A Uniqueness Theorem for Bessel Operator from Interior Spectral Data

Abstract

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.

1. 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 𝐿𝑦=−𝑦^′′+𝑞(𝑥)𝑦. For the case where it is considered on the whole line or half line, the Sturm-Liouville operator together with the function 𝑞(𝑥) 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 [4–6]. 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 𝐿 satisfying (1)–(3): 𝐿𝑦=−𝑦^′′+ℓ(ℓ+1)𝑥²+𝑞(𝑥)𝑦=𝜆𝑦,0<𝑥<1 (1) with boundary conditions, 𝑦(0)𝑦=0,_′(1,𝜆)+𝐻𝑦(1,𝜆)=0, (2)(3) where 𝑞(𝑥) is a real-valued function and 𝑞∈𝐿₂(0,1), 𝜆 spectral parameter, ℓ∈ℕ₀,  𝐻∈ℝ. The operator 𝐿 is self adjoint on the 𝐿₂(0,1) and has a discrete spectrum {𝜆_𝑛}.

Let us introduce the second singular Sturm-Liouville operator 𝐿^∼ satisfying 𝐿^∼𝑦=−𝑦^′′+ℓ(ℓ+1)𝑥²+𝑞^∼(𝑥)𝑦=𝜆𝑦,0<𝑥<1, (4) subject to the same boundary conditions (2), (3), where 𝑞^∼(𝑥) is a real-valued function and 𝑞^∼∈𝐿₂(0,1). The operator 𝐿^∼ is self adjoint on the 𝐿₂(0,1) and has a discrete spectrum {𝜆^∼_𝑛}.

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 𝐸=𝑝²/2𝑀=ℏ²𝑘²/2𝑀=𝑘², where 𝑝 is its initial or final momentum, and 𝑘 the corresponding wave number, ℏ planck constant, 𝑀 particle's mass, and 𝐸 energy. The reduced radial Schrödinger equation for the partial wave of angular momentum ℓ then reads [18] 𝑑²𝑑𝑟²Ψ₁(𝑘,𝑟)+𝑘²−ℓ(ℓ+1)𝑟²Ψ₁(𝑘,𝑟)=𝑉(𝑟)Ψ₁(𝑘,𝑟). (5) When 𝑉=0, the above equation reduces to the classical Bessel equation in the form 𝑑²𝑑𝑟²Ψ₁(𝑘,𝑟)+𝑘²−ℓ(ℓ+1)𝑟²Ψ₁(𝑘,𝑟)=0. (6) This equation has the solution 𝐽_ℓ(𝑟), 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]: 𝜆_𝑛=ℓ𝑛+2²𝜋²+¹₀𝑞(𝑥)𝑑𝑥−𝑙(𝑙+1)+𝑎_𝑛, (7) where the series ∑^∞_𝑛=1𝑎²_𝑛<∞. Next, we present the main results in this paper. When 𝑏=1/2, we get the following uniqueness Theorem 1.

Theorem 1. If for every 𝑛∈ℕ one has 𝜆_𝑛=𝜆^∼_𝑛,𝑦^′_𝑛(1/2)𝑦_𝑛()=1/2^∼𝑦^′_𝑛(1/2_∼)𝑦_𝑛(1/2) (8) then 𝑞(𝑥)=𝑞^∼(𝑥)𝑎.𝑒𝑜𝑛𝑡ℎ𝑒𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙(0,1). (9)

In the case 𝑏≠1/2, the uniqueness of 𝑞(𝑥) can be proved if we require the knowledge of a part of the second spectrum.

Let {𝑚(𝑛)}_𝑛∈ℕ be a sequence of natural numbers with a property 𝑚(𝑛)=𝑛𝜎1+𝜀_𝑛,0<𝜎≤1,𝜀_𝑛⟶0. (10)

Lemma 2. Let {𝑚(𝑛)}_𝑛∈ℕ be a sequence of natural numbers satisfying (10) and 𝑏∈(0,1/2) are so chosen that 𝜎>2𝑏. If for any 𝑛∈ℕ𝜆_𝑚(𝑛)=𝜆^∼_𝑚(𝑛),𝑦^′_𝑚(𝑛)(𝑏)𝑦_𝑚(𝑛)(𝑏)=^∼𝑦^′_𝑚(𝑛)(𝑏)_∼𝑦_𝑚(𝑛)(𝑏) (11) then 𝑞(𝑥)=𝑞^∼(𝑥)𝑎.𝑒𝑜𝑛(0,𝑏]. (12)

