 (2006), 13999-14014.
doi: 10.1088/0305-4470/39/45/009. |
[4] |
G. Berkolaiko, An elementary introduction to quantum graphs, arXiv: 1603.07356v2. |
[5] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, Rhode Island, 2013.
doi: 10.1090/surv/186. |
[6] |
R. Carlson,
Inverse eigenvalue problems on directed graphs, T. Am. Math. Soc., 351 (1999), 4069-4088.
doi: 10.1090/S0002-9947-99-02175-3. |
[7] |
W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-0979-9. |
[8] |
S. Gnutzmann and U. Smilansky, Quantum graphs: applications to quantum chaos and universal spectral statistics, arXiv: nlin/0605028v2. |
[9] |
C. Gordon, P. Perry and D. Schüth, Isospectral and isoscattering manifolds: a survey of techniques and examples, in Geometry, Spectral Theory, Groups and Dynamics, Contemporary Mathematics vol. 387 (eds. M. Entov, Y. Pinchover and M. Sageev), American Mathematical Society, (2005), 157–179.
doi: 10.1090/conm/387/07241. |
[10] |
C. Gordon, D. Webb and S. Wolpert,
One cannot hear the shape of a drum, Bull. Am. Mat. Soc. (N.S.), 27 (1992), 134-138.
doi: 10.1090/S0273-0979-1992-00289-6. |
[11] |
C. Gordon, D. Webb and S. Wolpert,
Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22.
doi: 10.1007/BF01231320. |
[12] |
B. Gutkin and U. Smilansky,
Can one hear the shape of a graph?, J. Phys. A: Math. Gen., 31 (2001), 6061-6068.
doi: 10.1088/0305-4470/34/31/301. |
[13] |
M. Kac,
Can one hear the shape of a drum?, Am. Math. Mon., 73 (1966), 1-23.
doi: 10.2307/2313748. |
[14] |
V. Kostrykin and R. Schrader,
Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595-630.
doi: 10.1088/0305-4470/32/4/006. |
[15] |
T. Kottos and U. Smilansky,
Quantum chaos on graphs, Phys. Rev. Lett., 79 (1997), 4794-4797.
doi: 10.1103/PhysRevLett.79.4794. |
[16] |
P. Kuchment, Quantum graphs: an introduction and a brief survey, arXiv: 0802.3442v1. |
[17] |
D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, Cham, 2014.
doi: 10.1007/978-3-319-04621-1. |
[18] |
O. Parzanchevski and R. Band,
Linear representations and isospectrality with boundary conditions, J. Geom. Anal., 20 (2010), 439-471.
doi: 10.1007/s12220-009-9115-6. |
[19] |
T. Shapira and U. Smilansky, Quantum graphs which sound the same, in Non-Linear Dynamics and Fundamental Interactions. NATO Science Series Ⅱ: Mathematics, Physics and Chemistry, vol 213. (eds. F. Khanna and D. Matrasulov), Springer, (2005), 17–29.
doi: 10.1007/1-4020-3949-2_2. |
[20] |
T. Sunada,
Riemanninan coverings and isospectral manifolds, Ann. Math., 121 (1985), 169-186.
doi: 10.2307/1971195. |
[21] |
J. von Below, Can one hear the shape of a network?, in Partial Differential Equations on Multistructures (Luminy, 1999), Lecture Notes in Pure and Appl. Math., vol. 219 (eds. F. Mehmeti, J. von Below and S. Nicaise), Dekker, (2001), 19–36. |
show all references
References:
[1] |
R. Band, G. Berkolaiko, C. H. Joyner and W. Liu, Quotients of finite-dimensional operators by symmetry representations, arXiv: 1711.00918v3. |
[2] |
R. Band, O. Parzanchevski and G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A: Math. Theor., 42 (2009), 175202.
doi: 10.1088/1751-8113/42/17/175202. |
[3] |
R. Band, T. Shapira and U. Smilansky,
Nodal domains on isospectral quantum graphs: the resolution of isospectrality?, J. Phys. A: Math. Gen., 39 (2006), 13999-14014.
doi: 10.1088/0305-4470/39/45/009. |
[4] |
G. Berkolaiko, An elementary introduction to quantum graphs, arXiv: 1603.07356v2. |
[5] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, Rhode Island, 2013.
doi: 10.1090/surv/186. |
[6] |
R. Carlson,
Inverse eigenvalue problems on directed graphs, T. Am. Math. Soc., 351 (1999), 4069-4088.
doi: 10.1090/S0002-9947-99-02175-3. |
[7] |
W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-0979-9. |
[8] |
S. Gnutzmann and U. Smilansky, Quantum graphs: applications to quantum chaos and universal spectral statistics, arXiv: nlin/0605028v2. |
[9] |
C. Gordon, P. Perry and D. Schüth, Isospectral and isoscattering manifolds: a survey of techniques and examples, in Geometry, Spectral Theory, Groups and Dynamics, Contemporary Mathematics vol. 387 (eds. M. Entov, Y. Pinchover and M. Sageev), American Mathematical Society, (2005), 157–179.
doi: 10.1090/conm/387/07241. |
[10] |
C. Gordon, D. Webb and S. Wolpert,
One cannot hear the shape of a drum, Bull. Am. Mat. Soc. (N.S.), 27 (1992), 134-138.
doi: 10.1090/S0273-0979-1992-00289-6. |
[11] |
C. Gordon, D. Webb and S. Wolpert,
Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22.
doi: 10.1007/BF01231320. |
[12] |
B. Gutkin and U. Smilansky,
Can one hear the shape of a graph?, J. Phys. A: Math. Gen., 31 (2001), 6061-6068.
doi: 10.1088/0305-4470/34/31/301. |
[13] |
M. Kac,
Can one hear the shape of a drum?, Am. Math. Mon., 73 (1966), 1-23.
doi: 10.2307/2313748. |
[14] |
V. Kostrykin and R. Schrader,
Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595-630.
doi: 10.1088/0305-4470/32/4/006. |
[15] |
T. Kottos and U. Smilansky,
Quantum chaos on graphs, Phys. Rev. Lett., 79 (1997), 4794-4797.
doi: 10.1103/PhysRevLett.79.4794. |
[16] |
P. Kuchment, Quantum graphs: an introduction and a brief survey, arXiv: 0802.3442v1. |
[17] |
D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, Cham, 2014.
doi: 10.1007/978-3-319-04621-1. |
[18] |
O. Parzanchevski and R. Band,
Linear representations and isospectrality with boundary conditions, J. Geom. Anal., 20 (2010), 439-471.
doi: 10.1007/s12220-009-9115-6. |
[19] |
T. Shapira and U. Smilansky, Quantum graphs which sound the same, in Non-Linear Dynamics and Fundamental Interactions. NATO Science Series Ⅱ: Mathematics, Physics and Chemistry, vol 213. (eds. F. Khanna and D. Matrasulov), Springer, (2005), 17–29.
doi: 10.1007/1-4020-3949-2_2. |
[20] |
T. Sunada,
Riemanninan coverings and isospectral manifolds, Ann. Math., 121 (1985), 169-186.
doi: 10.2307/1971195. |
[21] |
J. von Below, Can one hear the shape of a network?, in Partial Differential Equations on Multistructures (Luminy, 1999), Lecture Notes in Pure and Appl. Math., vol. 219 (eds. F. Mehmeti, J. von Below and S. Nicaise), Dekker, (2001), 19–36. |




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