Eigenparameter dependent inverse boundary value problem for a class of Sturm-Liouville operator
© Mamedov and Cetinkaya; licensee Springer 2014
Received: 10 March 2014
Accepted: 29 July 2014
Published: 25 September 2014
In this work a Sturm-Liouville operator with piecewise continuous coefficient and spectral parameter in the boundary conditions is considered. The eigenvalue problem is investigated; it is shown that the eigenfunctions form a complete system and an expansion formula with respect to the eigenfunctions is obtained. Uniqueness theorems for the solution of the inverse problem with a Weyl function and spectral data are proved.
MSC: 34L10, 34L40, 34A55.
Physical applications of the eigenparameter dependent Sturm-Liouville problems, i.e. the eigenparameter appears not only in the differential equation of the Sturm-Liouville problem but also in the boundary conditions, are given in –. Spectral analyses of these problems are examined as regards different aspects (eigenvalue problems, expansion problems with respect to eigenvalues, etc.) in –. Similar problems for discontinuous Sturm-Liouville problems are examined in –.
Inverse problems for differential operators with boundary conditions dependent on the spectral parameter on a finite interval have been studied in –. In particular, such problems with discontinuous coefficient are studied in –.
We investigate a Sturm-Liouville operator with discontinuous coefficient and a spectral parameter in boundary conditions. The theoretic formulation of the operator for the problem is given in a suitable Hilbert space in Section 2. In Section 3, an asymptotic formula for the eigenvalues is given. In Section 4, an expansion formula with respect to the eigenfunctions is obtained and Section 5 contains uniqueness theorems for the solution of the inverse problem with a Weyl function and spectral data.
2 Operator formulation
Taking , we find (12). □
3 Asymptotic formulas of the eigenvalues
It is clear from  (p.66) that . By virtue of this we have . The lemma is proved. □
4 Expansion formula with respect to eigenfunctions
Using maximum principle for module of analytic functions and Liouville theorem, we get . From this we obtain a.e. on . Thus we conclude the completeness of the eigenfunctions in . □
5 Uniqueness theorems
Thus, for every fixed x functions and are entire functions for λ. It can easily be seen from (37) that and . Consequently, we get and for every x and λ. Hence, we arrive at . □
From (38), it is clear that the function can be constructed by . Since for every , we can say that . Then from Theorem 5, it is obvious that . □
This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
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