Finite Spectrum of Sturm–Liouville Problems with Transmission Conditions Dependent on the Spectral Parameter

Abstract

In this paper, we mainly study the finite spectrum of Sturm–Liouville problems with transmission conditions dependent on the spectral parameter. By analyzing on the characteristic function, we prove that this kind of Sturm–Liouville problems consist of finite number of eigenvalue and these finite eigenvalues can be located anywhere in the complex plane. It is illustrated that the number of eigenvalues not only depends on the partition of the domain interval, but also depends on the transmission conditions dependent on the spectral parameter and the boundary conditions.

2010 Mathematics Subject Classification:

• Primary 34L10
• 34B20
• Secondary 47A45

Keywords:

• eigenvalues
• finite spectrum
• Sturm–Liouville problems
• transmission conditions with spectral parameter

Data availability

No data were used to support this study.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Additional information

Funding

This work was supported by National Natural Science Foundation of China (Grant Nos. 12261066 and 11661059), Natural Science Foundation of Inner Mongolia (Grant Nos. 2021MS01020 and 2017JQ07) and The Basic Scientific Research Program of Universities of Inner Mongolia (Grant No. YJ20220389).