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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 361989, 10 pages
Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse
1Department of Mathematics, Faculty of Sciences, Cumhuriyet University, 58140 Sivas, Turkey
2Department of Secondary Science and Mathematics Education, Faculty of Education, Cumhuriyet University, 58140 Sivas, Turkey
Received 22 November 2012; Revised 2 March 2013; Accepted 4 March 2013
Academic Editor: Juan J. Nieto
Copyright © 2013 Rauf Kh. Amırov and A. Adiloglu Nabıev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study some inverse problems for a boundary value problem generated with a quadratic pencil of Sturm-Liouville equations with impulse on a finite interval are considered. Some useful integral representations for the linearly independent solutions of a quadratic pencil of Sturm-Liouville equation have been derived and using these, important spectral properties of the boundary value problem are investigated; the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers are obtained. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.
The theory of inverse problems for differential operators occupies an important position in the current developments of the spectral theory of linear operators. Inverse problems of spectral analysis consist in the recovery of operators from their spectral data. One takes for the main spectral data, for instance, one, two, or more spectra, the spectral function, the spectrum, and the normalizing constants, the Weyl function. Different statements of inverse problems are possible depending on the selected spectral data. The already existing literature on the theory of inverse problems of spectral analysis is abundant. The most comprehensive account of the current state of this theory and its applications can be found in the monographs of Marchenko , Levitan , Beals et al. , and Yurko .
In the present work we consider some inverse problems for the boundary value problem generated by the differential equation with the boundary conditions and with the jump conditions where is the spectral parameter, are real functions, is a real number, and . Here we denote by the space of functions , , such that the derivatives are absolutely continuous and .
There exist many papers containing a fairly comprehensive analysis of direct and inverse problems of spectral analysis of the Sturm-Liouville equation a special case () of (1). For instance, inverse problems for a regular Sturm-Liouville operator with separated boundary conditions have been investigated in  (see also [1–4]).
Some versions of inverse problems for (1) which is a natural generalization of the Sturm-Liouville equation were fully studied in [6–14]. Namely, the inverse problems for a pencil on the half axis and the entire axis were considered in [6–8], where the scattering data, the spectral function, and the Weyl function, respectively, were taken for the spectral data. The problem of the recovery of (1) from the spectra of two boundary value problems with certain separated boundary conditions was solved in . The analysis of inverse spectral problems for (1) with other kinds of separated boundary conditions as well as with periodic and antiperiodic boundary conditions was the subject of  (see also ) where the corresponding results of the monograph  were extended to the case . The inverse periodic problem for the pencil was solved in  using another approach. We also point out the paper , in which the uniqueness of the recovery of the pencil from three spectra was investigated.
Boundary value problems with discontinuities inside the interval often appear in mathematics, physics, and other fields of natural sciences. The inverse problems of reconstructing the material properties of a medium from data collected outside of the medium give solutions to many important problems in engineering and geosciences. For example, in electronics, the problem of constructing parameters of heterogeneous electronic lines is reduced to a discontinuous inverse problem [15, 16]. The reduced mathematical model exhibits the boundary value problem for the equation of type (1) with given spectral information which is described by the desirable amplitude and phase characteristics. Note that the problem of reconstructing the permittivity and conductivity profiles of a one-dimensional discontinuous medium is also closed to the spectral information [17, 18]. Geophysical models for oscillations of the Earth are also reduced to boundary value problems with discontinuity in an interior point .
Direct and inverse spectral problems for differential operators without discontinuities have been extensively studied by many authors [20–25]. Some classes of direct and inverse problems for discontinuous boundary value problems in various statements have been considered in [18, 26–32] and other works. Boundary value problems with singularity have been studied in [33–37], and for further discussion see the references therein. Note that the inverse spectral problem for the boundary problem (1)–(3) has never been considered before.
In what follows we denote the boundary value problem (1)–(3) by In Section 2 we derive some integral representations for the linearly independent solutions of (1), and using these, we investigate important spectral properties of the boundary value problem . In Section 3 the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers of are obtained. Finally, in Section 4 three inverse problems of reconstructing the boundary value problem from the Weyl function, from the spectral data, and from two spectra are considered and the uniqueness theorems are proved.
