Influence of quantum graph parameters on the asymptotics of the number of resonances
Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 4, pp. 682-688.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study quantum graphs with the Dirac operator on edges and the Kirchhoff coupling condition at the vertices. The number of resonances is determined numerically. It is revealed how the parameters of the quantum graph (particularly, the topological structure and the volume of the quantum graph as well as the parameters of the Dirac operator) affect the number of resonances.
Keywords: quantum graph, Dirac operator, asymptotics, resonance.
@article{CHFMJ_2024_9_4_a12,
     author = {A. G. Belolipetskaia and I. Yu. Popov},
     title = {Influence of quantum graph parameters on the asymptotics of the number of resonances},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {682--688},
     publisher = {mathdoc},
     volume = {9},
     number = {4},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2024_9_4_a12/}
}
TY  - JOUR
AU  - A. G. Belolipetskaia
AU  - I. Yu. Popov
TI  - Influence of quantum graph parameters on the asymptotics of the number of resonances
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2024
SP  - 682
EP  - 688
VL  - 9
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2024_9_4_a12/
LA  - en
ID  - CHFMJ_2024_9_4_a12
ER  - 
%0 Journal Article
%A A. G. Belolipetskaia
%A I. Yu. Popov
%T Influence of quantum graph parameters on the asymptotics of the number of resonances
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2024
%P 682-688
%V 9
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2024_9_4_a12/
%G en
%F CHFMJ_2024_9_4_a12
A. G. Belolipetskaia; I. Yu. Popov. Influence of quantum graph parameters on the asymptotics of the number of resonances. Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 4, pp. 682-688. http://geodesic.mathdoc.fr/item/CHFMJ_2024_9_4_a12/

[1] Exner P., et al., Analysis on Graphs and Its Applications, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 2008

[2] Berkolaiko G., Kuchment P., Introduction to Quantum Graphs, American Mathematical Society, Providence, 2012

[3] Lax P.D., Phillips R.S., Scattering theory, Academic Press, New York, 1967

[4] Sjostrand J., Zworski M., “Complex scaling and the distribution of scattering poles”, Journal of American Mathematical Society, 4:4 (1991), 729–769

[5] Rouleux M., “Resonances for a semi-classical Schrödinger operator near a non trapping energy level”, Publ. RIMS, 34 (1998), 487–523, Kyoto University

[6] Exner P., Lotoreichik V., Tater M., “On resonances and bound states of Smilansky Hamiltonian”, Nanosystems: Physics, Chemistry, Mathematics, 7 (2016), 789–802

[7] Bonnet-Ben Dhia A.-S., Chesnel L., Pagneux V., “Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem”, Proceedings of the Royal Society A, 474:2213 (2018)

[8] Khrushchev S.V., Nikol'skii N.K., Pavlov B.S., Unconditional bases of exponentials and of reproducing kernels, Springer-Verlag, Berlin, New York, 1981, 214–335 pp.

[9] Gadylshin R.R., “Existence and asymptotics of poles with small imaginary part for the Helmholtz resonator”, Russian Mathematical Surveys, 52:1 (1997), 1–72

[10] Blinova I.V., Popov A.I., Bosova A.A., “Spectral gaps for star-like quantum graph and for two coupled rings”, Nanosystems: Physics, Chemistry, Mathematics, 13:3 (2022), 425–429

[11] Popov I.Y., Yurova T.S., “Resonances for a solvable model of ultrasound scattering by a cell membrane”, JETP Letters, 118:2 (2023), 146–151

[12] Popov I., Yurova T.S., “Resonances for Laplacian perturbed on surface and cell membrane model”, Boletin de la Sociedad Matematica Mexicana, 29:3 (2023), 85

[13] Trifanova E.S., Bagmutov A.S., Katasonov V.G., Popov I.Y., “Asymptotic expansions of resonances for waveguides coupled through converging windows”, Chelyabinsk Physical and Mathematical Journal, 8:1 (2023), 72–82

[14] Davies E.B., Pushnitski A., “Non-Weyl resonance asymptotics for quantum graphs”, Analysis and PDE, 4 (2011), 729–756

[15] Davies E.B., Exner P., Lipovsky J., “Non-Weyl asymptotics for quantum graphs with general coupling conditions”, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 474013

[16] Exner P., Lipovsky J., “Resonances from perturbations of quantum graphs with rationally related edges”, Journal of Physics A: Mathematical and Theoretical, 43:10 (2010), 105301

[17] Exner P., Lipovsky J., “Non-Weyl resonance asymptotics for quantum graphs in a magnetic field”, Physics Letters A, 375 (2011), 805–807

[18] Belolipetskaia A. G., Popov I.Y., “Dirac operator with different potentials on edges of quantum graph: resonance asymptotics”, Nanosystems: Physics, Chemistry, Mathematics, 12:4 (2021), 425–429

[19] Popov I.Y., Popov A.I., “Quantum dot with attached wires: Resonant states completeness”, Reports on Mathematical Physics, 80:1 (2017), 1–10

[20] Blinova I.V., Popov I.Y., “Quantum graph with the Dirac operator and resonance states completeness”, Operator Theory: Advances and Applications, 268 (2018), 111–124

[21] Gerasimov D.A., Popov I.Y., “Completeness of resonance states for quantum graph with two semi-infinite edges”, Complex Variables and Elliptic Equations, 63:7–8 (2018), 996–1010

[22] Blinova I.V., Popov I.Y., Popov A.I., “Resonance states completeness for relativistic particle on a sphere with two semi-infinite lines attached”, Journal of King Saud University — Science, 32:1 (2019), 836–841

[23] Popov I., Gerasimov D., Blinova I., Popov A., “Incompleteness of resonance states for quantum ring with two semi-infinite edges”, Analysis and Mathematical Physics, 9:3 (2019), 1287–1302