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

and Applied Analysis 3 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 usingTheorem 1, we can formulate the following assertion. Theorem 2. Let q(x) ∈ L2[0, π], p(x) ∈ W 2 [0, π]. Then there are the functionsA(x, t), B(x, t)whose first order partial derivatives are summable on [0, π] for each x ∈ [0, π] such that the representation φ (x, λ)=φ0 (x, λ)+∫ x 0 A (x, t) cos λt dt+∫ x 0 B (x, t) sin λt dt (13) is satisfied, where φ0 (x, λ) = l + (x) cos [λx − β (x)] + l − (x) cos [λ (2a − x) − β (x)] . (14) Moreover, the relations α + β + (x) = α + xp (0)


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 with the boundary conditions 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 ( ( ) ≡ 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][2][3][4]). Some versions of inverse problems for (1) which is a natural generalization of the Sturm-Liouville equation were 2 Abstract and Applied Analysis fully studied in [6][7][8][9][10][11][12][13][14]. Namely, the inverse problems for a pencil on the half axis and the entire axis were considered in [6][7][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 ( ) ̸ = 0. The inverse periodic problem for the pencil was solved in [12] using another approach. We also point out the paper [14], 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 [19].
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.
Then there are the functions ( , ), ( , ) whose first order partial derivatives are summable on [0, ] for each ∈ [0, ] such that the representation is satisfied, where Moreover, the relations are held.
One here supposes that the function ( ) satisfies the additional condition Definition 3. A complex number 0 is called an eigenvalue of the boundary value problem ( , ) if (1) with = 0 has a nontrivial solution 0 ( ) satisfying the boundary conditions (2) and the jump conditions (3). In this case 0 ( ) is called the eigenfunction of the problem ( , ) corresponding to the eigenvalue 0 . The number of linearly independent solutions of the problem ( , ) for a given eigenvalue 0 is called the multiplicity of 0 .
The following lemmas can be proved analogously to the corresponding assertions in [11].
Since condition (17) holds, it follows that ( 0 , ) > 0. Let 0 be an eigenvalue of the boundary value problem ( , ) and 0 ( ) an eigenfunction corresponding to this eigenvalue and normalized by the condition ( 0 , 0 ) = 1. By taking the inner product of both sides of the relation The desired assertion follows from the last relation by virtue of ( 0 0 , 0 ) > 0 with regard to the fact that ( ) is real.
Let us show that 0 is a simple eigenvalue. Assume that this is not true. Suppose that 1 ( ) and 2 ( ) are linearly independent eigenfunctions corresponding to the eigenvalue 0 . Then for a given value of 0 , each solution 0 ( ) of (1) will be given as linear combination of solutions 1 ( ) and 2 ( ). Moreover it will satisfy boundary conditions (2) and discontinuity conditions (3). However, it is impossible. Proof. Let 0 ( ) be an eigenfunction corresponding to eigenvalue 0 and normalized by the condition ( 0 , 0 ) = 1 of the problem (1)- (3). Suppose that 1 ( ) is an associated function of eigenfunction 0 ( ), that is, the following equalities hold: 4 Abstract and Applied Analysis If these equations are multiplied by 1 ( ) and 0 ( ), respectively, as inner product, subtracting them side by side and taking into our account that operator 0 is symmetric, the function ( ) and 0 are real, we get 0 = ( 0 , 0 ). Due to the condition (6), 0 = ( 0 , 0 ) does not agree with (19 ). Therefore, the assertion is not true.
Moreover, the sign of the left-hand side of (22) is similar to the sign of .
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 11. For sufficiently large values of , one has
Proof. As it is shown in [38], |Δ ( )| ≥ | Im | for all ∈ , where > 0 is some constant. On the other hand, since for sufficiently large values of (see [1]) we get (47). The lemma is proved.

Inverse Problems
Together with ( , ), we consider the boundary value prob-lem̃( , ) 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 ( , ), theñwill denote the corresponding object related to the problem̃( , ).
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 { , } ≥0 , and from two spectra { , } ≥0 and prove that the following two lemmas can be easily obtained from asymptotic behavior (49) of the eigenvalues . (62) The following theorem shows that the Weyl function uniquely determines the potentials and the coefficients of the boundary value problem ( , ).
it is easy to observe that The following two theorems show that two spectra and spectral data also uniquely determine the potentials and the coefficients of the boundary value problem ( , ).