Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse

Abstract

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.

1. Introduction

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 [1], Levitan [2], Beals et al. [3], and Yurko [4].

In the present work we consider some inverse problems for the boundary value problem generated by the differential equation 𝐿_𝜆𝑦∶=𝑦^′′+𝜆²(𝑥)(𝑥[)∪(]−2𝜆𝑝−𝑞)𝑦=0,𝑥∈0,𝑎𝑎,𝜋 (1) with the boundary conditions 𝑈𝑦∶=𝑦^′(0)=0,𝑉𝑦∶=𝑦(𝜋)=0 (2) and with the jump conditions 𝑦(𝑎+0)=𝛼𝑦(𝑎−0),𝑦^′(𝑎+0)=𝛼⁻¹𝑦^′(𝑎−0), (3) where 𝜆 is the spectral parameter, 𝑝(𝑥)∈𝑊¹₂[0,𝜋],𝑞(𝑥)∈𝐿₂[0,𝜋] are real functions, 𝛼 is a real number, and 𝛼>0,𝛼≠1,𝑎∈(𝜋/2,𝜋). Here we denote by 𝑊^𝑚₂[0,𝜋] the space of functions 𝑓(𝑥), 𝑥∈[0,𝜋], such that the derivatives 𝑓^(𝑚)(𝑥) (𝑚=0,𝑛−1) are absolutely continuous and 𝑓^(𝑛)(𝑥)∈𝐿₂[0,𝜋].

There exist many papers containing a fairly comprehensive analysis of direct and inverse problems of spectral analysis of the Sturm-Liouville equation 𝑙𝑦∶=−𝑦^′′+𝑞(𝑥)𝑦=𝜆²𝑦, (4) a special case (𝑝(𝑥)≡0) of (1). For instance, inverse problems for a regular Sturm-Liouville operator with separated boundary conditions have been investigated in [5] (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 𝐿_𝜆 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 [9]. 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 [10] (see also [11]) where the corresponding results of the monograph [1] were extended to the case 𝑝(𝑥)≠0. The inverse periodic problem for the pencil 𝐿_𝜆 was solved in [12] using another approach. We also point out the paper [14], in which the uniqueness of the recovery of the pencil 𝐿_𝜆 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 [19].

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 𝐿(𝑎,𝛼). 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 𝐿(𝑎,𝛼). In Section 3 the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers of 𝐿(𝑎,𝛼) are obtained. Finally, in Section 4 three inverse problems of reconstructing the boundary value problem 𝐿(𝛼,𝑎) 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

Let 𝑓_𝜈(𝑥,𝜆)(𝜈=1,2) be solution of (1) under the initial condition 𝑓_𝜈(0,𝜆)=1,𝑓^′_𝜈(0,𝜆)=𝜆𝑤_𝜈 (5) and discontinuity conditions (3), where 𝑤_𝜈=(−1)^𝜈+1𝑖.

It is obvious that the functions 𝑓_𝜈(𝑥,𝜆) satisfy the integral equations 𝑓_𝜈(𝑥,𝜆)=𝑙⁺(𝑥)𝑒^{𝜆𝑤𝜈𝑥}+𝑙⁻(𝑥)𝑒^{𝜆𝑤𝜈(2𝑎−𝑥)}+𝑙⁺(𝑥)^𝑎₀sin𝜆(𝑥−𝑡)𝜆𝑞(𝑡)+2𝜆𝑝(𝑡)𝑓_𝜈(𝑡,𝜆)𝑑𝑡−𝑙⁻(𝑥)^𝑎₀sin𝜆(𝑥+𝑡−2𝑎)𝜆𝑞(𝑡)+2𝜆𝑝(𝑡)𝑓_𝜈(𝑡,𝜆)𝑑𝑡+^𝑥_𝑎sin𝜆(𝑥−𝑡)𝜆𝑞(𝑡)+2𝜆𝑝(𝑡)𝑓_𝜈(𝑡,𝜆)𝑑𝑡, (6) where 𝑙^±(𝑥)=(1/2)(𝑙(𝑥)±(1/𝑙(𝑥)) and 𝑙(𝑥)=1,0≤𝑥<𝑎𝛼,𝑎<𝑥≤𝜋. (7)

Using the integral equations (6) and standard successive approximation methods [7, 9, 11], the following theorem is proved.

Theorem 1. If 𝑞(𝑥)∈𝐿₂[−𝑏,𝑏],𝑝(𝑥)∈𝑊¹₂[−𝑏,𝑏](0<𝑏<𝜋), then the solution 𝑓_𝜈(𝑥,𝜆) has the form 𝑓_𝜈(𝑥,𝜆)=𝑓_0𝜈(𝑥,𝜆)+^𝑥_−𝑥𝐴_𝜈(𝑥,𝑡)𝑒^{𝜆𝑤𝜈𝑡}𝑑𝑡, (8) where 𝑓_0𝜈(𝑥,𝜆)=𝑙⁺(𝑥)𝑒^{𝜆𝑤𝜈𝑥}𝑅⁺_𝜈(𝑥)+𝑙⁻(𝑥)𝑒^{𝜆𝑤𝜈(2𝑎−𝑥)}𝑅⁻_𝜈(𝑥),𝑅^±_𝜈(𝑥)=𝑒^{∓𝑤𝜈𝛽±(𝑥)},𝛽^±(𝑥)=^𝑥_{(𝑎∓𝑎)/2}𝑝(𝑡)𝑑𝑡, (9) and the function 𝐴_𝜈(𝑥,𝑡) satisfies the inequality ^𝑥_−𝑥𝐴_𝜈(𝑥,𝑡)𝑑𝑡≤𝑒^{𝑐𝜎(𝑥)}−1 (10) with 𝜎(𝑥)=^𝑥₀𝑞(𝑡𝑝(𝑡𝛼(𝑥−𝑡))+2)𝑑𝑡,𝑐=2⁺+|𝛼⁻|+1,𝛼^±=121𝛼±𝛼. (11)

Let 𝜑(𝑥,𝜆) be the solution of (1) that satisfies the initial conditions 𝜑(0,𝜆)=1,𝜑^′(0,𝜆)=0, (12) and the jump condition (3).

Then by using Theorem 1, we can formulate the following assertion.