2. Integral Representations of Solutions and the Spectral Characteristics
It is obvious that the functions satisfy the integral equations where and
Theorem 1. If , then the solution has the form where and the function satisfies the inequality with
Then by using Theorem 1, we can formulate the following assertion.
Theorem 2. Let . Then there are the functions whose first order partial derivatives are summable on for each such that the representation is satisfied, where Moreover, the relations are held.
If one additionally supposes that , then the functions and satisfy the system of partial differential equations Conversely, if the second order derivatives of functions are summable on for each and satisfy equalities (16) and conditions (15), then the function which is defined by (13) is a solution of (1) satisfying initial conditions (12) and discontinuity conditions (3).
One here supposes that the function ) satisfies the additional condition for all such that and
Definition 3. A complex number is called an eigenvalue of the boundary value problem if (1) with has a nontrivial solution satisfying the boundary conditions (2) and the jump conditions (3). In this case is called the eigenfunction of the problem corresponding to the eigenvalue . The number of linearly independent solutions of the problem for a given eigenvalue is called the multiplicity of .
The following lemmas can be proved analogously to the corresponding assertions in .
Lemma 4. The eigenvalues of the boundary value problem are real, nonzero, and simple.
Proof. We define a linear operator in the Hilbert space as follows. The domain consists of all functions satisfying the boundary conditions (2) and the jump conditions (3). For , we set . Integration by part yields
Since condition (17) holds, it follows that .
Let be an eigenvalue of the boundary value problem and an eigenfunction corresponding to this eigenvalue and normalized by the condition . By taking the inner product of both sides of the relation by , we obtain and hence
The desired assertion follows from the last relation by virtue of with regard to the fact that is real.
Let us show that is a simple eigenvalue. Assume that this is not true. Suppose that and are linearly independent eigenfunctions corresponding to the eigenvalue . Then for a given value of , each solution of (1) will be given as linear combination of solutions and . Moreover it will satisfy boundary conditions (2) and discontinuity conditions (3). However, it is impossible.
Proof. Let be an eigenfunction corresponding to eigenvalue and normalized by the condition of the problem (1)–(3). Suppose that is an associated function of eigenfunction , that is, the following equalities hold:
If these equations are multiplied by and , respectively, as inner product, subtracting them side by side and taking into our account that operator is symmetric, the function and are real, we get . Due to the condition (6), does not agree with . Therefore, the assertion is not true.
Lemma 6. Eigenfunctions corresponding to different eigenvalues of the problem are orthogonal in the sense of the equality where denotes the inner product in .
3. Properties of the Spectrum
In this section we investigate some spectral properties of the boundary value problem .
Denote , where . By virtue of Liouville's formula, the Wronskian does not depend on . The function is called the characteristic function of . Obviously, the function is entire in and it has at most a countable set of zeros .
Lemma 8. The zeros of the characteristic function coincide with the eigenvalues of the boundary value problem . The functions and are eigenfunctions corresponding to the eigenvalue , and there exists a sequence such that
Proof. Let . Then by virtue of , for some constant . Hence is an eigenvalue and are eigenfunctions related to .
Conversely, let be an eigenvalue of , show that . Assuming the converse suppose that . In this case the functions and are linearly independent. Then is a general solution of the problem . If , we can write Then we have which is a contradiction.
Note that we have also proved that for each eigenvalue there exists only one eigenfunction (up to a multiplicative constant). Therefore there exists sequence such that .
Let us denote
The numbers are called normalized numbers of the boundary value problem .
Lemma 9. The equality holds. Here .
Proof. If we differentiate the equalities
with respect to , we get
By virtue of these equalities we have
If the last equations are integrated from to and from to , respectively, by the discontinuity conditions we obtain If we add the last equalities side by side, we get or For , this yields The lemma is proved.
Let and are zeros of .
Lemma 10. The roots of the characteristic equation are separate, that is
Proof. Let . Then, , where . Since , then . Using these notations we can rewrite the equation in the following form:
Here which implies that . Preliminarily show that there are no multiple roots of (35). Assuming the converse we suppose to be a multiple root of (35). Then
holds. Now (35) and (36) imply that which is a contradiction. Therefore, (35) has no multiple roots.
