OPTIMIZATION SPECTRAL PROBLEM FOR THE STURM-LIOUVILLE OPERATOR IN THE SPACE OF VECTOR FUNCTIONS

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription or Fee Access

Abstract

An inverse spectral optimization problem is considered: for a given matrix potential Q0(x) it is required to find the matrix function Q^(x) closest to it, such that the k-th eigenvalue of the Sturm–Liouville matrix operator with potential Q^(x) matched the given value λ∗. The main result of the paper is the proof of existence and uniqueness theorems. Explicit formulas for the optimal potential are established through solutions to systems of nonlinear differential equations of the second order, known in mathematical physics as systems of nonlinear Schrödinger equations

About the authors

V. A. Sadovnichii

M.V. Lomonosov Moscow State University

Author for correspondence.
Email: info@rector.msu.ru
Russian Federation, Moscow

Ya. T. Sultanaev

Bashkir State Pedagogical University n.a. M. Akmulla; Moscow Center for Fundamental and Applied Mathematics

Author for correspondence.
Email: sultanaevyt@gmail.com
Russian Federation, Ufa; Russian Federation, Moscow

N. F. Valeev

Institute of Mathematics with Computing Centre

Author for correspondence.
Email: valeevnf@yandex.ru
Russian Federation, Ufa

References

  1. Möller M., Zettl A. Differentiable dependence of eigenvalues of operators in Banach spaces, Journal of Operator Theory. 1996. P. 335–355.
  2. Pöschel J., Trubowitz E. Inverse spectral theory, volume 130 of Pure and Applied Mathematics, 1987.
  3. Yurko V.A. Inverse Spectral Problems and their Applications, Saratov, PI Press, 2001. 499 p.
  4. Chu M., Golub G.H. Inverse eigenvalue problems: theory, algorithms, and applications, Vol. 13. Oxford University Press, 2005.
  5. Gladwell G.M.L. Inverse Problems in Scattering: An Introduction, Kluwer Academic Publishers, 1993. https://doi.org/10.1007/978-94-011-2046-3
  6. Садовничий В.А., Султанаев Я.Т., Валеев Н.Ф. Многопараметрические обратные спектральные задачи и их приложения // Доклады академии наук. 2009. Т. 426. № 4. С. 457–460.
  7. Садовничий В.А., Султанаев Я.Т., Валеев Н.Ф. Оптимизационная обратная спектральная задача для векторного оператора Штурма–Лиувилля // Дифференциальные уравнения. 2022. Т. 58. № 12. С. 1707–1711.
  8. Ilyasov Y.Sh., Valeev N.F. On nonlinear boundary value problem corresponding to -dimensional inverse spectral problem // J. Diff. Eq. 2019. V. 266. № 8. P. 4533–4543. https://doi.org/10.1016/j.jde.2018.10.00310.1016/j.jde.2018.10.003
  9. Yavdat Ilyasov, Nur Valeev. Recovery of the nearest potential field from the m observed eigenvalues // Physica D: Nonlinear Phenomena. 2021. V. 426. 5 p. https://doi.org/10.1016/j.physd.2021.132985
  10. Egorov Y.V., Kondrat’ev V.A. Estimates for the first eigenvalue in some Sturm-Liouville problems // Russian Math. Surv. 1996. V. 51. № 3. P. 439.
  11. Wei Q., Meng G., Zhang M. Extremal values of eigenvalues of Sturm–Liouville operators with potentials in L1 balls // J. Diff. Eq. 2009. V. 247. № 2. P. 364–400.
  12. Shuyuan Guo, Zhang Meirong. A Variational Approach to the Optimal Locations of the Nodes of the Second Dirichlet Eigenfunctions. Mathematical Methods in the Applied Sciences. 2022. https://doi.org/10.1002/mma.8930
  13. Guo H., Qi J. Extremal norm for potentials of Sturm-Liouville eigenvalue problems with separated boundary conditions // EJDE. 2017. V. 99. P. 1–11. http://ejde.math.unt.edu
  14. Като Т. Теория возмущений линейных операторов. М.: Наука.1972. 740 с.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2023 В.А. Садовничий, Я.Т. Султанаев, Н.Ф. Валеев