Non-Weyl resonance asymptotics for quantum graphs in a magnetic field. (English) Zbl 1241.81078
Summary: We study asymptotical behaviour of resonances for a quantum graph consisting of a finite internal part and external leads placed into a magnetic field, in particular, the question whether their number follows the Weyl law. We prove that the presence of a magnetic field cannot change a non-Weyl asymptotics into a Weyl one and vice versa. On the other hand, we present examples demonstrating that for some non-Weyl graphs the ”effective size” of the graph, and therefore the resonance asymptotics, can be affected by the magnetic field.
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 35B34 | Resonance in context of PDEs |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
References:
| [1] | (Exner, P.; Keating, J. P.; Kuchment, P.; Sunada, T.; Teplyaev, A., Analysis on Graphs and Applications, Proceedings of a Isaac Newton Institute Programme. Analysis on Graphs and Applications, Proceedings of a Isaac Newton Institute Programme, January 8-June 29, 2007. Analysis on Graphs and Applications, Proceedings of a Isaac Newton Institute Programme. Analysis on Graphs and Applications, Proceedings of a Isaac Newton Institute Programme, January 8-June 29, 2007, Proceedings of Symposia in Pure Mathematics, vol. 77 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 670 · Zbl 1190.01017 |
| [2] | Davies, E. B.; Exner, P.; Lipovský, J., J. Phys. A: Math. Theor., 43, 474013 (2010) · Zbl 1204.81078 |
| [3] | Davies, E. B.; Pushnitski, A., Non-Weyl resonance asymptotics for quantum graphs · Zbl 1268.34056 |
| [4] | Exner, P.; Lipovský, J., J. Phys. A: Math. Theor., 43, 105301 (2010) · Zbl 1192.81160 |
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| [6] | Kuchment, P., Quantum graphs: an introduction and a brief survey, (Exner, P.; Keating, J. P.; Kuchment, P.; Sunada, T.; Teplyaev, A., Analysis on Graphs and Applications, Proceedings of a Isaac Newton Institute Programme. Analysis on Graphs and Applications, Proceedings of a Isaac Newton Institute Programme, January 8-June 29, 2007. Analysis on Graphs and Applications, Proceedings of a Isaac Newton Institute Programme. Analysis on Graphs and Applications, Proceedings of a Isaac Newton Institute Programme, January 8-June 29, 2007, Proceedings of Symposia in Pure Mathematics, vol. 77 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 291-314 · Zbl 1210.05169 |
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