Eigenvalue properties of Sturm-Liouville problems with transmission conditions dependent on the eigenparameter

Lanfang Zhang; Jijun Ao; Na Zhang; Lanfang Zhang; Jijun Ao; Na Zhang

doi:10.3934/era.2024084

Electronic Research Archive

2024, Volume 32, Issue 3: 1844-1863. doi: 10.3934/era.2024084

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Research article

Eigenvalue properties of Sturm-Liouville problems with transmission conditions dependent on the eigenparameter

College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, China

Received: 20 December 2023 Revised: 15 February 2024 Accepted: 22 February 2024 Published: 01 March 2024

This paper studies a discontinuous Sturm-Liouville problem in which the spectral parameter appears not only in the differential equation but also in the transmission conditions. By constructing an appropriate Hilbert space and inner product, the eigenvalue and eigenfunction problems of the Sturm-Liouville problem are transformed into an eigenvalue problem of a certain self-adjoint operator. Next, the eigenfunctions of the problem and some properties of the eigenvalues are given via construction of the basic solution. The Green's function for the Sturm-Liouville problem is also given. Finally, the continuity of the eigenvalues and eigenfunctions of the problem is discussed. Especially, the differential expressions of the eigenvalues for some parameters have been obtained, including the parameters in the eigenparameter-dependent transmission conditions.
- Sturm-Liouville problems,
- eigenparameter-dependent transmission conditions,
- Green's function,
- dependence of eigenvalues
Citation: Lanfang Zhang, Jijun Ao, Na Zhang. Eigenvalue properties of Sturm-Liouville problems with transmission conditions dependent on the eigenparameter[J]. Electronic Research Archive, 2024, 32(3): 1844-1863. doi: 10.3934/era.2024084

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Abstract

This paper studies a discontinuous Sturm-Liouville problem in which the spectral parameter appears not only in the differential equation but also in the transmission conditions. By constructing an appropriate Hilbert space and inner product, the eigenvalue and eigenfunction problems of the Sturm-Liouville problem are transformed into an eigenvalue problem of a certain self-adjoint operator. Next, the eigenfunctions of the problem and some properties of the eigenvalues are given via construction of the basic solution. The Green's function for the Sturm-Liouville problem is also given. Finally, the continuity of the eigenvalues and eigenfunctions of the problem is discussed. Especially, the differential expressions of the eigenvalues for some parameters have been obtained, including the parameters in the eigenparameter-dependent transmission conditions.

References

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