Let {𝑙(𝑛)}_𝑛∈ℕ and {𝑟(𝑛)}_𝑛∈ℕ be a sequence of natural numbers such that 𝑙(𝑛)=𝑛𝜎₁1+𝜀_1,𝑛,0<𝜎₁≤1,𝜀_1,𝑛𝑟(𝑛)=𝑛⟶0,𝜎₂1+𝜀_2,𝑛,0<𝜎₂≤1,𝜀_2,𝑛⟶0 (13)(14) and let 𝜇_𝑛 be the eigenvalues of (1), (2), and (15) and 𝜇^∼_𝑛 be the eigenvalues of (4), (2), and (15) 𝑦^′(1,𝜆)+𝐻₁𝑦(1,𝜆)=0,𝐻≠𝐻₁. (15) Using Mochizuki and Trooshin's method from Lemma 2 and Theorem 1, we will prove that the following Theorem 3 holds.

Theorem 3. Let {𝑙(𝑛)}_𝑛∈ℕ and {𝑟(𝑛)}_𝑛∈ℕ be a sequence of natural numbers satisfying (13) and (14), and 1/2<𝑏<1 are so chosen that 𝜎₁>2𝑏−1,𝜎₂>2−2𝑏. If for any 𝑛∈ℕ one has 𝜆_𝑛=𝜆^∼_𝑛,𝜇_𝑙(𝑛)=𝜇^∼_𝑙(𝑛),𝑦^′_𝑟(𝑛)(𝑏)𝑦_𝑟(𝑛)(𝑏)=^∼𝑦^′_𝑟(𝑛)(𝑏)_∼𝑦_𝑟(𝑛)(𝑏) (16) then 𝑞(𝑥)=𝑞^∼(𝑥)𝑎.𝑒𝑜𝑛(0,1). (17)