Theorem 2. Let 𝑞(𝑥)∈𝐿₂[0,𝜋],𝑝(𝑥)∈𝑊¹₂[0,𝜋]. Then there are the functions 𝐴(𝑥,𝑡),𝐵(𝑥,𝑡) whose first order partial derivatives are summable on [0,𝜋] for each 𝑥∈[0,𝜋] such that the representation𝜑(𝑥,𝜆)=𝜑₀(𝑥,𝜆)+^𝑥₀𝐴(𝑥,𝑡)cos𝜆𝑡𝑑𝑡+^𝑥₀𝐵(𝑥,𝑡)sin𝜆𝑡𝑑𝑡 (13) is satisfied, where 𝜑₀(𝑥,𝜆)=𝑙⁺(𝑥)cos𝜆𝑥−𝛽⁺(𝑥)+𝑙⁻(𝑥)cos𝜆(2𝑎−𝑥)−𝛽⁻(𝑥). (14) Moreover, the relations 𝛼⁺𝛽⁺(𝑥)=𝛼⁺𝑥𝑝(0)+2^𝑥₀𝐴(𝜉,𝜉)sin𝛽⁺(𝜉)−𝐵(𝜉,𝜉)cos𝛽⁺(𝜉2𝑑)𝑑𝜉,𝐴()𝑑𝑥𝑥,𝑥cos𝛽⁺(𝑥)+𝐵(𝑥,𝑥)sin𝛽⁺(𝑥)=𝛼⁺𝑞(𝑥)+𝑝²(𝑥,2𝑑)𝐴()𝑑𝑥𝑥,𝑡cos𝛽⁻(𝑥)−𝐵(𝑥,𝑡)sin𝛽⁻(𝑥)^{𝑡=2𝑎−𝑥+0}_{𝑡=2𝑎−𝑥−0}=𝛼⁻𝑞(𝑥)+𝑝²(𝑥,𝐴_𝑡)(𝑥,𝑡)_𝑡=0=𝐵(𝑥,0)=0 (15) are held.

If one additionally supposes that 𝑞(𝑥)∈𝑊¹₂[0,𝜋],𝑝(𝑥)∈𝑊²₂[0,𝜋], then the functions 𝐴(𝑥,𝑡) and 𝐵(𝑥,𝑡) satisfy the system of partial differential equations 𝐴_𝑥𝑥(𝑥,𝑡)−𝑞(𝑥)𝐴(𝑥,𝑡)−2𝑝(𝑥)𝐵_𝑡(𝑥,𝑡)=𝐴_𝑡𝑡(),𝐵𝑥,𝑡_𝑥𝑥(𝑥,𝑡)−𝑞(𝑥)𝐵(𝑥,𝑡)+2𝑝(𝑥)𝐴_𝑡(𝑥,𝑡)=𝐵_𝑡𝑡(𝑥,𝑡). (16) Conversely, if the second order derivatives of functions 𝐴(𝑥,𝑡),𝐵(𝑥,𝑡) are summable on [0,𝜋] for each 𝑥∈[0,𝜋] and 𝐴(𝑥,𝑡),𝐵(𝑥,𝑡) satisfy equalities (16) and conditions (15), then the function 𝜑(𝑥,𝜆) which is defined by (13) is a solution of (1) satisfying initial conditions (12) and discontinuity conditions (3).

One here supposes that the function 𝑞(𝑥) satisfies the additional condition ^𝜋₀𝑦^′(𝑥)²+𝑞(𝑥)𝑦(𝑥)²𝑑𝑥>0 (17) for all 𝑦(𝑥)∈𝑊²₂[0,𝑎)∪(𝑎,𝜋] such that 𝑦(𝑥)≠0 and 𝑦^′(0)𝑦(0)−𝑦^′(𝜋)𝑦(𝜋)=0. (18)

Definition 3. A complex number 𝜆₀ is called an eigenvalue of the boundary value problem 𝐿(𝛼,𝑎) if (1) with 𝜆=𝜆₀ has a nontrivial solution 𝑦₀(𝑥) satisfying the boundary conditions (2) and the jump conditions (3). In this case 𝑦₀(𝑥) is called the eigenfunction of the problem 𝐿(𝛼,𝑎) corresponding to the eigenvalue 𝜆₀. The number of linearly independent solutions of the problem 𝐿(𝛼,𝑎) for a given eigenvalue 𝜆₀ is called the multiplicity of 𝜆₀.
The following lemmas can be proved analogously to the corresponding assertions in [11].

Lemma 4. The eigenvalues of the boundary value problem 𝐿(𝛼,𝑎) are real, nonzero, and simple.

Proof. We define a linear operator 𝐿₀ in the Hilbert space 𝐿₂[0,𝜋] as follows. The domain 𝐷(𝐿₀) consists of all functions 𝑦(𝑥)∈𝑊²₂[0,𝜋] satisfying the boundary conditions (2) and the jump conditions (3). For 𝑦∈𝐷(𝐿₀), we set 𝐿₀𝑦=−𝑦^′′+𝑞(𝑥)𝑦. Integration by part yields 𝐿₀𝑦,𝑦=^𝜋₀𝐿₀𝑦𝑦(𝑥)=𝑑𝑥_𝜋0𝑦^′(𝑥)²+𝑞(𝑥)𝑦(𝑥)²𝑑𝑥. (19)
Since condition (17) holds, it follows that (𝐿₀𝑦,𝑦)>0.
Let 𝜆₀ be an eigenvalue of the boundary value problem 𝐿(𝛼,𝑎) and 𝑦₀(𝑥) an eigenfunction corresponding to this eigenvalue and normalized by the condition (𝑦₀,𝑦₀)=1. By taking the inner product of both sides of the relation 𝑦₀^′′(𝑥)+[𝜆²₀−2𝜆₀𝑝(𝑥)−𝑞(𝑥)]𝑦₀(𝑥)=0 by 𝑦₀(𝑥), we obtain 𝜆²₀−2𝜆₀(𝑝𝑦₀,𝑦₀)−(𝐿₀𝑦₀,𝑦₀)=0 and hence 𝜆₀=𝑝𝑦₀,𝑦₀±𝑝𝑦₀,𝑦₀²+𝐿₀𝑦₀,𝑦₀. 19^′
The desired assertion follows from the last relation by virtue of (𝐿₀𝑦₀,𝑦₀)>0 with regard to the fact that 𝑝(𝑥) is real.
Let us show that 𝜆₀ is a simple eigenvalue. Assume that this is not true. Suppose that 𝑦₁(𝑥) and 𝑦₂(𝑥) are linearly independent eigenfunctions corresponding to the eigenvalue 𝜆₀. Then for a given value of 𝜆₀, each solution 𝑦₀(𝑥) of (1) will be given as linear combination of solutions 𝑦₁(𝑥) and 𝑦₂(𝑥). Moreover it will satisfy boundary conditions (2) and discontinuity conditions (3). However, it is impossible.

