Abstract
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.
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1 Introduction
We consider the boundary value problem
where is a real valued function, λ is a complex parameter, , , , are positive real numbers and
where .
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 [1]–[4]. Spectral analyses of these problems are examined as regards different aspects (eigenvalue problems, expansion problems with respect to eigenvalues, etc.) in [5]–[13]. Similar problems for discontinuous Sturm-Liouville problems are examined in [14]–[18].
Inverse problems for differential operators with boundary conditions dependent on the spectral parameter on a finite interval have been studied in [19]–[23]. In particular, such problems with discontinuous coefficient are studied in [24]–[27].
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
For the solution of (1), the following integral representation as is obtained similar to [28] for all λ:
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 :
where
Let us define
which is independent from . Substituting and into (9) we get
The function is entire and has zeros at the eigenvalues of the problem (1)-(3).
In the Hilbert space let an inner product be defined by
where
We define the operator
with
where
The boundary value problem (1)-(3) is equivalent to the equation . When are the eigenvalues, the eigenfunctions of operator L are in the form of
For any eigenvalue the solutions (4), (5) satisfy the relation
and the normalized numbers of the boundary value problem (1)-(3) are given below:
Lemma 1
The eigenvalues of the boundary value problem (1)-(3) are simple, i.e.
Proof
Since
we get
With the help of (2), (3) we get
Adding
to both sides of the last equation and using the relations (10), (11) we have
Taking , we find (12). □
3 Asymptotic formulas of the eigenvalues
The solution of (1) satisfying the initial conditions (4) when is in the following form:
where
and
The eigenvalues () of the boundary value problem (1)-(3) when can be found by using the equation
and can be represented in the following way:
where .
Roots of the function are separated, i.e.,
Lemma 2
The eigenvalues of the boundary value problem (1)-(3) are in the form of
whereis a bounded sequence,
and.
Proof
From (8), it follows that
The expressions of and let us calculate :
where
Therefore, for sufficiently large n, on the contours
we have
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
Substituting (16) into (15), as taking into account the equality and the relations , , integrating by parts and using the properties of the kernels and we have
where
Let us show that . It is obvious that can be reduced to the integral
where . Now, take
It is clear from [28] (p.66) that . By virtue of this we have . The lemma is proved. □
4 Expansion formula with respect to eigenfunctions
Denote
and consider the function
Theorem 3
The eigenfunctionsof the boundary value problem (1)-(3) form a complete system in.
Proof
With the help of (10) and (12), we can write
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 . □
Theorem 4
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:
Proof
Since and are the solutions of the boundary value problem (1)-(3), we have
Integrating by parts and taking into account the boundary conditions (2), (3) we obtain
where
If we consider the following contour integral where is a counter-clockwise oriented contour:
and then taking into consideration (20) we get
where
On the other hand, with the help of (25) we get
Comparing (26) and (27) we obtain
where
The relations below hold for sufficiently large
The validity of
can easily be seen from (28) and (29). The last equation gives us the expansion formula
Since the system of is complete and orthogonal in , the Parseval equality
holds. □
5 Uniqueness theorems
We consider the statement of the inverse problem of the reconstruction of the boundary value problem (1)-(3) from the Weyl function.
Let the functions and denote the solutions of (1) satisfying the conditions , , and , respectively, and and be the solutions of (1) under the initial conditions (4), (5).
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
Hence,
In view of (4) and (5), we get for
Using (31) we obtain
where is the characteristic function of the boundary value problem :
It is clear that
Theorem 5
The boundary value problem of (1)-(3) is identically denoted by the Weyl function.
Proof
Let us denote the matrix as
Then we have
or
Taking (31) into consideration in (35) we get
From the estimates as
we have from (36)
for .
Now, if we take into consideration (32) and (35), we have
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 [29]:
Theorem 6
The spectral data identically define the boundary value problem (1)-(3).
Proof
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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Acknowledgements
This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
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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
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DOI: https://doi.org/10.1186/s13661-014-0194-3