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 −𝑦^′′+ℓ(ℓ+1)𝑥²(𝑥𝑦(0)−+𝑞)𝑦=𝜆𝑦,0<𝑥<1,=0,_∼𝑦^′′+ℓ(ℓ+1)𝑥²+𝑞^∼(𝑥)𝑦^∼=𝜆𝑦^∼,0<𝑥<1,_∼𝑦(0)=0. (18)(19)(20)(21) As known from [18], Bessel's functions of the first kind of order 𝑣=ℓ−1/2 are 𝐽_𝑣(𝑥)=^∞_𝑘=0(−1)^𝑘𝑥^𝑣+2𝑘2^𝑣+2𝑘𝑘!Γ(𝑣+𝑘+1) (22) and asymptotic formulas for large argument 𝐽_𝑣(𝑥)=2𝜋𝑥cos𝑥−𝑣𝜋2−𝜋41+𝑂𝑥,𝐽^′_𝑣(𝑥)=−2𝜋𝑥sin𝑥−𝑣𝜋2−𝜋41+𝑂𝑥. (23)
It can be shown [19] that there exists a kernel 𝐻(𝑥,𝑡)(𝐻^∼(𝑥,𝑡)) continuous in the triangle 0≤𝑡≤𝑥≤1 such that by using the transformation operator every solution of (18), (19) and (20), (21) can be expressed in the form [8, 21], 𝑦()=√𝑥,𝜆𝑥√𝜆^𝑣𝐽_𝑣√𝜆𝑥+^𝑥₀𝐻()√𝑥,𝑡𝑡√𝜆^𝑣𝐽_𝑣√𝜆𝑡𝑑𝑡,_∼𝑦()=√𝑥,𝜆𝑥√𝜆^𝑣𝐽_𝑣√𝜆𝑥+𝐻^𝑥₀^∼()√𝑥,𝑡𝑡√𝜆^𝑣𝐽_𝑣√𝜆𝑡𝑑𝑡, (24)(25) respectively, where the kernel 𝐻(𝑥,𝑡)  (𝐻^∼(𝑥,𝑡)) is the solution of the equation 𝜕²𝐻(𝑥,𝑡)𝜕𝑥²+ℓ(ℓ+1)𝑥²𝐻()=𝜕𝑥,𝑡₂𝐻(𝑥,𝑡)𝜕𝑡²+ℓ(ℓ+1)𝑡²+𝑞(𝑡)𝐻(𝑥,𝑡) (26) subject to the boundary conditions 2𝑑𝐻(𝑥,𝑥)(𝑥),𝑑𝑥=𝑞lim_𝑡→0𝐻(𝑥,𝑡)𝑡^𝑣−1/2=0,𝐽^′_𝑣(𝑡,𝜆)=𝑂𝑡^𝑣−1/2. (27) After the transformations 1𝜉=4(𝑥+𝑡)²1,𝜂=4(𝑥−𝑡)²,𝐻(𝑥,𝑡)=𝜉−𝜂^−𝑣+1/2𝑈𝜉,𝜂, (28) we obtain the following problem: 𝜕²𝑈−1𝜕𝜉𝜕𝜂4𝜉−𝜂𝜕𝑈+1𝜕𝜉4𝜉−𝜂𝜕𝑈=1𝜕𝜂4𝑞𝜉𝜂𝜉+√𝜂𝑈()𝑈,𝜉,𝜉=0,𝜕𝑈+𝛼𝜕𝜉𝜉1𝑈=4𝑞𝜉𝜉^𝑣−11,𝛼=−𝑣+2. (29)
This problem can be solved by using the Riemann method [21].
Multiplying (18) by 𝑦^∼(𝑥,𝜆) and (20) by 𝑦(𝑥,𝜆), subtracting and integrating from 0 to 1/2, we obtain ₀^1/2𝑞(𝑥)−𝑞^∼(𝑥)𝑦(𝑥,𝜆)𝑦^∼()=𝑥,𝜆𝑑𝑥_∼𝑦(𝑥,𝜆)𝑦^′(𝑥,𝜆)−𝑦(𝑥,𝜆)𝑦^∼′(𝑥,𝜆)₀^1/2. (30) The functions 𝑦(𝑥,𝜆) and 𝑦^∼(𝑥,𝜆) satisfy the same initial conditions (19) and (21), that is, 𝑦^∼(0,𝜆)𝑦^′(0,𝜆)−𝑦(0,𝜆)𝑦^∼′(0,𝜆)=0. (31) Let 𝑄(𝑥)=𝑞(𝑥)−𝑞^∼(𝑥),𝐾(𝜆)=₀^1/2𝑄(𝑥)𝑦(𝑥,𝜆)𝑦^∼(𝑥,𝜆)𝑑𝑥. (32)(33) If the properties of 𝑦(𝑥,𝜆) and 𝑦^∼(𝑥,𝜆) are considered, the function 𝐾(𝜆) is an entire function.
Therefore the condition of Theorem 1 implies 𝑦^∼12,𝜆_𝑛𝑦^′12,𝜆_𝑛1−𝑦2,𝜆𝑦_𝑛^∼′12,𝜆_𝑛=0 (34) and hence 𝐾𝜆_𝑛=0,𝑛∈ℕ. (35)
In addition, using (24) and (33) for 0<𝑥<1, |𝐾(𝜆1)|≤𝑀𝜆^𝑣, (36) where 𝑀 is constant.
Introduce the function 𝑊(𝜆)=𝑦^′(1,𝜆)+𝐻𝑦(1,𝜆). (37) By using the asymptotic forms of 𝑦 and 𝑦^′, we obtain 𝑊(𝜆)=√𝜆sin√𝜆−𝑣𝜋2−𝜋4+𝑂(1). (38) The zeros of 𝑊(𝜆) are the eigenvalues of 𝐿 and hence it has only simple zeros 𝜆_𝑛 because of the seperated boundary conditions. From (38), 𝑊(𝜆) is an entire function of order 1/2 of 𝜆. Since the set of zeros of the entire function 𝑊(𝜆) is contained in the set of zeros of 𝐾(𝜆), we see that the function Ψ(𝜆)=𝐾(𝜆)𝑊(𝜆) (39) is an entire function on the parameter 𝜆. From (36), (38), and (39), we get |Ψ(𝜆1)|=𝑂𝜆^𝑣+1/2. (40) So, for all 𝜆, from the Liouville theorem, Ψ(𝜆)=0, (41) or 𝐾(𝜆)=0. (42)
It was proved in [19] that there exists absolutely continuous function 𝐻^∼∼(𝑥,𝜏) such that we have 𝑦(𝑥,𝜆)𝑦^∼()=1𝑥,𝜆21+cos2√𝜆𝑥−𝑣𝜋2−𝜋4+𝐻^𝑥₀^∼∼()√𝑥,𝜏×cos2𝜆𝜏−𝑣𝜋2−𝜋4𝑑𝜏, (43) where 𝐻^∼∼(𝑥,𝑡)=2𝐻(𝑥,𝑥−2𝜏)+𝐻^∼(𝑥,𝑥−2𝜏)+2^𝑥_{−𝑥+2𝜏}𝐻(𝑥,𝑠)𝐻^∼()+𝑥,𝑠−2𝜏𝑑𝑠_{𝑥−2𝜏−𝑥}𝐻(𝑥,𝑠)𝐻^∼(𝑥,𝑠+2𝜏)𝑑𝑠. (44) We are now going to show that 𝑄(𝑥)=0 a.e. on (0,1/2]. From (33), (43) we have 12₀^1/2𝑄(𝑥)1+cos2√𝜆𝑥−𝑣𝜋2−𝜋4+𝐻^𝑥₀^∼∼()√𝑥,𝜏×cos2𝜆𝜏−𝑣𝜋2−𝜋4𝑑𝜏𝑑𝑥=0. (45) This can be written as ₀^1/2𝑄(𝑥)𝑑𝑥+₀^1/2cos2√𝜆𝜏−𝑣𝜋2−𝜋4×𝑄(𝜏)+₀^1/2𝑄(𝑥)×𝐻^∼∼(𝑥,𝜏)𝑑𝑥𝑑𝜏=0. (46) Let 𝜆→∞ along the real axis, by the Riemann-Lebesgue lemma, one should have ₀^1/2𝑄(𝑥)𝑑𝑥=0,_01/2cos2√𝜆𝜏−𝑣𝜋2−𝜋4×𝑄(𝜏)+_𝜏^1/2𝑄(𝑥)𝐻^∼∼(𝑥,𝜏)𝑑𝑥𝑑𝜏=0. (47) Thus from the completeness of the functions cos, it follows that Q(𝜏)+_𝜏^1/2𝑄(𝑥)𝐻^∼∼()1𝑥,𝜏𝑑𝑥=0,0<𝑥<2. (48) But this equation is a homogeneous Volterra integral equation and has only the zero solution. Thus we have obtained 𝑄(𝑥)=𝑞(𝑥)−𝑞^∼(𝑥)=0, (49) or 𝑞^∼(𝑥)=𝑞(𝑥) (50) almost everywhere on (0,1/2]. Therefore Theorem 1 is proved.