Lemma 5. The problem (1)–(3) does not have associated functions.

Proof. Let 𝑦₀(𝑥) be an eigenfunction corresponding to eigenvalue 𝜆₀ and normalized by the condition (𝑦₀,𝑦₀)=1 of the problem (1)–(3). Suppose that 𝑦₁(𝑥) is an associated function of eigenfunction 𝑦₀(𝑥), that is, the following equalities hold: 𝜆²₀𝑦₀−2𝜆₀𝑝(𝑥)𝑦₀−𝐿₀𝑦₀𝜆=0,₂₀𝑦₁−2𝜆₀𝑝(𝑥)𝑦₁−𝐿₀𝑦₁+2𝜆₀−𝑝(𝑥)𝑦₀=0. (20)
If these equations are multiplied by 𝑦₁(𝑥) and 𝑦₀(𝑥), respectively, as inner product, subtracting them side by side and taking into our account that operator 𝐿₀ is symmetric, the function 𝑝(𝑥) and 𝜆₀ are real, we get 𝜆₀=(𝑝𝑦₀,𝑦₀). Due to the condition (6), 𝜆₀=(𝑝𝑦₀,𝑦₀) does not agree with 19^′. Therefore, the assertion is not true.

Lemma 6. Eigenfunctions corresponding to different eigenvalues of the problem 𝐿(𝛼,𝑎) are orthogonal in the sense of the equality 𝜆₁+𝜆₂𝑦₁,𝑦₂−2𝑝𝑦₁,𝑦₂=0, (21) where(⋅,⋅) denotes the inner product in 𝐿₂[0,𝜋].

Lemma 7. Let 𝑦(𝑥,𝜆) be a solution of (1) satisfying the condition (18) and the jump conditions (3). Then 𝜆 is real and nonzero and 𝜆−𝑝(^𝜋₀𝑥)𝑦(𝑥,𝜆)²𝑑𝑥≠0. (22) Moreover, the sign of the left-hand side of (22) is similar to the sign of 𝜆.

3. Properties of the Spectrum

In this section we investigate some spectral properties of the boundary value problem 𝐿(𝛼,𝑎).

Let 𝜓(𝑥,𝜆) be a solution of (1) with the conditions 𝜓(𝜋,𝜆)=0,𝜓^′(𝜋,𝜆)=1 and the jump conditions (3). It is clear that function 𝜓(𝑥,𝜆) is entire in 𝜆 for each fixed 𝑥.

Denote Δ(𝜆)=⟨𝜓(𝑥,𝜆),𝜑(𝑥,𝜆)⟩, where ⟨𝑦,𝑧⟩∶= 𝑦^′𝑧−𝑦𝑧^′. By virtue of Liouville's formula, the Wronskian ⟨𝜓(𝑥,𝜆),𝜑(𝑥,𝜆)⟩ does not depend on 𝑥. The function Δ(𝜆) is called the characteristic function of 𝐿(𝛼,𝑎). Obviously, the function Δ(𝜆) is entire in 𝜆 and it has at most a countable set of zeros {𝜆_𝑛}.

Lemma 8. The zeros {𝜆_𝑛} of the characteristic function Δ(𝜆) coincide with the eigenvalues of the boundary value problem 𝐿(𝛼,𝑎). The functions 𝜑(𝑥,𝜆_𝑛) and 𝜓(𝑥,𝜆_𝑛) are eigenfunctions corresponding to the eigenvalue 𝜆_𝑛, and there exists a sequence {𝛽_𝑛} such that 𝜓𝑥,𝜆_n=𝛽_𝑛𝜑𝑥,𝜆_𝑛,𝛽_𝑛≠0. (23)

Proof. Let Δ(𝜆₀)=0. Then by virtue of ⟨𝜓(𝑥,𝜆₀),𝜑(𝑥,𝜆₀)⟩=0, 𝜑(𝑥,𝜆₀)=𝐶𝜓(𝑥,𝜆₀) for some constant 𝐶. Hence 𝜆₀ is an eigenvalue and 𝜑(𝑥,𝜆₀),𝜓(𝑥,𝜆₀) are eigenfunctions related to 𝜆₀.
Conversely, let 𝜆₀ be an eigenvalue of 𝐿(𝛼,𝑎), show that Δ(𝜆₀)=0. Assuming the converse suppose that Δ(𝜆₀)≠0. In this case the functions 𝜑(𝑥,𝜆₀) and 𝜓(𝑥,𝜆₀) are linearly independent. Then 𝑦(𝑥,𝜆₀)=𝑐₁𝜑(𝑥,𝜆₀)+𝑐₂𝜓(𝑥,𝜆₀) is a general solution of the problem 𝐿(𝛼,𝑎). If 𝑐₁≠0, we can write 𝜑𝑥,𝜆₀=1𝑐₁𝑦𝑥,𝜆₀−𝑐₂𝑐₁𝜓𝑥,𝜆₀. (24) Then we have 𝜑𝑥,𝜆₀,𝜓𝑥,𝜆₀=1𝑐₁𝑦^′𝜋,𝜆₀𝜓𝜋,𝜆₀−𝑦𝜋,𝜆₀𝜓^′𝜋,𝜆₀1=−𝑐₁𝑦𝜋,𝜆₀=0 (25) 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 𝛽_𝑛 such that 𝜓(𝑥,𝜆_𝑛)=𝛽_𝑛𝜑(𝑥,𝜆_𝑛).

Let us denote 𝛼_𝑛∶=^𝜋₀𝜑²𝑥,𝜆_𝑛1𝑑𝑥−𝜆_𝑛^𝜋₀𝑝(𝑥)𝜑²𝑥,𝜆_𝑛𝑑𝑥. (26)

The numbers {𝛼_𝑛} are called normalized numbers of the boundary value problem 𝐿(𝛼,𝑎).

Lemma 9. The equality Δ̇(𝜆_𝑛)=−2𝜆_𝑛𝛼_𝑛𝛽_𝑛 holds. Here Δ̇(𝜆)=(𝑑/𝑑𝜆)Δ(𝜆).

Proof. If we differentiate the equalities −𝜑^′′(𝑥,𝜆)+2𝜆𝑝(𝑥)+𝑞(𝑥)𝜑(𝑥,𝜆)=𝜆²𝜑(𝑥,𝜆),−𝜓^′′(𝑥,𝜆)+2𝜆𝑝(𝑥)+𝑞(𝑥)𝜓(𝑥,𝜆)=𝜆²𝜓(𝑥,𝜆) (27) with respect to 𝜆, we get −𝜑̇^′′(𝑥,𝜆)+2𝜆𝑝(𝑥)+𝑞(𝑥)𝜑̇(𝑥,𝜆)=𝜆²𝜑̇()(𝑥𝜑(),−𝜓̇𝑥,𝜆+2𝜆−𝑝)𝑥,𝜆_′′(𝑥,𝜆)+2𝜆𝑝(𝑥)+𝑞(𝑥)𝜓̇(𝑥,𝜆)=𝜆²𝜓̇(𝑥,𝜆)+2𝜆−𝑝(𝑥)𝜓(𝑥,𝜆). (28)
By virtue of these equalities we have 𝑑𝜑()𝜓̇𝑑𝑥𝑥,𝜆^′(𝑥,𝜆)−𝜑^′()𝜓̇((𝑥𝜑()𝜓(),𝑑𝑥,𝜆𝑥,𝜆)=2𝜆−𝑝)𝑥,𝜆𝑥,𝜆𝜑̇()𝜓𝑑𝑥𝑥,𝜆^′(𝑥,𝜆)−𝜑̇^′()𝜓((𝑥𝜑()𝜓().𝑥,𝜆𝑥,𝜆)=2𝜆−𝑝)𝑥,𝜆𝑥,𝜆 (29)
If the last equations are integrated from 𝑥 to 𝜋 and from 0 to 𝑥, respectively, by the discontinuity conditions we obtain 𝜑(𝜉,𝜆)𝜓̇^′(𝜉,𝜆)−𝜑^′(𝜉,𝜆)𝜓̇(𝜉,𝜆)^𝜋_𝑥=−2^𝜋_𝑥(𝜉𝜑()𝜓()𝜑̇()𝜓𝜆−𝑝)𝜉,𝜆𝜉,𝜆𝑑𝜉,𝜉,𝜆^′(𝜉,𝜆)−𝜑̇^′(𝜉,𝜆)𝜓(𝜉,𝜆)^𝑥₀=−2^𝑥₀𝜆−𝑝(𝜉)𝜑(𝜉,𝜆)𝜓(𝜉,𝜆)𝑑𝜉. (30) If we add the last equalities side by side, we get 𝑊𝜑̇(𝜉,𝜆),𝜓(𝜉,𝜆)+𝑊𝜑(𝜉,𝜆),𝜓̇(𝜉,𝜆)=−2^𝜋₀𝜆−𝑝(𝜉)𝜑(𝜉,𝜆)𝜓(𝜉,𝜆)𝑑𝜉 (31) or Δ̇(𝜆)=−2^𝜋₀𝜆−𝑝(𝜉)𝜑(𝜉,𝜆)𝜓(𝜉,𝜆)𝑑𝜉. (32) For 𝜆→𝜆_𝑛, this yields Δ̇𝜆_𝑛=−2^𝜋₀𝜆_𝑛−𝑝(𝜉)𝜑𝜉,𝜆_𝑛𝜓𝜉,𝜆_𝑛𝑑𝜉=−2𝛽_𝑛^𝜋₀𝜆_𝑛−𝑝(𝜉)𝜑²𝜉,𝜆_𝑛𝑑𝜉=−2𝜆_𝑛𝛽_𝑛^𝜋₀𝜑²𝜉,𝜆_𝑛−1𝑑𝜉𝜆_𝑛^𝜋₀𝑝(𝜉)𝜑²𝜉,𝜆_𝑛𝑑𝜉=−2𝜆_𝑛𝛽_𝑛𝛼_𝑛. (33) The lemma is proved.

Let Δ₀(𝜆)=𝛼⁺cos[𝜆𝜋−𝛽⁺(𝜋)]+𝛼⁻cos[𝜆(2𝑎−𝜋)+𝛽⁻(𝜋)] and {𝜆⁰_𝑛} are zeros of Δ₀(𝜆).

Lemma 10. The roots of the characteristic equation Δ₀(𝜆)=0 are separate, that is inf_𝑛≠𝑚𝜆⁰_𝑛−𝜆⁰_𝑚=𝛽>0. (34)

