Solvability and stability of the inverse problem for the quadratic differential pencil

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the nonlinear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

The paper is concerned with the theory of inverse spectral problems, which consist in recovery of operators from their spectral characteristics. The most complete results in inverse problem theory have been obtained for the Sturm-Liouville equation (1.1) with q 1 (x) ≡ 0 (see the monographs [2][3][4][5] and references therein). In particular, Sturm-Liouville inverse problems with singular potentials of class W −1 2 were studied in [6,7] and other papers. Investigation of inverse problems for differential pencils induced by equation (1.1) with nonlinear dependence on the spectral parameter causes principal difficulties comparing with the classical Sturm-Liouville problems. Therefore, a number of open questions still remain in this direction. At the same time, inverse problems for equation (1.1) are used in various applications, e.g., for modeling interactions between colliding relativistic particles in quantum mechanics [8] and for studying vibrations of mechanical systems in viscous media [9]. For the quadratic differential pencil (1.1) on a finite interval with the regular potentials q j ∈ W j 2 [0, π], j = 0, 1, and the Robin boundary conditions y (0)−hy(0) = 0, y (π)+Hy(π) = 0, the solvability conditions for the inverse spectral problem were obtained by Gasymov and Guseinov [10]. Later on, their approach was applied for investigation of inverse problems for the pencils with non-separated boundary conditions [11][12][13]. Hryniv and Pronska [14][15][16][17] developed an approach to inverse problems for the pencils of form (1.1)-(1.2) with the singular potentials q j ∈ W j−1 2 [0, π], j = 0, 1. Their approach is based on the reduction of equation (1.1) to a first-order system. In the recent paper [18], the analogous reduction was applied to the inverse scattering problem for the quadratic differential pencil on the half-line. However, the results of the mentioned papers have the common disadvantage that consists in the requirement of real-valued potentials and positivity of some operator. Under this requirement, the eigenvalues of the pencil are real and simple, which makes the situation similar to the classical Sturm-Liouville operators and significantly simplifies the investigation of inverse problems. However, in the general case, the pencil (1.1)-(1.2) can have multiple and/or non-real eigenvalues even if the potentials q j are real-valued.
Buterin and Yurko [19,20] developed another approach, which allows to solve inverse problems for quadratic differential pencils with the complex-valued potentials q j ∈ W j 2 [0, π], j = 0, 1, and without any additional restrictions. The approach of [19,20] is based on the method of spectral mappings [5,21]. This method allows to reduce a nonlinear inverse spectral problem to a linear equation in an appropriate Banach space, by using contour integration in the λ-plane and the theory of analytic functions. The approach based on the method of spectral mappings was also applied to the pencils of the matrix Sturm-Liouville operators [22,23], to the scalar pencils on the half-line [24], to the half inverse problem [25], and to the pencils on graphs (see [26,27] and references therein). However, the results obtained by using this approach for differential pencils include only uniqueness theorems and constructive procedures for solving inverse problems. The most principal questions of solvability and stability for the general case of complex-valued potentials were open. The present paper aims to fill this gap.
It is also worth mentioning that, in recent years, a number of scholars have been actively studying inverse problems for quadratic differential pencils (see [28][29][30][31][32][33][34][35][36] and other papers of these authors). The majority of those results are concerned with partial inverse problems, inverse nodal problems, and recovery of the pencils from the interior spectral data.
The aim of this paper is to study solvability and stability of the inverse spectral problem for the pencil (1.1)-(1.2). Developing the ideas of the method of spectral mappings [5,[19][20][21], we reduce the inverse problem to the so-called main equation, which is a linear equation in the Banach space of bounded infinite sequences. The most important difficulties of our investigation are related with eigenvalue multiplicities. The multiplicities influence on the definition of the spectral data and on the construction of the main equation. Moreover, under a small perturbation of the spectrum, multiple eigenvalues can split into smaller groups, which complicates the analysis of local solvability and stability. Nevertheless, we take this effect into account and obtain the results valid for arbitrary multiplicities. For dealing with multiple eigenvalues, we use some ideas previously developed for the non-self-adjoint Sturm-Liouville operators in [37][38][39].
Thus, the following main results are obtained.
• In Section 2, the spectral data of the quadratic differential pencil are defined and a constructive solution of the inverse problem is obtained for the case of the complexvalued singular potentials q j ∈ W j−1
• In Section 3, we construct infinitely differentiable approximations q N j of the potentials q j , by using finite spectral data (Theorem 3.4). This theorem plays an auxiliary role in the further sections, but also can be treated as a separate result.
• In Section 4, we prove Theorem 4.1, which provides sufficient conditions for the global solvability of the inverse problem. Theorem 4.1 implies Corollary 4.3 on the local solvability and stability of the inverse problem for spectrum perturbations that do not change eigenvalue multiplicities.
• In Section 5, we prove Theorem 5.1 and Corollary 5.3 on the local solvability and stability of the inverse problem in the general case, taking splitting of multiple eigenvalues into account.
• In Section 6, our theoretical results are illustrated by numerical examples. Namely, we approximate a pencil having a double eigenvalue by a family of pencils with simple eigenvalues.

