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.
Keywords:Sturm-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].
We consider the boundary value problem
2 Special solutions
For the solution of equation (1), the following integral representation is obtained in  for all λ:
3 Some spectral properties
Proof The proof can be done in a similar way to . Indeed, let us assume that is an eigenvalue of the function . Then
holds, i.e., the functions and are linearly dependent (), and they satisfy the boundary conditions (2), (3). Hence is an eigenvalue, and are eigenfunctions, related to this eigenvalue. Conversely, let be an eigenvalue of the operator A, and let , be the corresponding eigenfunctions. Then the boundary conditions (2), (3) hold both for the eigenfunctions and . Additionally, if the functions and satisfy the conditions , , then , . According to boundary conditions (2), (3), we have
Let us define
The eigenfunctions of operator A are in the form of
Integrating it from 0 to π, and using the boundary condition (3), we obtain
Corollary 3The eigenvalues of the boundary value problem (1)-(3) are real.
are called the norming constants of the boundary value problem (1)-(3).
Lemma 4The following equality holds
With the help of (2) and (3),
4 Asymptotic formulas of eigenvalues
Lemma 7The eigenvalues of the boundary value problem (1)-(3) are in the form of
Proof From (6), it follows that
Therefore, for sufficiently large n, on the contours
By the Rouche theorem, we obtain that the number of zeros of the function inside the contour coincides with the number of zeros of the function . Furthermore, applying the Rouche theorem to the circle , we get that, for sufficiently large n there exists only one zero of the function in . Owing to the arbitrariness of we have
Substituting (18) into (16), we get
can be reduced to
5 Expansion formula
Proof To construct the resolvent operator of A, we need to solve the boundary value problem
Substituting (24) and (25) into (23) and taking into account the boundary conditions (21) and (22), we have
Proof With the help of (9) and (11), we can write
Using (19) and (26), we get
Let us denote that
where δ is a sufficiently small positive number. (16) is valid from Theorem 12.4 in  for .
On the other hand, we can say from Lemma 1.3.1 in  that for every , the following relation holds
From the estimates (31)-(34), it is obvious that
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 . □
is valid, where
Integrating by parts and taking into account the boundary conditions (2), (3), we obtain
On the other hand, taking into account (38), we have
Comparing (39) and (40), we obtain
Thus, we obtain
Now, let us show that
From the arbitrariness of ϵ, we reach
The validity of (42) can be easily seen from (43) and (44). Thus, we obtain
If we take
the last equation gives us the expansion formula
6 Weyl solution, Weyl function
It is clear that
The functions and are respectively called the Weyl solution and the Weyl function of the boundary value problem (1)-(3). The Weyl function is a meromorphic function having simple poles at points eigenvalues of boundary value problem (1)-(3). Relations (46), (48) yield
It can be shown that
Validity of the equation below can be shown analogously to 
From (50) and (52), we have
Taking equation (49) into consideration in (54), we get
Now, from the estimates
we have from equation (55)
Theorem 12The spectral data identically define the boundary value problem (1)-(3).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
This work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK).
Fulton, CT: Two-point boundary-value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb.. 77, 293–308 (1977). Publisher Full Text
Mamedov, SG: Determination of a second order differential equation with respect to two spectra with a spectral parameter entering into the boundary conditions (Russian). Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Tekh. Mat. Nauk. 3, 15–22 (1982)
Browne, PJ, Sleeman, BD: Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. Inverse Probl.. 12(4), 377–381 (1996). Publisher Full Text
Binding, PA, Browne, PJ, Watson, BA: Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. J. Lond. Math. Soc.. 62(1), 161–182 (2000). Publisher Full Text
McCarthy, CM, Rundell, W: Eigenparameter dependent inverse Sturm-Liouville problems. Numer. Funct. Anal. Optim.. 24(1-2), 85–105 (2003). Publisher Full Text
Guliyev, NJ: Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions. Inverse Probl.. 21(4), 1315–1330 (2005). Publisher Full Text
Amirov, RK, Ozkan, AS, Keskin, B: Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions. Integral Transforms Spec. Funct.. 20(8), 607–618 (2009). Publisher Full Text
Mamedov, KR, Kosar, NP: Inverse scattering problem for Sturm-Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition. Math. Methods Appl. Sci.. 34(2), 231–241 (2011). Publisher Full Text
Akhmedova, EN, Huseynov, HM: On eigenvalues and eigenfunctions of one class of Sturm-Liouville operators with discontinuous coefficients. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci.. XXIII(4), 7–18 (2003)
Nabiev, AA, Amirov, RK: On the boundary value problem for the Sturm-Liouville equation with the discontinuous coefficient. Math. Methods Appl. Sci. (2012). Publisher Full Text