Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition
© Mamedov and Cetinkaya; licensee Springer 2013
Received: 5 April 2013
Accepted: 29 July 2013
Published: 14 August 2013
This work aims to examine a Sturm-Liouville operator with a piece-wise continuous coefficient and a spectral parameter in boundary condition. The orthogonality of the eigenfunctions, realness and simplicity of the eigenvalues are investigated. The asymptotic formula of the eigenvalues is found, and the resolvent operator is constructed. It is shown that the eigenfunctions form a complete system and the expansion formula with respect to eigenfunctions is obtained. Also, the evolution of the Weyl solution and Weyl function is discussed. Uniqueness theorems for the solution of the inverse problem with Weyl function and spectral data are proved.
MSC:34L10, 34L40, 34A55.
KeywordsSturm-Liouville operator expansion formula inverse problem Weyl function
In recent years, there has been a growing interest in physical applications of boundary value problems with a spectral parameter, contained in the boundary conditions. The relationship between diffusion processes and Sturm-Liouville problem with eigen-parameter in the boundary conditions has been shown in . Another example of this relationship between the same problem and the wave equation has been examined in [2, 3]. Sturm-Liouville problems with a discontinuous coefficient arise upon non-homogeneous material properties.
In a finite interval, inverse problems for the Sturm-Liouville operator with spectral parameter, contained in the boundary conditions, have been investigated, and the uniqueness of the solution of these problems has been shown in [4–9]. The inverse problem has been analyzed by zeros of the eigenfunctions in , by numerical methods in  and by two spectra, consisting of sequences of eigenvalues and the normed constants in . In [11, 12], eigenvalue-dependent inverse problem with the discontinuities inside the interval was examined by the Weyl function. In a finite interval, discontinuous and no eigenvalue parameter containing direct problem and inverse problem with the Weyl function were discussed in [13, 14]. The similar problem was investigated in the half line by scattering data in [15, 16].
2 Special solutions
where is a real function.
is an entire function of λ and is called the characteristic function of the boundary value problem (1)-(3).
3 Some spectral properties
where and are the eigenfunctions of the boundary value problem (1)-(3), corresponding to the eigenvalue .
Therefore, we have proved that for each eigenvalue , there exists only one (up to a multiplicative constant) eigenfunction. □
The boundary value problem (1)-(3) is equivalent to the equation .
Lemma 2 The eigenfunctions and , corresponding to different eigenvalues , are orthogonal.
Since , the lemma is proved. □
Corollary 3 The eigenvalues of the boundary value problem (1)-(3) are real.
are called the norming constants of the boundary value problem (1)-(3).
Now, let us agree to denote differentiation with respect to λ with a dot ().
Taking into consideration (9) and (10), for , we arrive (11). □
Corollary 5 All zeros of are simple, i.e., .
4 Asymptotic formulas of eigenvalues
Consequently, holds uniformly on . Now, passing to the limit in equality (14), as , we have . This is a contradiction, and it proves the validity of lemma’s statement. □
the validity of can be seen directly. Lemma is proved. □
5 Expansion formula
Assume that is not a spectrum point of operator A. Then, there exists resolvent operator . Let us find the expression of the operator .
where is as in (19). □
Theorem 9 The eigenfunctions of the boundary value problem (1)-(3) form a complete system in .
Then from (28), we have . Consequently, for fixed the function is entire with respect to λ.
where δ is a sufficiently small positive number. (16) is valid from Theorem 12.4 in  for .
Using maximum principle for module of analytic functions and Liouville theorem, we get . From this and the expression of the boundary value problem (20)-(22), we obtain that a.e. on . Thus, we reach the completeness of the eigenfunctions in . □
holds. Extension of the Parseval equality to an arbitrary vector-function of the class can be carried out by usual methods. □
6 Weyl solution, Weyl function
Let us take into consideration a boundary value problem with the coefficient similar to (1)-(3) and assume that if an element α belongs to boundary value problem (1)-(3), then belongs to one with .
Theorem 11 The boundary value problem (1)-(3) is identically denoted by the Weyl function . (If , then .)
Therefore, if , then and are entire functions for every fixed x. It can easily be seen from equation (56) that and . Consequently, we get and for every x and λ. Hence, we arrive at . □
Theorem 12 The spectral data identically define the boundary value problem (1)-(3).
Proof From (51), it is clear that the function can be constructed by . Since for every , from Theorem 10, we can say that . Then from Theorem 11, it is obvious that . □
This work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK).
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