Proof. Let 𝜆𝜋−𝛽⁺(𝜋)=𝑥. Then, 𝜆(2𝑎−𝜋)+𝛽⁻(𝜋)=𝑘𝑥+𝑏, where 𝑘=(2𝑎−𝜋)/𝜋,𝑏=𝛽⁺(𝜋)((2𝑎−𝜋)/𝜋)+𝛽⁻(𝜋). Since 𝑎∈(𝜋/2,𝜋), then 𝑘∈(0,1). Using these notations we can rewrite the equation Δ₀(𝜆)=0 in the following form: 𝐴cos𝑥=cos(𝑘𝑥+𝑏). (35) Here 𝐴=−(𝛼⁺/𝛼⁻) which implies that |𝐴|>1. Preliminarily show that there are no multiple roots of (35). Assuming the converse we suppose 𝑥₀ to be a multiple root of (35). Then 𝐴sin𝑥₀=𝑘sin𝑘𝑥₀+𝑏 (36) holds. Now (35) and (36) imply that 𝐴²=1−(1−𝑘²)sin²(𝑘𝑥₀+𝑏)≤1 which is a contradiction. Therefore, (35) has no multiple roots.
Further assuming (34) not to be true let {𝑥^′_𝑝} and {𝑥_𝑝^′′} be increasing sequences of roots of (35) such that 𝑥^′_𝑝≠𝑥_𝑝^′′ and lim_𝑝→∞𝑥^′_𝑝−𝑥_𝑝^′′=0. (37) If we assume that 𝑥^′_𝑝=2𝑛_𝑝𝜋+𝑟^′_𝑝, where 𝑛∈ℕand{𝑟^′_𝑝} is a bounded sequence (0<𝑟^′_𝑝<2𝜋), then from (37) we find that 𝑥_𝑝^′′=2𝑛_𝑝𝜋+𝑟_𝑝^′′, where {𝑟_𝑝^′′} is a bounded sequence such that lim_𝑝→∞|𝑟^′_𝑝−𝑟_𝑝^′′|=0. It is obvious that 𝑘𝑥^′_𝑝=2𝜋[𝑘𝑛_𝑝]+𝑠^′_𝑝,𝑘𝑥_𝑝^′′=2𝜋[𝑘𝑛_𝑝]+𝑠_𝑝^′′, where 𝑠^′_𝑝=2𝜋{𝑘𝑛_𝑝}+𝑟^′_𝑝𝑘,𝑠_𝑝^′′=2𝜋{𝑘𝑛_𝑝}+𝑟_𝑝^′′𝑘 and lim_𝑝→∞|𝑠^′_𝑝−𝑠_𝑝^′′|=0. Here [⋅] and {⋅} denote the integer and fractional parts of a real number, respectively. Since sequences {𝑟^′_𝑝}_𝑝≥1,{𝑟_𝑝^′′}_𝑝≥1,{𝑠^′_𝑝}_𝑝≥1 and {𝑠_𝑝^′′}_𝑝≥1 will be bounded, without loss of generality we can assume that these sequences are convergent. Then let lim_𝑝→∞𝑟^′_𝑝=lim_𝑝→∞𝑟_𝑝^′′=𝑥₀,lim_𝑝→∞𝑠^′_𝑝=lim_𝑝→∞𝑠_𝑝^′′=𝑦₀. (38)(39) Therefore, we can write the equality 𝐴cos𝑥^′_𝑝=cos(𝑘𝑥^′_𝑝+𝑏) as 𝐴cos𝑟^′_𝑝=cos𝑠^′_𝑝+𝑏. (40) Then by virtue of (38) and (39), from (40) we get 𝐴cos𝑥₀=cos𝑦₀+𝑏. (41) Similarly we can obtain 𝐴cos𝑟_𝑝^′′=cos𝑠_𝑝^′′+𝑏. (42) Further, from (40) and (42), we have 𝑟𝐴sin^′𝑝+𝑟_𝑝^′′2𝑟sin^′𝑝−𝑟_𝑝^′′2𝑠=sin^′𝑝+𝑠_𝑝^′′2𝑠+𝑏sin^′𝑝−𝑠_𝑝^′′2. (43) Let us write this equality as 𝑟𝐴sin^′𝑝+𝑟_𝑝^′′2𝑟sin^′𝑝−𝑟_𝑝^′′2𝑠=sin^′𝑝+𝑠_𝑝^′′2𝑘𝑟+𝑏sin^′𝑝−𝑟_𝑝^′′2. (44) Now dividing both sides of equality (44) by (𝑟^′_𝑝−𝑟_𝑝^′′)/2≠0 and taking limit as 𝑝→∞, by virtue of (3) and (39), we obtain 𝐴sin𝑥₀=𝑘sin𝑦₀+𝑏. (45) Finally, from (41) and (45), we conclude that 𝐴²=1−(1−𝑘²)sin²(𝑦₀+𝑏)≤1 which is a contradiction. Hence roots of equation Δ₀(𝜆)=0 are separate. The lemma is proved.

Denote Γ_𝑛=𝜆∶|𝜆|=𝜆⁰_𝑛+𝛽2𝐺,𝑛=0,1,…,_𝛿∶=𝜆∶𝜆−𝜆⁰_𝑛≥𝛿(𝛿>0), (46) where 𝛿 is sufficiently small positive number (𝛿≪𝛽/2).

Lemma 11. For sufficiently large values of 𝑛, one has Δ(𝜆)−Δ_𝑜(𝜆<𝐶)^𝛿2𝑒^|Im𝜆|𝜋,𝜆∈Γ_𝑛. (47)

Proof. As it is shown in [38], |Δ_𝑜(𝜆)|≥𝐶_𝛿𝑒^|Im𝜆|𝜋 for all 𝜆∈𝐺_𝛿, where 𝐶_𝛿>0 is some constant. On the other hand, since lim_|𝜆|→∞𝑒^{−|Im𝜆|𝜋}Δ(𝜆)−Δ_𝑜(𝜆)=lim_|𝜆|→∞𝑒^{−|Im𝜆|𝜋𝜋}₀𝐴()+𝜋,𝑡cos𝜆𝑡𝑑𝑡_𝜋0𝐵(𝜋,𝑡)sin𝜆𝑡𝑑𝑡=0 (48) for sufficiently large values of 𝑛 (see [1]) we get (47). The lemma is proved.

Lemma 12. The problem 𝐿(𝛼,𝑎) has countable set of eigenvalues. If one denotes by 𝜆₁,𝜆₂,… the positive eigenvalues arranged in increasing order and by 𝜆₋₁,𝜆₋₂,… the negative eigenvalues arranged in decreasing order, then eigenvalues of the problem 𝐿(𝛼,𝑎) have the asymptotic behavior 𝜆_𝑛=𝜆⁰_𝑛+𝑑_𝑛𝜆⁰_𝑛+𝛿_𝑛𝜆⁰_𝑛,𝑛⟶±∞, (49) where 𝛿_𝑛∈𝑙₂ and 𝑑_𝑛 is a bounded sequence, 𝜆⁰_𝑛=𝑛+(1/𝜋)𝛽⁺(𝜋)+ℎ_𝑛,sup_𝑛|ℎ_𝑛|<∞.

