Overview
This research investigates the root functions of a Sturm-Liouville problem where the spectral parameter is linearly incorporated into one of the boundary conditions. The primary focus is on the properties of these root functions, encompassing both eigenfunctions and associated functions, particularly those corresponding to multiple eigenvalues. The study develops explicit representations for the biorthogonal system and for specific associated functions, which are critical components in spectral analysis.
Approach
The methodology employed centers on using the characteristic function of the boundary value problem. By leveraging this function, the researchers derived explicit representations. These representations facilitate the direct reformulation of established criteria regarding the basis and minimality properties of the system of root functions. This reformulation expresses these criteria in terms of the characteristic function and its derivatives evaluated at the eigenvalues. The proposed approach was then applied to analyze several illustrative examples.
Findings
- Explicit representations were obtained for the biorthogonal system in Sturm-Liouville problems with a boundary condition linearly dependent on an eigenparameter.
- Explicit representations were established for several special associated functions, vital for spectral analysis, within these problems.
- Criteria for the basis and minimality properties of the system of root functions can be directly reformulated using the characteristic function and its derivatives at the eigenvalues.
- The derived representations lead to a simplification in the investigation of particular boundary value problems.
- Analysis of illustrative examples confirmed the effectiveness of the proposed approach and demonstrated agreement with existing results in the literature.
Why This Matters
The ability to reformulate basis and minimality criteria directly through the characteristic function simplifies the analysis of specific boundary value problems. This methodological simplification streamlines the process of investigating these problems, providing a more straightforward analytical framework.