Asymptotics of the eigenvalues of a boundary value problem for the operator Schrödinger equation with boundary conditions nonlinearly dependent on the spectral parameter

Cover Page


Cite item

Full Text

Abstract

On the space H1 = L2(H, [0, 1]), where H is a separable Hilbert space, we study the asymptotic behavior of the eigenvalues of a boundary value problem for the operator Schrödinger equation for the case when one, and the same, spectral parameter participates linearly in the equation and quadratically in the boundary condition. Asymptotic formulae are obtained for the eigenvalues of the considered boundary value problem.

About the authors

Ilyas F. Hashimoglu

Karabük University

Author for correspondence.
Email: i.hasimoglu@karabuk.edu.tr
ORCID iD: 0000-0002-1690-2186
Scopus Author ID: 37115452500
http://www.mathnet.ru/person180012

PhD; Associate Professor; Faculty of Business, Dept. of Business Administration

Karabük, 78050, Turkey

References

  1. Hashimoglu I., Akın Ö., Mamedov K. R. The discreteness of the spectrum of the Schrödinger operator equation and some properties of the s-numbers of the inverse Schrödinger operator, Math. Methods Appl. Sci., 2019, vol. 42, no. 7, pp. 2231–2243. https://doi.org/10.1002/mma.5489.
  2. Gorbachuk V. I., Rybak M. A. On the boundary-value problems for the Sturm–Liouville equation with a spectral parameter in the equation and boundary condition, In: Direct and Inverse Problems of the Scattering Theory. Kiev, 1981, pp. 3–16 (In Russian).
  3. Rybak M. A. Asymptotic distribution of the eigenvalues of some boundary value problems for Sturm–Liouville operator equations, Ukr. Math. J., 1980, vol. 32, no. 2, pp. 159–162. https://doi.org/10.1007/BF01092795.
  4. Aliev B. A. Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation, Ukr. Math. J., 2006, vol. 58, no. 8, pp. 1298–1306. https://doi.org/10.1007/s11253-006-0134-1.
  5. Aliev B. A. Asymptotic behavior of eigenvalues of a boundary value problem for a second-order elliptic differential-operator equation with spectral parameter quadratically occurring in the boundary condition, Diff. Equ., 2018, vol. 54, no. 9, pp. 1256–1260. https://doi.org/10.1134/S0012266118090124.
  6. Aliev B. A. On eigenvalues of a boundary value problem for a second order elliptic differential-operator equation, Proc. Inst. Math. Mech., Natl. Acad. Sci. Azerb., 2019, vol. 45, no. 2, pp. 213–221. https://doi.org/10.29228/proc.5.
  7. Kapustin N. Yu. On a spectral problem in the theory of the heat operator, Diff. Equ., 2009, vol. 45, no. 10, pp. 1509–1511. https://doi.org/10.1134/S001226610910019X.
  8. Kapustin N. Yu. On the uniform convergence in C 1 of Fourier series for a spectral problem with squared spectral parameter in a boundary condition, Diff. Equ., 2011, vol. 47, no. 10, pp. 1408–1413. https://doi.org/10.1134/S001226611110003X.
  9. Kerimov N. B., Mamedov Kh. R. On one boundary value problem with a spectral parameter in the boundary conditions, Sib. Math. J., 1999, vol. 40, no. 2, pp. 281–290. https://doi.org/10.1007/s11202-999-0008-5.
  10. Aslanova N. M. Study of the asymptotic eigenvalue distribution and trace formula of a second order operator-differential equation, Bound. Value Probl., 2011, vol. 2011, no. 7, pp. 1–22. https://doi.org/10.1186/1687-2770-2011-7.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2021 Authors; Samara State Technical University (Compilation, Design, and Layout)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies