The Green function for simplest quantum graphs
https://doi.org/10.17586/2220-8054-2015-6-6-762-766
Abstract
We treat the problem of the Green function for quantum graphs by focusing on such topologies as star and tree graphs. The exact Green function for the Schr¨odinger equation on primary star graphs is derived in the form of 3 × 3− matrix using the vertex boundary conditions providing continuity and current conservation. Extension of the approach for the derivation for the Green function on tree graph is presented. Possible practical applications of the obtained results are discussed.
About the Authors
K. K. Sabirov
Physics Faculty, National University of Uzbekistan
Uzbekistan
Uzbekistan
Tashkent
U. A. Aminov
Urganch State University
Uzbekistan
Uzbekistan
Urganch
Kh. Sh. Saparov
Urganch State University
Uzbekistan
Uzbekistan
Urganch
M. K. Karimov
Urganch State University
Uzbekistan
Uzbekistan
Urganch
Kh. Abdikarimov
Urganch State University
Uzbekistan
Uzbekistan
Urganch
References
1. T. Kottos and U. Smilansky. Periodic orbit theory and spectral statistics for quantum graphs, Ann.Phys., 1999, 76, P. 274.
2. S. Gnutzmann, U. Smilansky. Quantum graphs: Applications to quantum chaos and universal spectral statistics, Adv.Phys., 2006, 55, P. 527.
3. S. GnutzmannJ.P., Keating b, F. Piotet. Eigenfunction statistics on quantum graphs, Ann.Phys., 2010, 325, P. 2595.
4. L. Pauling. The Diamagnetic Anisotropy of Aromatic Molecules, J. Chem. Phys., 1936, 4, P. 673.
5. P. Exner, P. Seba, P. Stovicek. Quantum interference on graphs controlled by an external electric field. J. Phys. A: Math. Gen., 1988, 21, P. 4009{4019.
6. P. Exner, P. Seba. Free quantum motion on a branching graph, Rep. Math. Phys., 1989, 28, P. 7.
7. R. Kostrykin, R. Schrader. Kirchhoff’s rule for quantum wires, J. Phys. A: Math. Gen., 1999, 32, P. 595.
8. J. Boman, P. Kurasov. Symmetries of quantum graphs and the inverse scattering problem, Adv. Appl. Math., 2005, 35, P. 58.
9. T. Cheon, P. Exner, O. Turek. Approximation of a general singular vertex coupling in quantum graphs, Ann.Phys., 2010, 325, P. 548.
10. J. P. Keating. Fluctuation statistics for quantum star graphs. Quantum graphs and their applications, Contemp. Math., 2006, 415, P. 191.
11. O. Hul et al. Experimental simulation of quantum graphs by microwave networks, Phys. Rev. E, 2004, 69, P. 056205.
12. W. J. Parnell. Greens functions, integral equations and applications, MATH 34032, Spring 2013.
13. F. Barra, P. Gaspard. Transport and dynamics on open quantum graphs, Physical review E, 2001, 65, P. 016205.
14. A. G. M. Schmidt, B. K. Cheng, M.G.E. da Luz. Green function approach for general quantum graphs, J. Phys. A: Math. Gen., 2003, 36, P. L545L551.
15. P. Calabrese, M. Mintchev, E. Vicari. Entanglement entropy of quantum wire junctions, J. Phys. A: Math. Theor., 2012, 45, P. 105206.
16. B. Bellazzini, M. Burrello, M. Mintchev, P. Sorba. Quantum field theory on star graphs, Proc. Symp. Pure Math., 2008, 77, P. 639.
17. B. Bellazzini, M. Mintchev. Quantum fields on star graphs, J. Phys. A, 2006, 39, P. 11101.
18. B. Bellazzini, M. Mintchev, P. Sorba. Quantum fields on star graphs with bound states at the vertex, J. Math. Phys., 2010, 51, P. 032302.
19. S. A. Fulling, L. Kaplan, J. H. Wilson. Vacuum energy and repulsive Casimir forces in quantum star graphs. Physical review A, 2007, 76, P. 012118.
20. G. Plunien, B. Muller, W. Greiner, The Casimir Effect, Phys.Rep., 1986, 134, P. 87.
Review
For citations:
Sabirov K.K., Aminov U.A., Saparov Kh.Sh., Karimov M.K., Abdikarimov Kh. The Green function for simplest quantum graphs. Nanosystems: Physics, Chemistry, Mathematics. 2015;6(6):762-766. https://doi.org/10.17586/2220-8054-2015-6-6-762-766
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