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Murat Sat, Etibar S. Panakhov, "A Uniqueness Theorem for Bessel Operator from Interior Spectral Data", Abstract and Applied Analysis, vol. 2013, Article ID 713654, 6 pages, 2013. https://doi.org/10.1155/2013/713654
A Uniqueness Theorem for Bessel Operator from Interior Spectral Data
Inverse problem for the Bessel operator is studied. A set of values of eigenfunctions at some internal point and parts of two spectra are taken as data. Uniqueness theorems are obtained. The approach that was used in investigation of problems with partially known potential is employed.
Inverse spectral analysis involves the problem of restoring a linear operator from some of its spectral parameters. Currently, inverse problems are being studied for certain special classes of ordinary differential operators. The simplest of these is the Sturm-Liouville operator . For the case where it is considered on the whole line or half line, the Sturm-Liouville operator together with the function has been called a potential. In this direction, Borg  gave important results. He showed that, in general, one spectrum does not determine a Sturm-Liouville operator, so the result of Ambarzumyan  is an exception to the general rule. In the same paper, Borg showed that two spectra of a Sturm-Liouville operator determine it uniquely. Later, Levinson , Levitan , and Hochstadt  showed that when the boundary condition and one possible reduced spectrum are given, then the potential is uniquely determined. Using spectral data, that is, the spectral function, spectrum, and norming constant, different methods have been proposed for obtaining the potential function in a Sturm-Liouville problem. Such problems were subsequently investigated by other authors [4–6]. On the other hand, inverse problems for regular and singular Sturm-Liouville operators have been extensively studied by [7–15].
The inverse problem for interior spectral data of the differential operator consists in reconstruction of this operator from the known eigenvalues and some information on eigenfunctions at some internal point. Similar problems for the Sturm-Liouville operator and discontinuous Sturm-Liouville problem were formulated and studied in [16, 17].
The main goal of the present work is to study the inverse problem of reconstructing the singular Sturm-Liouville operator on the basis of spectral data of a kind: one spectrum and some information on eigenfunctions at the internal point.
Consider the following singular Sturm-Liouville operator satisfying (1)–(3): with boundary conditions, where is a real-valued function and , spectral parameter, , . The operator is self adjoint on the and has a discrete spectrum .
Let us introduce the second singular Sturm-Liouville operator satisfying subject to the same boundary conditions (2), (3), where is a real-valued function and . The operator is self adjoint on the and has a discrete spectrum .
2. Main Results
Before giving some results concerning the Bessel equation, we should give its physical properties. The total energy of the particle is given by , where is its initial or final momentum, and the corresponding wave number, planck constant, particle's mass, and energy. The reduced radial Schrödinger equation for the partial wave of angular momentum then reads  When , the above equation reduces to the classical Bessel equation in the form This equation has the solution , called the Bessel function.
Eigenvalues of the problem (1)–(3) are the roots of (3). This spectral characteristic satisfies the following asymptotic expression [19, 20]: where the series . Next, we present the main results in this paper. When , we get the following uniqueness Theorem 1.
Theorem 1. If for every one has then
In the case , the uniqueness of can be proved if we require the knowledge of a part of the second spectrum.
Let be a sequence of natural numbers with a property
Lemma 2. Let be a sequence of natural numbers satisfying (10) and are so chosen that . If for any then
Let and be a sequence of natural numbers such that and let be the eigenvalues of (1), (2), and (15) and be the eigenvalues of (4), (2), and (15) Using Mochizuki and Trooshin's method from Lemma 2 and Theorem 1, we will prove that the following Theorem 3 holds.
3. Proof of the Main Results
In this section, we present the proofs of main results in this paper.
Proof of Theorem 1. Before proving Theorem 1, we will mention some results, which will be needed later. We get the initial value problems
As known from , Bessel's functions of the first kind of order are
and asymptotic formulas for large argument
It can be shown  that there exists a kernel continuous in the triangle such that by using the transformation operator every solution of (18), (19) and (20), (21) can be expressed in the form [8, 21], respectively, where the kernel is the solution of the equation subject to the boundary conditions After the transformations we obtain the following problem:
This problem can be solved by using the Riemann method .
Multiplying (18) by and (20) by , subtracting and integrating from to , we obtain The functions and satisfy the same initial conditions (19) and (21), that is, Let If the properties of and are considered, the function is an entire function.
Therefore the condition of Theorem 1 implies and hence
In addition, using (24) and (33) for , where is constant.
Introduce the function By using the asymptotic forms of and , we obtain The zeros of are the eigenvalues of and hence it has only simple zeros because of the seperated boundary conditions. From (38), is an entire function of order of . Since the set of zeros of the entire function is contained in the set of zeros of , we see that the function is an entire function on the parameter . From (36), (38), and (39), we get So, for all , from the Liouville theorem, or
It was proved in  that there exists absolutely continuous function such that we have where We are now going to show that a.e. on . From (33), (43) we have This can be written as Let along the real axis, by the Riemann-Lebesgue lemma, one should have Thus from the completeness of the functions , it follows that But this equation is a homogeneous Volterra integral equation and has only the zero solution. Thus we have obtained or almost everywhere on . Therefore Theorem 1 is proved.
Theorem 4. To prove that on almost everywhere, we should repeat the above arguments for the supplementary problem subject to the boundary conditions
Next, we show that Lemma 2 holds.
Proof of Lemma 2. As in the proof of Theorem 1 we can show that
where and . From the assumption
together with the initial condition at it follows that,
Next, we will show that on the whole plane. The asymptotics (23) imply that the entire function is a function of exponential type .
Define the indicator of function by Since , from (23) it follows that Let us denote by the number of zeros of in the disk . According to  set of zeros of every entire function of the exponential type, not identically zero, satisfies the inequality where is the number of zeros of in the disk . By (58), From the assumption and the known asymptotic expression (7) of the eigenvalues we obtain
For the case , The inequalities (59) and (62) imply that on the whole plane.
Similar to the proof of Theorem 1, we have This completes the proof of Lemma 2.
Now we prove that Theorem 3 is valid.
Proof of Theorem 3. From
where satisfies (14) and . Similar to the proof of Lemma 2, we get
Thus, it needs to be proved that a.e on . The eigenfunctions and satisfy the same boundary condition at . It means that
on for any where are constants.
Let , . From (54) and (66) we obtain
We are going to show that inequality (59) fails and consequently, the entire function of exponential type vanishes on the whole -plane. The and have the same asymptotics (7). Counting the number of and located inside the disc of radius , we have of 's and of 's.
This means that Repeating the last part of the proof of Lemma 2, and considering the condition , we can show that identically on the whole -plane which implies that and consequently
Hence the proof of Theorem 3 is completed.
The authors would like to thank the referees for valuable comments in improving the original paper.
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