Proof. According to Lemma 11, if 𝑛 is a sufficiently large natural number and 𝜆∈Γ_𝑛, we have |Δ_𝑜(𝜆)|≥𝐶_𝛿𝑒^|Im𝜆|𝜋>(𝐶_𝛿/2)𝑒^|Im𝜆|𝜋>|Δ(𝜆)−Δ_𝑜(𝜆)|. Applying Rouche's theorem we conclude that for sufficiently large 𝑛 inside the contour Γ_𝑛 the functions Δ_𝑜(𝜆) and Δ_𝑜(𝜆)+{Δ(𝜆)−Δ_𝑜(𝜆)}=Δ(𝜆) have the same number of zeros counting their multiplicities. That is, there are exactly (𝑛+1) zeros 𝜆₀,𝜆₁,…,𝜆_𝑛. in Γ_𝑛. Analogously, it is shown by Rouche's theorem that, for sufficiently large values of 𝑛, the function Δ(𝜆) has a unique zero inside each circle |𝜆−𝜆⁰_𝑛|<𝛿. Since 𝛿>0 is arbitrary, it follows that 𝜆_𝑛=𝜆⁰_𝑛+𝜀_𝑛, where lim_𝑛→∞𝜀_𝑛=0. Further according to Δ(𝜆_𝑛)=0, we have Δ₀𝜆⁰_𝑛+𝜀_𝑛+∫^𝜋₀𝐴(𝜋,𝑡)cos𝜆⁰_𝑛+𝜀_𝑛+𝑡𝑑𝑡_𝜋0𝐵(𝜋,𝑡)sin𝜆⁰_𝑛+𝜀_𝑛𝑡𝑑𝑡=0. (50) On the other hand, since Δ₀(𝜆)=𝛼⁺cos𝜆𝜋−𝛽⁺(𝜋)+𝛼⁻cos𝜆(2𝑎−𝜋)+𝛽⁻(𝜋),Δ₀𝜆⁰_𝑛+𝜀_𝑛=Δ̇_𝑜𝜆⁰_𝑛𝜀_𝑛+𝑜𝜀_𝑛,𝑛⟶∞, (51) (50) takes the form of Δ̇₀𝜆⁰_𝑛𝜀_𝑛+^𝜋₀𝐴(𝜋,𝑡)cos𝜆⁰_𝑛+𝜀_𝑛+𝑡𝑑𝑡_𝜋0𝐵(𝜋,𝑡)sin𝜆⁰_𝑛+𝜀_𝑛𝑡𝑑𝑡+𝑜𝜀_𝑛=0,𝑛⟶∞. (52) It is easy to see that the function Δ₀(𝜆) is type of “Sine” [39], so there exists 𝛾_𝛿>0 such that |Δ̇_𝑜(𝜆⁰_𝑛)|≥𝛾_𝛿>0 is satisfied for all 𝑛. We also have 𝜆⁰_𝑛1=𝑛+𝜋𝛽⁺(𝜋)+ℎ_𝑛, (53) where sup_𝑛|ℎ_𝑛|≤𝑀 for some constant 𝑀>0 [40] (see also [41]). Further, substituting (53) into (52) after certain transformations [1, page 67], we reach 𝜀_𝑛∈𝑙₂. We can obtain more precisely 𝜀_𝑛=12Δ̇_𝑜𝜆⁰_𝑛𝜆⁰_𝑛𝛼⁺sin𝜆⁰_𝑛𝜋−𝛽₁(𝜋)+𝛼⁻sin𝜆⁰_𝑛(2𝑎−𝜋)+𝛽₂(𝜋×_𝜋0)𝑞(𝑥)+𝑝²(𝑥)−𝛼𝑑𝑥₊cos𝜆⁰_𝑛𝜋−𝛽₁(𝜋)+𝛼⁻cos𝜆⁰_𝑛(2𝑎−𝜋)+𝛽₂(𝜋×𝑝(𝜋)(0)+1)−𝑝Δ̇_𝑜𝜆⁰_𝑛𝜆⁰_𝑛^𝜋0𝐵^′_𝑡(𝜋,𝑡)cos𝜆⁰_𝑛−𝑡𝑑𝑡_𝜋0𝐴^′_𝑡(𝜋,𝑡)sin𝜆⁰_𝑛+𝑜𝜀𝑡𝑑𝑡_𝑛𝜆⁰_𝑛,𝑛⟶∞. (54)
Since ∫^𝜋₀𝐵^′_𝑡(𝜋,𝑡)cos𝜆⁰_𝑛𝑡𝑑𝑡∈𝑙₂,∫^𝜋₀𝐴^′_𝑡(𝜋,𝑡)sin𝜆⁰_𝑛𝑡𝑑𝑡∈𝑙₂, we have 𝜀_𝑛=12Δ̇_𝑜𝜆⁰_𝑛𝜆⁰_𝑛𝛼⁺sin𝜆⁰_𝑛𝜋−𝛽₁(𝜋)+𝛼⁻sin𝜆⁰_𝑛(2𝑎−𝜋)+𝛽₂(𝜋×_𝜋0)𝑞(𝑥)+𝑝²(𝑥)−𝛼𝑑𝑥₊cos𝜆⁰_𝑛𝜋−𝛽₁(𝜋)+𝛼⁻cos𝜆⁰_𝑛(2𝑎−𝜋)+𝛽₂(𝜋×𝑝(𝜋)(0)+𝛿)−𝑝^𝑛𝜆⁰_𝑛, (55) where 𝛿_𝑛∈𝑙₂. Hence we obtain 𝜆_𝑛=𝜆⁰_𝑛+𝑑_𝑛𝜆⁰_𝑛+𝛿_𝑛𝜆⁰_𝑛, (56) where 𝑑_𝑛=12Δ̇_𝑜𝜆⁰_𝑛𝛼⁺sin𝜆⁰_𝑛𝜋−𝛽⁺(𝜋)+𝛼⁻sin𝜆⁰_𝑛(2𝑎−𝜋)+𝛽⁻(𝜋)×∫^𝜋₀𝑞(𝑥)+𝑝²(𝑥)−𝛼𝑑𝑥₊cos𝜆⁰_𝑛𝜋−𝛽⁺(𝜋)+𝛼⁻cos𝜆⁰_𝑛(2𝑎−𝜋)+𝛽⁻(𝜋)×𝑝(𝜋)−𝑝(0) (57) is a bounded sequence. The proof is completed.

Lemma 13. Normalizing numbers 𝛼_𝑛 of the problem 𝐿(𝛼,𝑎) are positive and the formula 𝛼_𝑛=𝜋2𝛼⁺²+(𝛼⁻)²+𝑑₁₁𝜆⁰_𝑛+𝑑_1𝑛𝑛 (58) holds, where 𝑑₁₁=−(𝛼𝜋/2)𝑝(0),𝑑_1𝑛∈𝑙₂.

Proof. The formula (58) can be easily obtained from the equalities 𝐴(𝑥,𝑥)sin𝜆_𝑛𝑥−𝐵(𝑥,𝑥)cos𝜆_𝑛𝑥=𝛼⁺2𝑝(𝑥)−𝑝(0)cos𝜆_𝑛𝑥−𝛽⁺(𝑥)+^𝑥₀𝑞(𝑡)+𝑝²(𝑡)𝑑𝑡sin𝜆_𝑛𝑥−𝛽⁺(𝑥),𝐴(𝑥,2𝑎−𝑥+0)−𝐴(𝑥,2𝑎−𝑥−0)sin𝜆_𝑛()−𝐵()(2𝑎−𝑥𝑥,2𝑎−𝑥+0−𝐵𝑥,2𝑎−𝑥−0)cos𝜆_𝑛()=𝛼2𝑎−𝑥₋2^𝑥₀𝑞(𝑡)+𝑝²(𝑡)𝑑𝑡sin𝜆_𝑛(2𝑎−𝑥)+𝛽⁻(𝑥)−𝑝(𝑥)−𝑝(0)cos𝜆_𝑛(2𝑎−𝑥)+𝛽⁻(𝑥) (59) by using the asymptotic formula (49) for 𝜆_𝑛.

4. Inverse Problems

Together with 𝐿(𝛼,𝑎), we consider the boundary value problem 𝐿(𝛼,𝑎) of the same form but with different coefficients (𝑞,𝑝,𝛼,𝑎). It is assumed in what follows that if a certain symbol 𝛾 denotes an object related to the problem 𝐿(𝛼,𝑎), then 𝛾 will denote the corresponding object related to the problem 𝐿(𝛼,𝑎).

In the present section, we investigate some inverse spectral problem of the reconstruction of a boundary value problem 𝐿(𝛼,𝑎) of type (1)–(4) from its spectral characteristics. Namely, we consider the inverse problems of reconstruction of the boundary value problem 𝐿(𝛼,𝑎) from the Weyl function, from the spectral data {𝜆_𝑛,𝛼_𝑛}_𝑛≥0, and from two spectra {𝜆_𝑛,𝜇_𝑛}_𝑛≥0 and prove that the following two lemmas can be easily obtained from asymptotic behavior (49) of the eigenvalues 𝜆_𝑛.

Lemma 14. If 𝜆_𝑛=𝜆_𝑛,𝑛=0,±1,±2,…, then 𝛽⁺(𝜋)=𝛽⁺(𝜋),𝛽⁻(𝜋)=𝛽⁻(𝜋), that is, the sequence {𝜆_𝑛} uniquely determines 𝛽^±(𝜋).

Lemma 15. If 𝜆_𝑛=𝜆_𝑛,𝑛=0,±1,±2,…, then 𝑎=𝑎,𝛼=𝛼, that is, the sequence {𝜆_𝑛} uniquely determines numbers 𝑎 and 𝛼.

Let Φ(𝑥,𝜆) be the solution of (1) under the conditions 𝑈(Φ)=1,𝑉(Φ)=0 and under the jump conditions (3). One sets 𝑀(𝜆)∶=Φ(0,𝜆). The functions Φ(𝑥,𝜆) and 𝑀(𝜆) are called the Weyl solution and Weyl function for the boundary value problem 𝐿(𝛼,𝑎), respectively. Using the solution 𝜑(𝑥,𝜆) defined in the previous sections one has Φ()𝜓()𝑥,𝜆=−𝑥,𝜆Δ(𝜆)()(𝜆)𝜑(),𝑀(𝜆)𝜓()=𝑆𝑥,𝜆+𝑀𝑥,𝜆=−0,𝜆Δ(𝜆), (60) where 𝜓(𝑥,𝜆) is a solution of (1) satisfying the conditions 𝜓(𝜋,𝜆)=0,𝜓^′(𝜋,𝜆)=−1, and the jump conditions (3), and 𝑆(𝑥,𝜆) is defined from the equality 𝜓(𝑥,𝜆)=𝜓(0,𝜆)𝜑(𝑥,𝜆)−Δ(𝜆)𝑆(𝑥,𝜆). (61) Note that, by virtue of equalities ⟨𝜑(𝑥,𝜆),𝑆(𝑥,𝜆)⟩≡1 and (60), one has Φ()(𝜑()((𝜆)𝑥,𝜆,𝜑𝑥,𝜆)≡1,𝑥,𝜆,𝜓𝑥,𝜆)≡−Δfor𝑥≠𝑎. (62)

The following theorem shows that the Weyl function uniquely determines the potentials and the coefficients of the boundary value problem 𝐿(𝛼,𝑎).

Theorem 16. If 𝑀(𝜆)=𝑀(𝜆), then 𝐿(𝛼,𝑎)=𝐿(𝛼,𝑎). Thus, the boundary value problem 𝐿(𝛼,𝑎) is uniquely defined by the Weyl function.

