Transparent vertex boundary conditions for quantum graphs: Simplified approach
https://doi.org/10.17586/2220-8054-2019-10-5-505-510
Abstract
We consider quantum graphs providing reflectionless wave transmission at the vertices. Imposing Kuska’s version of so-called absorbing boundary conditions we derive the constraints, which make usual continuity and Kirchhoff conditions equivalent to transparent boundary conditions.
About the Authors
M. M. Aripov
National University of Uzbekistan
Uzbekistan
Uzbekistan
M. M. Aripov
Vuzgorodok, Tashkent
K. K. Sabirov
Tashkent University of Information Technologies
Uzbekistan
Uzbekistan
K. K. Sabirov
108 Amir Temur Str., 100200, Tashkent
J. R. Yusupov
Turin Polytechnic University in Tashkent
Uzbekistan
Uzbekistan
J. R. Yusupov
17 Niyazov Str., 100095, Tashkent
References
1. Engquist B., Majda A. Absorbing boundary conditions for the numerical simulation of waves. Math. Comput., 1977, 31, P. 629–651.
2. Kosloff R., Kosloff D., Absorbing boundaries for wave propagation problems. J. Comput. Phys., 1986, 63(2), P. 363–376.
3. Arnold A., Ehrhardt M. Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equations in Underwater Acoustics. J. Comput. Phys., 1998, 145(2), P. 611–638.
4. Ehrhardt M. Discrete Transparent Boundary Conditions for General Schrödinger-type Equations. VLSI Design, 1999, 9(4), P. 325–338.
5. Ehrhardt M., Arnold A. Discrete transparent boundary conditions for the Schrödinger equation. Riv. di Math. Univ. di Parma, 2001, 6(4), P. 57–108.
6. Ehrhardt M. Efficient Numerical Implementation of Discrete Transparent Boundary Conditions for Parabolic Equations in Underwater Acoustics. Acta Acustica united with Acustica, 2002, 88, P. 711–713.
7. Arnold A., Ehrhardt M. and Sofronov I. Discrete transparent boundary conditions for the Schrdinger equation: fast calculation, approximation, and stability. Comm. Math. Sci., 2003, 1(3), P. 501–556.
8. Jiang S. and Greengard L. Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension. Comput. Math. Appl., 2004, 47, P. 955–966.
9. Antoine X., Arnold A., Besse C., Ehrhardt M. and Schädle A., A Review of artificial boundary conditions for the Schr¨odinger equation. Commun. Comput. Phys., 2008, 4(4), P. 729–796.
10. Ehrhardt M. Discrete transparent boundary conditions for Schrödinger-type equations for non-compactly supported initial data. Appl. Numer. Math., 2008, 58(5), P. 660–673.
11. Šumichrast L. and Ehrhardt M. Comparison of the continuous, semi-discrete and fully-discrete transparent boundary conditions (TBC) for the parabolic wave equation: 1 theory. J. Electr. Engineering, 2009, 60(6), P. 301–312.
12. Antoine X., Besse C., Klein P. Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential. J. Comput. Phys., 2009, 228(2), P. 312–335.
13. Ehrhardt M. Absorbing Boundary Conditions for Hyperbolic Systems. Numer. Math.: Theor. Meth. Appl., 2010, 3(3), P. 295–337.
14. Klein P., Antoine X., Besse C. and Ehrhardt M. Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schr¨odinger Equations with Unbounded Potentials and Nonlinearities. Commun. Comput. Phys., 2011, 10(5), P. 1280–1304.
15. Arnold A., Ehrhardt M., Schulte M. and Sofronov I. Discrete transparent boundary conditions for the Schr¨odinger equation on circular domains. Commun. Math. Sci., 2012, 10(3), P. 889–916.
16. Feshchenko R.M. and Popov A.V. Exact transparent boundary condition for the three-dimensional Schrödinger equation in a rectangular cuboid computational domain. Phys. Rev. E, 2013, 88, P. 053308.
17. Antoine X. et al., Absorbing boundary conditions for relativistic quantum mechanics equations. J. Comput. Phys., 2014, 277, P. 268–304.
18. Petrov P. and Ehrhardt M. Transparent boundary conditions for iterative high-order parabolic equations. J. Comput. Phys., 2016, 313, P. 144– 158.
19. Shibata T. Absorbing boundary conditions for the finite-difference time-domain calculation of the one-dimensional Schrödinger equation. Phys. Rev. B, 1991, 43(8), P. 6760.
20. Kuska J.-P. Absorbing boundary conditions for the Schrödinger equation on finite intervals. Phys. Rev. B, 1992, 46(8), P. 5000.
21. Kottos T. and Smilansky U. Periodic Orbit Theory and Spectral Statistics for Quantum Graphs. Ann. Phys., 1999, 76, P. 274.
22. Gnutzmann S. and Smilansky U. Quantum graphs: Applications to quantum chaos and universal spectral statistics. Adv. Phys., 2006, 55, P. 527–625.
23. Gnutzmann S., Schanz H. and Smilansky U. Topological Resonances in Scattering on Networks (Graphs). Phys. Rev. Lett., 2013, 110, P. 094101.
24. Gnutzmann S., Waltner D. Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory. Phys. Rev. E, 2016, 93, P. 032204.
25. Gnutzmann S., Waltner D. Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures. Phys. Rev. E, 2016, 94, P. 062216.
26. Yusupov J.R., Sabirov K.K., Ehrhardt M. and Matrasulov D.U. Transparent quantum graphs. Phys. Lett. A, 2019, 383, P. 2382–2388.
27. Crank J. and Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Advances in Comput. Math., 1996, 6(1), P. 207–226.
28. Dubeibe F.L. Solving the time-dependent Schrödinger equation with absorbing boundary conditions and source terms in Mathematica 6.0. Int. J. Mod. Phys. C, 2010, 21(11), P. 1391–1406.
29. Jiang Sh. and Greengard L. Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension. Computers & Mathematics with Applications, 2004, 47(6), P. 955–966.
30. Neuhauser D. and Baer M. The time-dependent Schrödinger equation: Application of absorbing boundary conditions. J. Chem. Phys., 1989, 90(8), P. 4351–4355.
Review
For citations:
Aripov M.M., Sabirov K.K., Yusupov J.R. Transparent vertex boundary conditions for quantum graphs: Simplified approach. Nanosystems: Physics, Chemistry, Mathematics. 2019;10(5):505-510. https://doi.org/10.17586/2220-8054-2019-10-5-505-510
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