The Basis Property of the System of Weak Eigenfunctions of a Discontinuous Sturm–Liouville Problem

The purpose of this study is to investigate some important spectral properties of one discontinuous Sturm–Liouville problem. By applying our own approaches the considered problem is transformed into an eigenvalue problem for suitable integral equation in terms of which it is defined as a concept of...

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Bibliographic Details
Published in:Mediterranean journal of mathematics 2017-06, Vol.14 (3), p.1-13, Article 114
Main Authors: Olǧar, H., Muhtarov, F. S.
Format: Article
Language:English
Subjects:
Discontinuity
Eigenvectors
Hilbert space
Integral equations
Linear operators
Mathematics
Mathematics and Statistics
Spectra
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Summary:The purpose of this study is to investigate some important spectral properties of one discontinuous Sturm–Liouville problem. By applying our own approaches the considered problem is transformed into an eigenvalue problem for suitable integral equation in terms of which it is defined as a concept of weak eigenfunctions. Then we introduce for consideration some compact operators in such a way that this integral equation can be reduced to the appropriate operator-pencil equation and prove that this operator-pencil is self-adjoint and positive definite for sufficiently large negative values of the eigenparameter. Finally, it is established that the spectrum is discrete and the system of corresponding weak eigenfunctions forms an orthonormal basis of the appropriate Hilbert space.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-017-0915-9