Abstract
In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author’s knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance.
Similar content being viewed by others
Data availability
This manuscript has no associated data.
References
Bondarenko N.P. Reconstruction of higher-order differential operators by their spectral data, Mathematics 10 (2022), no. 20, Article ID 3882 (32 pp.).
Mirzoev, K.A.; Shkalikov, A.A. Differential operators of even order with distribution coefficients, Math. Notes 99 (2016), no. 5, 779–784.
Mirzoev, K.A.; Shkalikov, A.A. Ordinary differential operators of odd order with distribution coefficients, preprint (2019), arXiv:1912.03660 [math.CA].
Bondarenko N.P. Linear differential operators with distribution coefficients of various singularity orders, Mathematical Methods in the Applied Sciences (2022). https://doi.org/10.1002/mma.8929.
Naimark, M.A. Linear Differential Operators, 2nd ed., Nauka, Moscow (1969); English transl. of 1st ed., Parts I,II, Ungar, New York (1967, 1968).
Akhmerova, E.F. Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order 2m, Math. Notes 90 (2011), no. 6, 813–823.
Badanin, A.; Korotyaev, E. Even order periodic operator on the real line, Int. Math. Res. Not. 2012 (2012), no. 5, 1143–1194.
Badanin, A.; Korotyaev, E.L. Third-order operators with three-point conditions associated with Boussinesq’s equation, Appl. Anal. 100 (2021), no. 3, 527–560.
Polyakov, D.M. Spectral asymptotics for the fourth-order operator with periodic coefficients, preprint (2022), arXiv:2202.03764 [math.SP].
Savchuk, A.M. On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential, Math. Notes 69 (2001), no. 2, 245–252.
Hryniv, R.O.; Mykytyuk, Ya.V. Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra, North-Holland Mathematics Studies 197 (2004), 97–114.
Mikhailets, V.; Molyboga, V. Uniform estimates for the semi-periodic eigenvalues of the singular differential operators, Methods Funct. Anal. Topology 10 (2004), no. 4, 30–57.
Mikhailets, V.A.; Molyboga, V.M. On the spectrum of singular perturbations of operators on the circle, Math. Notes 91 (2012), no. 4, 588–591.
Vladimirov, A.A. On one approach to definition of singular differential operators, preprint (2017), arXiv:1701.08017 [math.SP].
Bondarenko, N.P. Inverse spectral problems for arbitrary-order differential operators with distribution coefficients, Mathematics 9 (2021), no. 22, Article ID 2989.
Savchuk, A.M.; Shkalikov, A.A. Asymptotic analysis of solutions of ordinary differential equations with distribution coefficients, Sb. Math. 211 (2020), no. 11, 1623–1659.
Rykhlov, V.S. Asymptotical formulas for solutions of linear differential systems of the first order, Results Math. 36 (1999), no. 3–4, 342–353.
Savchuk, A.M. Direct and Inverse Spectral Problems for the Sturm-Liouville Operator and the Dirac System, Doctor of Science Thesis, Moscow State University, Moscow (2018) [in Russian].
Marchenko, V.A. Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev (1977) [in Russian]; English transl., Birkhauser (1986).
Freiling, G.; Yurko, V. Inverse Sturm-Liouville Problems and Their Applications, Huntington, NY: Nova Science Publishers (2001)
Buterin, S.A. On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval, J. Math. Anal. Appl. 335 (2007), no. 1, 739–749.
Buterin, S.A.; Shieh, C.-T.; Yurko, V.A. Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions, Boundary Value Problems (2013), 2013:180.
Funding
This work was supported by Grant 21-71-10001 of the Russian Science Foundation, https://rscf.ru/en/project/21-71-10001/.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bondarenko, N.P. SPECTRAL DATA ASYMPTOTICS FOR THE HIGHER-ORDER DIFFERENTIAL OPERATORS WITH DISTRIBUTION COEFFICIENTS. J Math Sci 266, 794–815 (2022). https://doi.org/10.1007/s10958-022-06118-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-06118-x
Keywords
- Higher-order differential operators
- Distribution coefficients
- Regularization
- Eigenvalue asymptotics
- Weight numbers