Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On the Spectrum Discreteness for the Magnetic Schrödinger Operator on Quantum Graphs

  • Popov, Igor Y. (Department of Mathematics, ITMO University) ;
  • Belolipetskaia, Anna G. (Department of Mathematics, ITMO University)
  • Received : 2019.12.24
  • Accepted : 2020.11.23
  • Published : 2021.06.30
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Abstract

The aim of this work is to study the discreteness of the spectrum of the Schrödinger operator on infinite quantum graphs in a magnetic field. The problem was solved on a set of quantum graphs of a special kind.

Keywords

Acknowledgement

This work was supported by the Government of the Russian Federation (grant 08-08), grant 16-11-10330 of Russian Science Foundation.

References

  1. S.Akduman and A. Pankov, Schrodinger operators with locally integrable potentials on infinite metric graphs, Appl. Anal., 96(12)(2017), 2149-2161. https://doi.org/10.1080/00036811.2016.1207247
  2. D. Barseghyan, et al., Spectral analysis of a class of Schrodinger operators exhibiting a parameter-dependent spectral transition, J. Phys. A, 49(16)(2016), 165302, 19 pp. https://doi.org/10.1088/1751-8113/49/16/165302
  3. M. Bellassoued, Stable determination of coefficients in the dynamical Schrodinger equation in a magnetic field, Inverse Problems, 33(5)(2017), 055009, 36 pp. https://doi.org/10.1088/0266-5611/33/5/055009
  4. G. Berkolaiko (ed.), Quantum graphs and their applications, Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their Applications, June 19-23, 2005, Snowbird, UT, Contemporary Mathematics 415, American Mathematical Soc., 2006.
  5. G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, American Mathematical Society, Providence, 2013.
  6. M. Bonnefont, et al., Magnetic sparseness and Schrodinger operators on graphs, Annales Henri Poincare, 21(2020), 1489-1516. https://doi.org/10.1007/s00023-020-00885-6
  7. A. Chatterjee, I. Y. Popov and M. O. Smolkina, Persistent current in a chain of two Holstein-Hubbard rings in the presence of Rashba spin-orbit interaction, Nanosystems: Physics, Chemistry, Mathematics, 10(2019), 50-62. https://doi.org/10.17586/2220-8054-2019-10-1-50-62
  8. D. A. Eremin, E. N. Grishanov, D. S. Nikiforov and I. Y. Popov, Wave dynamics on time-depending graph with Aharonov-Bohm ring, Nanosystems: Physics, Chemistry, Mathematics, 9(2018), 457-463.
  9. P. Exner, et al. (ed.) Analysis on graphs and its applications, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007, Proceedings of Symposia in Pure Mathematics 77, American Mathematical Soc., 2008.
  10. E. Korotyaev and N. Saburova, Magnetic Schrodinger operators on periodic discrete graphs, J. Funct. Anal., 272(4)(2017), 1625-1660. https://doi.org/10.1016/j.jfa.2016.12.015
  11. M. O. Kovaleva and I. Y. Popov, On Molchanov's condition for the spectrum discreteness of a quantum graph Hamiltonian with δ-coupling, Rep. Math. Phys., 76(2)(2015), 171-178. https://doi.org/10.1016/S0034-4877(15)30027-6
  12. A. M. Molchanov, On conditions of spectrum discreteness for self-adjoint differential operators of second order, Proc. Moscow Math. Soc., 2(1953), 169-199.
  13. R. Oinarov and M. Otelbaev, A criterion for a general Sturm-Liouville operator to have a discrete spectrum, and a related imbedding theorem, Differential Equations, 24(4)(1988), 402-408.
  14. N. Raymond, Bound states of the magnetic Schrodinger operator, EMS Tracts in Mathematics 27, European Mathematical Society, Berlin, 2017.
  15. K. Ruedenberg and C. W. Scherr, Free electron network model for conjugated systems. I. theory, J. Chem. Phys., 21(9)(1953), 1565-1581.
  16. J. Zhao, G. Shi and J. Yan, Discreteness of spectrum for Schrodinger operators with δ-type conditions on infinite regular trees, Proc. Roy. Soc. Edinburgh Sect. A, 147(5)(2017), 1091-1117.