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.


Introduction
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 [1] gave important results. He showed that, in general, one spectrum does not determine a Sturm-Liouville operator, so the result of Ambarzumyan [2] 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 [3], Levitan [4], and Hochstadt [5] 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][5][6]. On the other hand, inverse problems for regular and singular Sturm-Liouville operators have been extensively studied by [7][8][9][10][11][12][13][14][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.

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 = 2 /2 = ℎ 2 2 /2 = 2 , where is its initial or final momentum, and the corresponding wave number, ℎ planck constant, particle's mass, and energy. The reduced radial Schrödinger equation for the partial wave of angular momentum ℓ then reads [18] This equation has the solution ℓ ( ), called the Bessel function.
In the case ̸ = 1/2, the uniqueness of ( ) can be proved if we require the knowledge of a part of the second spectrum.

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 (0) = 0.
As known from [18], Bessel's functions of the first kind of order V = ℓ − 1/2 are Abstract and Applied Analysis 3 and asymptotic formulas for large argument It can be shown [19] that there exists a kernel ( , )(̃( , )) continuous in the triangle 0 ≤ ≤ ≤ 1 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 we obtain the following problem: This problem can be solved by using the Riemann method [21].
The functions ( , ) and̃( , ) satisfy the same initial conditions (19) and (21), that is, Let If the properties of ( , ) and̃( , ) are considered, the function ( ) is an entire function. Therefore the condition of Theorem 1 implies and hence In addition, using (24) and (33) for 0 < < 1, 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 1/2 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, It was proved in [19] that there exists absolutely continuous functioñ( , ) such that we have We are now going to show that ( ) = 0 a.e. on (0, 1/2]. From (33), (43) we have This can be written as Thus from the completeness of the functions cos, it follows that But this equation is a homogeneous Volterra integral equation and has only the zero solution. Thus we have obtained or̃( almost everywhere on (0, 1/2]. Therefore Theorem 1 is proved. where = √ = and ( ) = ( ) −̃( ). From the assumption together with the initial condition at 0 it follows that, Next, we will show that ( ) = 0 on the whole plane. The asymptotics (23) imply that the entire function ( ) is a function of exponential type ≤ 2 . Define the indicator of function ( ) by Since | Im √ | = | sin |, = arg √ from (23) it follows that ℎ ( ) ≤ 2 |sin | .
Let us denote by ( ) the number of zeros of ( ) in the disk {| | ≤ }. According to [22] 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 > 2 , The inequalities (59) This completes the proof of Lemma 2.
Now we prove that Theorem 3 is valid.

Proof of Theorem 3. From
where { ( )} ∈N satisfies (14) 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 Repeating the last part of the proof of Lemma 2, and considering the condition 1 > 2 − 1, we can show that ( ) = 0 identically on the whole -plane which implies that ( ) =̃( ) a.e on (0, ] and consequently ( ) =̃( ) a.e on (0, 1) .
Hence the proof of Theorem 3 is completed.