Proof. Since 𝜓^(𝜈)(𝑥,𝜆)=𝑂|𝜆|^𝜈−1exp(|Im𝜆|(𝜋−𝑥)),𝜆∈𝐺_𝛿,|Δ(𝜆)|≥𝐶_𝛿exp(|Im𝜆|𝜋),𝜆∈𝐺_𝛿,𝐶_𝛿<0,𝜈=0,1, (63)(64) it is easy to observe that Φ^(𝜈)(𝑥,𝜆)≤𝐶_𝛿|𝜆|^𝜈−1exp(−|Im𝜆|𝑥),𝜆∈𝐺_𝛿. (65) Let us define the matrix 𝑃(𝑥,𝜆)=[𝑃_𝑗𝑘(𝑥,𝜆)]_{𝑗,𝑘=1,2}, where 𝑃_𝑗1(𝑥,𝜆)=𝜑^(𝑗−1)(𝑥,𝜆)Φ^′(𝑥,𝜆)−Φ^(𝑗−1)(𝑥,𝜆)𝜑^′(),𝑃𝑥,𝜆_𝑗2(𝑥,𝜆)=Φ^(𝑗−1)(𝑥,𝜆)𝜑^′(𝑥,𝜆)−𝜑^(𝑗−1)(𝑥,𝜆)Φ(𝑥,𝜆). (66) Then we have 𝜑(𝑥,𝜆)=𝑃₁₁(𝑥,𝜆)𝜑(𝑥,𝜆)+𝑃₁₂(𝑥,𝜆)𝜑^′(),Φ()𝑥,𝜆𝑥,𝜆=𝑃₁₁(𝑥,𝜆)Φ(𝑥,𝜆)+𝑃₁₂(𝑥,𝜆)Φ^′(𝑥,𝜆). (67) According to (60) and (65), for each fixed 𝑥, the functions 𝑃_𝑗𝑘(𝑥,𝜆) are meromorphic in 𝜆 with poles at points 𝜆_𝑛 and 𝜆_𝑛. Denote𝐺^∘_𝛿=𝐺_𝛿∩𝐺_𝛿. By virtue of (65), (66), and 𝜑^(𝜈)(𝑥,𝜆)=𝑂|𝜆|^𝜈exp(|Im𝜆|𝑥),𝜆∈𝐺^∘_𝛿, (68) we get 𝑃₁₂(𝑥,𝜆)≤𝐶_𝛿|𝜆|⁻¹,𝑃₁₁(𝑥,𝜆)≤𝐶_𝛿,𝜆∈𝐺^∘_𝛿. (69) It follows from (60) and (66) that if 𝑀(𝜆)≡𝑀(𝜆), then for each fixed 𝑥 the functions 𝑃_1𝑘(𝑥,𝜆) are entire in 𝑘. Together with (69) this yields 𝑃₁₂(𝑥,𝜆)≡0,𝑃₁₂(𝑥,𝜆)≡𝐴(𝑥). Now using (67), we obtain 𝜑(𝑥,𝜆)≡𝐴(𝑥)𝜑(𝑥,𝜆),Φ(𝑥,𝜆)≡𝐴(𝑥)Φ(𝑥,𝜆). (70) Therefore, for |𝜆|→∞,arg𝜆∈[𝜀,𝜋−𝜀](𝜀>0), we have 𝜑()=𝑏𝑥,𝜆2exp−𝑖𝜆𝑥−𝛽₁(𝑥)11+𝑂𝜆, (71) where 𝑏=1 for 𝑥<𝑎 and 𝑏=𝛼⁺ for 𝑥>𝑎. Similarly, one can calculate Φ(𝑥,𝜆)=(𝑖𝜆𝑏)⁻¹exp𝑖𝜆𝑥−𝛽₁(𝑥)11+𝑂𝜆,|𝜆|⟶∞,arg𝜆∈[𝜀,𝜋−𝜀]. (72) Finally, taking into account the relations ⟨Φ(𝑥,𝜆),𝜑(𝑥,𝜆)⟩≡1 and (65), we have 𝛼⁺=𝛼⁺,𝐴(𝑥)≡1, that is, 𝜑(𝑥,𝜆)≡𝜑(𝑥,𝜆),Φ(𝑥,𝜆)≡Φ(𝑥,𝜆) for all 𝑥 and 𝜆. Consequently, 𝐿(𝛼,𝑎)=𝐿(𝛼,𝑎). 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 𝐿(𝛼,𝑎).

Theorem 17. If 𝜆_𝑛=𝜆_𝑛,𝜇_𝑛=𝜇_𝑛,𝑛=0,±1,±2,…, then 𝐿(𝛼,𝑎)=𝐿(𝛼,𝑎).

Proof. It is obvious that characteristic functions Δ(𝜆) and 𝜓(0,𝜆) are uniquely determined by the sequences {𝜆²_𝑛} and {𝜇²_𝑛}(𝑛=0,±1,±2,…), respectively. If 𝜆_𝑛=𝜆_𝑛,𝜇_𝑛=𝜇_𝑛,𝑛=0,±1,±2,…, then Δ(𝜆)≡Δ(𝜆),𝜓(0,𝜆)≡𝜓(0,𝜆). It follows from (60) that 𝑀(𝜆)≡𝑀(𝜆). Therefore, applying Theorem 16, we conclude that 𝐿(𝛼,𝑎)=𝐿(𝛼,𝑎). The proof is completed.

Theorem 18. If 𝜆_𝑛=𝜆_𝑛,𝛼_𝑛=𝛼_𝑛,𝑛=0,±1,±2,…, then 𝐿(𝛼,𝑎)=𝐿(𝛼,𝑎), that is, spectral data {𝜆_𝑛,𝛼_𝑛} uniquely determines the problem 𝐿(𝛼,𝑎).

Proof. It is obvious that the Weyl function 𝑀(𝜆) is meromorphic with simple poles at points 𝜆²_𝑛. Using the expression Δ(𝜆)=Δ₀(𝜆)+^𝜋₀𝐴(𝜋,𝜆)cos𝜆𝑡𝑑𝑡+^𝜋₀𝐵(𝜋,𝜆)sin𝜆𝑡𝑑𝑡 (73) and equalities 2𝜆_𝑛𝛽_𝑛𝛼_𝑛=−Δ̇(𝜆_𝑛),𝜓(𝑥,𝜆_𝑛)=𝛽_𝑛𝜑(𝑥,𝜆_𝑛), we have Re_{𝜆=𝜆𝑛}(𝜆)𝜓𝑠𝑀=−0,𝜆_𝑛Δ̇𝜆_𝑛𝛽=−^𝑛Δ̇𝜆_𝑛=12𝜆_𝑛𝛼_𝑛. (74) Since the Weyl function 𝑀(𝜆) is regular for 𝜆∈Γ_𝑛, applying the Rouche theorem, we conclude that 𝑀(𝜆)=12𝜋𝑖_Γ𝑛𝑀𝜇𝜇−𝜆𝑑𝜇,𝜆∈intΓ_𝑛. (75) Taking (60) and (63) into account, we arrive at |𝑀(𝜆)|≤𝐶_𝛿|𝜆|⁻¹,𝜆∈𝐺_𝛿. Therefore 𝑀(𝜆)=lim_𝑛→∞12𝜋𝑖_Γ𝑛1𝑀𝜇𝜇−𝜆𝑑𝜇, (76) where Γ_𝑛1={𝜆∶|𝜆|=|𝜆⁰_𝑛|},𝑛=0,±1,±2,…. Hence, by the residue theorem, we have 𝑀(𝜆)=^∞_𝑛=−∞12𝜆_𝑛𝛼_𝑛𝜆−𝜆_𝑛. (77) Finally, from the equality 𝑀(𝜆)≡𝑀(𝜆), applying Theorem 16, we conclude that 𝐿(𝛼,𝑎)=𝐿(𝛼,𝑎). The theorem is proved.