Constructive solution
In this section, we define the spectral data of the problem L(q 0 , q 1 ) and develop Algorithm 2.8 for recovery of the potentials q j ∈ W j−1 2 (0, π), j = 0, 1, from the spectral data. The nonlinear inverse problem is reduced to the linear equation (2.18), which plays a crucial role in the constructive solution and also in study of solvability and stability for the inverse problem. In addition, relying on Algorithm 2.8, we obtain the uniqueness of the inverse problem solution (Theorem 2.9). We follow the strategy of [20], so some formulas and propositions of this section are provided without proofs. However, it is worth mentioning that our constructive solution is novel for the case of the singular potentials q j ∈ W j−1 2 (0, π), j = 0, 1. The most important difference from the regular case q j ∈ W j 2 [0, π], j = 0, 1, is the construction of the regularized series (2.20)-(2.22) and the analysis of their convergence in Lemma 2.6. The results of this section will be used in the further sections for investigation of solvability and stability issues.
Here and below, the same notation {κ n } is used for various sequences from l 2 . Introduce the notations S := {n ∈ Z 0 : ∀k < n λ k = λ n }, m n := #{k ∈ Z 0 : λ k = λ n }, that is, {λ n } n∈S is the set of all the distinct eigenvalues and m n is the multiplicity of the eigenvalue λ n . Without loss of generality, we impose the following assumption.
Further, we focus on Inverse Problem 2.3. For convenience, let us call the collection {λ n , M n } n∈Z 0 the spectral data of the problem L.
One can easily obtain the asymptotics In the regular case q j ∈ W j 2 [0, π], the asymptotics (2.1) and (2.4) can be improved (see [20]): where Note that the function σ = q 1 (x) dx is determined by q 1 uniquely up to an additive constant. However, this constant does not influence on the definitions of the Weyl function and the spectral data. Thus, in the regular case, we may assume that σ(x) = x 0 q 1 (t) dt, so σ(0) = 0 and y [1] (0) = y (0).
Along with L = L(q 0 , q 1 ), we consider another problemL = L(q 0 ,q 1 ) of the same form but with different coefficientsq j ∈ W j−1 2 (0, π), j = 0, 1. We agree that, if a symbol γ denotes an object related to L, then the symbolγ with tilde will denote the similar object related toL. Note that the quasi-derivatives for these two problems are supposed to be different: y [1] = y − σy for L and y [1] = y −σy forL. Without loss of generality, we suppose that the both eigenvalue sequences {λ n } n∈Z 0 and {λ n } n∈Z 0 satisfy Assumption (O).
Introduce the notationŝ By using the contour integration in the λ-plane, Buterin and Yurko [20] have derived the following relation: (P n,i;k,0 (x)S k,0 (x) −P n,i;k,1 (x)S k,1 (x)), n ∈ Z 0 , i = 0, 1. (2.10) However, it is inconvenient to use (2.10) as the main equation of the inverse problem, since the series converges only "with brackets". Therefore, below we transform (2.10) into an equation in the Banach space of infinite sequences.

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where I is the identity operator in B.
Finally, we arrive at the following algorithm for solving Inverse Problem 2.3.
Algorithm 2.8. Suppose that the data {λ n , M n } n∈Z 0 are given.
Note that the choice of the square root branch for Θ(x) and Λ(x) is uniquely specified by the continuity of these functions, the condition Θ(0) = 1, and (2.27). If Θ(x) = 0 for some x ∈ [0, π], one can apply the step-by-step process described in [20]. However, in our analysis of the inverse problem solvability and stability in the further sections, the condition Θ(x) = 0 is always fulfilled. Algorithm 2.8 implies the following uniqueness theorem for solution of Inverse Problem 2.3.