Theorem 4. To prove that 𝑞(𝑥)=0 on [1/2,1) almost everywhere, we should repeat the above arguments for the supplementary problem 𝐿𝑦=−𝑦^′′+ℓ(ℓ+1)(1−𝑥)²+𝑞(1−𝑥)𝑦,0<𝑥<1 (51) subject to the boundary conditions 𝑦(1)𝑦=0,_′(0,𝜆)+𝐻𝑦(0,𝜆)=0. (52)

Consequently 𝑞(𝑥)=𝑞^∼(𝑥)a.eontheinterval(0,1). (53)

Next, we show that Lemma 2 holds.

Proof of Lemma 2. As in the proof of Theorem 1 we can show that 𝐺𝜌=^𝑏₀𝑄(𝑥)𝑦(𝑥,𝜆)𝑦^∼()=𝑥,𝜆𝑑𝑥_∼𝑦(𝑥,𝜆)𝑦^′(𝑥,𝜆)−𝑦(𝑥,𝜆)𝑦^∼′(𝑥,𝜆)_𝑥=𝑏, (54) where 𝜌=√𝜆=𝑟𝑒^𝑖𝜃 and 𝑄(𝑥)=𝑞(𝑥)−𝑞^∼(𝑥). From the assumption 𝑦^′_𝑚(𝑛)(𝑏)𝑦_𝑚(𝑛)(𝑏)=^∼𝑦^′_𝑚(𝑛)(𝑏)_∼𝑦_𝑚(𝑛)(𝑏) (55) together with the initial condition at 0 it follows that, 𝐺𝜌_𝑚(𝑛)=0,𝑛∈ℕ. (56) Next, we will show that 𝐺(𝜌)=0 on the whole 𝜌 plane. The asymptotics (23) imply that the entire function 𝐺(𝜌) is a function of exponential type ≤2𝑏.
Define the indicator of function 𝐺(𝜌) by ℎ(𝜃)=lim_𝑟→∞𝐺supln𝑟𝑒^𝑖𝜃𝑟. (57) Since |Im√𝜆|=𝑟|sin𝜃|, 𝜃=arg√𝜆 from (23) it follows that ℎ(𝜃)≤2𝑏|sin𝜃|. (58) Let us denote by 𝑛(𝑟) the number of zeros of 𝐺(𝜌) in the disk {|𝜌|≤𝑟}. According to [22] set of zeros of every entire function of the exponential type, not identically zero, satisfies the inequality lim_𝑟→∞𝑛(𝑟)inf𝑟≤12𝜋₀^2𝜋ℎ(𝜃)𝑑𝜃, (59) where 𝑛(𝑟) is the number of zeros of 𝐺(𝜌) in the disk |𝜌|≤𝑟. By (58), 12𝜋₀^2𝜋ℎ(𝜃)𝑏𝑑𝜃≤𝜋₀^2𝜋|sin𝜃|𝑑𝜃=4𝑏𝜋. (60) From the assumption and the known asymptotic expression (7) of the eigenvalues 𝜆_𝑛 we obtain 𝑛(𝑟)≥2_{((𝜋𝑛/𝜎)[1+𝑂1/𝑛)]<𝑟}21=𝜋𝜎𝑟(1+𝑜(1)),𝑟⟶∞. (61)
For the case 𝜎>2𝑏, lim_𝑟→∞𝑛(𝑟)𝑟≥2𝜋𝜎>4𝑏𝜋=2𝑏₀^2𝜋||1sin𝜃𝑑𝜃≥2𝜋₀^2𝜋ℎ(𝜃)𝑑𝜃. (62) The inequalities (59) and (62) imply that 𝐺(𝜌)=0 on the whole 𝜌 plane.
Similar to the proof of Theorem 1, we have 𝑞(𝑥)=𝑞^∼(𝑥)a.eontheinterval(0,𝑏]. (63) This completes the proof of Lemma 2.