Further assuming (34) not to be true let and be increasing sequences of roots of (35) such that and If we assume that , where and is a bounded sequence , then from (37) we find that , where is a bounded sequence such that . It is obvious that , where and . Here and denote the integer and fractional parts of a real number, respectively. Since sequences and will be bounded, without loss of generality we can assume that these sequences are convergent. Then let Therefore, we can write the equality as Then by virtue of (38) and (39), from (40) we get Similarly we can obtain Further, from (40) and (42), we have Let us write this equality as Now dividing both sides of equality (44) by and taking limit as , by virtue of (3) and (39), we obtain Finally, from (41) and (45), we conclude that which is a contradiction. Hence roots of equation are separate. The lemma is proved.
Denote where is sufficiently small positive number .
Lemma 11. For sufficiently large values of , one has
Lemma 12. The problem has countable set of eigenvalues. If one denotes by the positive eigenvalues arranged in increasing order and by the negative eigenvalues arranged in decreasing order, then eigenvalues of the problem have the asymptotic behavior where and is a bounded sequence, .
Proof. According to Lemma 11, if is a sufficiently large natural number and , we have . Applying Rouche's theorem we conclude that for sufficiently large inside the contour the functions and have the same number of zeros counting their multiplicities. That is, there are exactly zeros . in . Analogously, it is shown by Rouche's theorem that, for sufficiently large values of , the function has a unique zero inside each circle . Since is arbitrary, it follows that , where . Further according to , we have
On the other hand, since
(50) takes the form of
It is easy to see that the function is type of “Sine” , so there exists such that is satisfied for all . We also have
where for some constant  (see also ). Further, substituting (53) into (52) after certain transformations [1, page 67], we reach . We can obtain more precisely
Since , we have where . Hence we obtain where is a bounded sequence. The proof is completed.
Lemma 13. Normalizing numbers of the problem are positive and the formula holds, where .
4. Inverse Problems
Together with , we consider the boundary value problem of the same form but with different coefficients . It is assumed in what follows that if a certain symbol denotes an object related to the problem , then will denote the corresponding object related to the problem .
In the present section, we investigate some inverse spectral problem of the reconstruction of a boundary value problem of type (1)–(4) from its spectral characteristics. Namely, we consider the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data , and from two spectra and prove that the following two lemmas can be easily obtained from asymptotic behavior (49) of the eigenvalues .
Lemma 14. If , then , that is, the sequence uniquely determines .
Lemma 15. If , then , that is, the sequence uniquely determines numbers and .
Let be the solution of (1) under the conditions and under the jump conditions (3). One sets . The functions and are called the Weyl solution and Weyl function for the boundary value problem , respectively. Using the solution defined in the previous sections one has where is a solution of (1) satisfying the conditions , and the jump conditions (3), and is defined from the equality Note that, by virtue of equalities and (60), one has
The following theorem shows that the Weyl function uniquely determines the potentials and the coefficients of the boundary value problem .
Theorem 16. If , then . Thus, the boundary value problem is uniquely defined by the Weyl function.
Proof. Since it is easy to observe that Let us define the matrix , where Then we have According to (60) and (65), for each fixed , the functions are meromorphic in with poles at points and . Denote. By virtue of (65), (66), and we get It follows from (60) and (66) that if , then for each fixed the functions are entire in . Together with (69) this yields . Now using (67), we obtain Therefore, for , we have where for and for . Similarly, one can calculate Finally, taking into account the relations and (65), we have , that is, for all and . Consequently, . The theorem is proved.
The following two theorems show that two spectra and spectral data also uniquely determine the potentials and the coefficients of the boundary value problem .
Theorem 17. If , then .
Proof. It is obvious that characteristic functions and are uniquely determined by the sequences and , respectively. If , then . It follows from (60) that . Therefore, applying Theorem 16, we conclude that . The proof is completed.
Theorem 18. If , then , that is, spectral data uniquely determines the problem .
Proof. It is obvious that the Weyl function is meromorphic with simple poles at points . Using the expression and equalities , we have Since the Weyl function is regular for , applying the Rouche theorem, we conclude that Taking (60) and (63) into account, we arrive at . Therefore where . Hence, by the residue theorem, we have Finally, from the equality , applying Theorem 16, we conclude that . The theorem is proved.
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