Estimates and approximation
This section plays an auxiliary role in studying solvability and stability of Inverse Problem 2.3. We impose the assumption of the uniform boundedness of the inverse operator (I −H(x)) −1 , and obtain auxiliary estimates for the values constructed by Algorithm 2.8. Further, by using the finite spectral data {λ n , M n } |n|≤N , we construct the infinitely differentiable approximations q N j of the potentials q j in Theorem 3.4. This theorem plays an auxiliary role in the proofs of global and local solvability, but also can be considered as a separate result.
In this section, we assume thatL = L(q 0 ,q 1 ),q j ∈ W j 2 [0, π], j = 0, 1, {λ n , M n } n∈Z 0 are complex numbers (not necessarily being the spectral data of some problem L) numbered according to Assumption (O). Suppose that the numbers {λ n , M n } n∈Z 0 and the spectral data {λ n ,M n } n∈Z 0 satisfy the following condition Ω := n∈Z 0 (nξ n ) 2 < ∞. where Therefore, we arrive at the following lemma.
where C does not depend on x and N . 3) Proof.
Step 1. Let us prove the continuity of R n,i;k,j (x). Clearly,H n,i;k,j ∈ C[0, π]. Fix ε > 0 and choose N such that the conclusion of Lemma 3.1 holds and where R N (x) = (I −H N (x)) −1 − I. Note that the inverse (I −H N (x)) −1 can be found by solving the system of finite linear equations a n,i + |k|≤N j=0,1H n,i;k,j (x)a k,j = b n,i , (n, i) ∈ J.
Step 2. Let us estimate R n,i;k,j (x). By definition, In the element-wise form, this implies R n,i;k,j (x) =H n,i;k,j (x) + Using this estimate together with (3.8) and (2.15), we arrive at (3.3).

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where the constant C does not depend on x and N . Proof.
Step 3. It remains to prove that {λ N n , M N n } n∈Z 0 are the spectral data of the problem L(q N 0 , q N 1 ). At this step, we assume that all the considered objects are related to the data {λ N n , M N n } n∈Z 0 for a sufficiently large fixed N , and the index N will be omitted for brevity.
Construct the function (3.24) where the summation range for k, j is |k| ≤ N , j = 0, 1, and Clearly, Φ(x, λ) is analytic in λ = λ n,i for each fixed x ∈ [0, π] and infinitely differentiable with respect to x for each fixed λ = λ n,i , (n, i) ∈ J.
Remark 3.7. In view of (2.3), Theorems 2.9 and 3.4 are valid for the spectral data {λ n , M n } n∈Z 0 being replaced by {λ n , α n } n∈Z 0 .

Solvability and stability
The goal of this section is to prove Theorem 4.1 on the global solvability of Inverse Problem 2.3. The proof is based on the constructive solution from Section 2, auxiliary estimates and the approximation by infinitely differentiable potentials obtained in Section 3. Theorem 4.1 implies Corollary 4.3 on the local solvability and stability without change of eigenvalue multiplicities. The latter result will be improved in Section 5.
The following corollary of Theorem 4.1 provides local solvability and stability of Inverse Problem 2.3.
In view of the definitions (2.14) and (3.1), the multiplicities in the sequences {λ n } n∈Z 0 and {λ n } n∈Z 0 coincide for sufficiently small Ω. In the next section, Corollary 4.3 will be generalized to the case of changing eigenvalue multiplicities.

Multiple eigenvalue splitting
In this section, we obtain the local solvability and stability of Inverse problem 2.3 in the general case, taking the possible splitting of multiple eigenvalues into account.
In addition, the estimate (5.2) is valid.

Numerical examples
In this section, we construct an example of a pencil having a double eigenvalue. Then, we approximate this pencil by pencils with simple eigenvalues.
Observe that, for sufficiently small δ > 0, the defined data fulfills the conditions of Corollary 5.3. An interesting feature of this example is that the eigenvalues λ ±1 are √ δ-close toλ ±1 and the absolute values of the residues M ±1 tend to infinity as δ → 0, but the corresponding potentials q 0 , q 1 are Cδ-close toq 0 ,q 1 in the sense of the estimate (5.2). This feature is confirmed by numerical computations. For δ = 0.02, the plots of the potentials q 1 (x),q 1 (x) and q 0 (x),q 0 (x) are presented in Figures 1 and 2   The method used for obtaining these results is based on the constructive solution of Inverse Problem 2.3 provided in Section 2. We use the model problem L(0, 0), so the inverse problem is reduced to a finite (4 × 4) system of linear algebraic equations.

Appendix
Here we provide auxiliary lemmas about rational functions.
Denote by R N the class of rational functions of form P N −1 (λ) Q N (λ) , where P N −1 (λ) is a polynomial of degree at most (N − 1) and Q N (λ) is a polynomial of degree N with the leading coefficient equal 1. where m n is the multiplicity of the corresponding zero λ n , and the constant C depends only on F (λ).
The proof of Lemma 7.1 is based on several auxiliary lemmas.
Lemma 7.2. Let {s j } 2N j=1 be distinct points in γ. Then a function F ∈ R N is uniquely specified by its valued at these points.
Proof. Suppose that, on the contrary, there exist two distinct functions , F (s j ) =F (s j ), j = 1, 2N .
Note that the radius r 0 > 0 can be chosen arbitrarily small by the choice of ε. Since Funding. This work was supported by Grant 20-31-70005 of the Russian Foundation for Basic Research.