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Eigenparameter dependent inverse boundary value problem for a class of Sturm-Liouville operator
Boundary Value Problems volume 2014, Article number: 194 (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.
We consider the boundary value problem
where is a real valued function, λ is a complex parameter, , , , are positive real numbers and
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
Let and be the solutions of (1) satisfying the initial conditions
where . The following properties hold for the kernel which has the partial derivative belonging to the space for every :
We obtain the integral representation of the solution :
Let us define
which is independent from . Substituting and into (9) we get
In the Hilbert space let an inner product be defined by
We define the operator
Taking , we find (12). □
3 Asymptotic formulas of the eigenvalues
and can be represented in the following way:
Roots of the function are separated, i.e.,
whereis a bounded sequence,
From (8), it follows that
The expressions of and let us calculate :
Therefore, for sufficiently large n, on the contours
By the Rouche theorem, we obtain the result that the number of zeros of the function
inside the contour coincides with the number of zeros of the function . Moreover, applying the Rouche theorem to the circle we find, for sufficiently large n, that there exists one zero of the function in . Owing to the arbitrariness of we have
Let us show that . It is obvious that can be reduced to the integral
where . Now, take
It is clear from  (p.66) that . By virtue of this we have . The lemma is proved. □
4 Expansion formula with respect to eigenfunctions
and consider the function
Now let and assume
Then from (20), we have . Consequently, for fixed the function is entire with respect to λ. Let us denote
where δ is sufficiently small positive number. It is clear that the relation below holds:
From (18) it follows that for fixed and sufficiently large we have
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 . □
If, then the expansion formula
is valid, where
and the series converges uniformly with respect to. For, the series converges in, moreover, the Parseval equality holds:
If we consider the following contour integral where is a counter-clockwise oriented contour:
and then taking into consideration (20) we get
On the other hand, with the help of (25) we get
The relations below hold for sufficiently large
The validity of
Since the system of is complete and orthogonal in , the Parseval equality
5 Uniqueness theorems
Further, let the function be the solution of (1) satisfying and . We set
The functions and are called the Weyl solution and the Weyl function for the boundary value problem (1)-(3), respectively. The Weyl function is a meromorphic function having simple poles at points , eigenvalues of the boundary value problem of (1)-(3). The Wronskian
does not depend on x. Taking , we get
Using (31) we obtain
where is the characteristic function of the boundary value problem :
It is clear that
Let us denote the matrix as
Then we have
From the estimates as
we have from (36)
Therefore if , one has
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 . □
The validity of the equation below can be seen analogously to :
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 . □
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This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Mamedov, K.R., Cetinkaya, F.A. Eigenparameter dependent inverse boundary value problem for a class of Sturm-Liouville operator. Bound Value Probl 2014, 194 (2014). https://doi.org/10.1186/s13661-014-0194-3
- Sturm-Liouville operator
- expansion formula
- inverse problem
- Weyl function