Now we prove that Theorem 3 is valid.

Proof of Theorem 3. From 𝜆_𝑟(𝑛)=𝜆^∼_𝑟(𝑛),𝑦^′_𝑟(𝑛)(𝑏)𝑦_𝑟(𝑛)(𝑏)=^∼𝑦^′_𝑟(𝑛)(𝑏)_∼𝑦_𝑟(𝑛)(𝑏), (64) where {𝑟(𝑛)}_𝑛∈ℕ satisfies (14) and 𝜎₂>2−2𝑏. Similar to the proof of Lemma 2, we get 𝑞(𝑥)=𝑞^∼(𝑥)a.eon[𝑏,1). (65) Thus, it needs to be proved that 𝑞(𝑥)=𝑞^∼(𝑥) a.e on (0,𝑏]. The eigenfunctions 𝑦_𝑛(𝑥,𝜆_𝑛) and 𝑦^∼_𝑛(𝑥,𝜆_𝑛) satisfy the same boundary condition at 1. It means that 𝑦_𝑛𝑥,𝜆_𝑛=𝜉𝑦_𝑛^∼_𝑛𝑥,𝜆_𝑛 (66) on [𝑏,1] for any 𝑛∈ℕ where 𝜉_𝑛 are constants.
Let 𝜌_𝑛=𝜆_𝑛, 𝑠_𝑛=√𝜇_𝑛. From (54) and (66) we obtain 𝐺𝜌_𝑛𝐺𝑠=0,𝑛∈ℕ,_𝑙𝑛=0,𝑛∈ℕ. (67)
We are going to show that inequality (59) fails and consequently, the entire function of exponential type 𝐺(𝜌) vanishes on the whole 𝜌-plane. The 𝜌_𝑛 and 𝑠_𝑛 have the same asymptotics (7). Counting the number of 𝜌_𝑛 and 𝑠_𝑛 located inside the disc of radius 𝑟, we have 21+𝜋𝑟11+𝑂𝑛 (68) of 𝜌_𝑛's and 21+𝜋𝑟𝜎₁11+𝑂𝑛. (69) of 𝑠_𝑛's.
This means that 𝑛(𝑟)2=2+𝜋𝑟𝜎₁1+1+𝑂𝑛,lim_𝑟→∞𝑛(𝑟)𝑟=2𝜋𝜎₁+1. (70) Repeating the last part of the proof of Lemma 2, and considering the condition 𝜎₁>2𝑏−1, we can show that 𝐺(𝜌)=0 identically on the whole 𝜌-plane which implies that 𝑞(𝑥)=𝑞^∼(𝑥)a.eon(0,𝑏] (71) and consequently 𝑞(𝑥)=𝑞^∼(𝑥)a.eon(0,1). (72)
Hence the proof of Theorem 3 is completed.

Acknowledgment

The authors would like to thank the referees for valuable comments in improving the original paper.