Local solvability and stability of the inverse problem for the non-self-adjoint Sturm–Liouville operator

We consider the non-self-adjoint Sturm–Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local solvability and stability of this inverse problem, relying on the method of spectral mappings. Possible splitting of multiple eigenvalues is taken into account.


Introduction
The paper concerns the theory of inverse spectral problems for differential operators. Such problems consist in constructing operators by their spectral information.
Inverse problem 1.1 is equivalent to the inverse problem by the spectral function, studied by Marchenko [5].
In the non-self-adjoint case, some of the eigenvalues {λ n } ∞ n=0 can be multiple, so the problem becomes more difficult to investigate. There are significantly less studies on inverse problems for the non-self-adjoint operator (1.1)-(1.2). In particular, in [8] the classical results for the self-adjoint Sturm-Liouville operator are generalized to the non-self-adjoint case with simple eigenvalues. Tkachenko [12] developed the method for solving another inverse problem, which consists in recovering the non-self-adjoint Sturm-Liouville operator from two spectra corresponding to different boundary conditions.
Note that the spectral data {λ n , α n } ∞ n=0 defined above do not uniquely specify q, h, and H in the general case. Nevertheless, in [13] the so-called generalized spectral data (GSD) has been introduced in the following way. Without loss of generality, we assume that multiple eigenvalues are consecutive: λ n = λ n+1 = · · · = λ n+m n -1 , where m n is the multiplicity of the eigenvalue λ n . By virtue of the well-known asymptotics we have m n = 1 for sufficiently large n. Define The sequence {ϕ n } ∞ n=0 is a complete system of root-functions for the problem L. The generalized weight numbers are defined as follows: Clearly, definition (1.5) generalizes (1.3). Thus, Inverse problem 1.1 turns into the inverse problem by GSD {λ n , α n } ∞ n=0 . Buterin [13] has proved the uniqueness theorem for this inverse problem and obtained a constructive algorithm for its solution based on the method of spectral mappings [8,14]. The question of GSD characterization for Sturm-Liouville operators with complex-valued potentials was investigated in [15,16]. However, necessary and sufficient conditions on GSD from [15,16] require solvability of some main equations. Those requirements are difficult to verify.
The aim of this paper is to investigate local solvability and stability of Inverse problem 1.1 in the non-self-adjoint case. Note that, under a small perturbation of the spectrum, multiple eigenvalues can split into smaller groups, so the generalized weight numbers change their form. As far as we know, this effect has not been studied before.
Some fragmentary results on stability under splitting of multiple eigenvalues were obtained in [17][18][19] for various inverse problems. Recently Buterin and Kuznetsova [20] proved local solvability and stability for the inverse problem by two spectra for the nonself-adjoint Sturm-Liouville operator. They also took splitting of multiple eigenvalues into account. However, Inverse problem 1.1 appears to be more interesting for investigation because of generalized weight numbers changing their structure.
In [15], some results on local solvability and stability were obtained for the inverse problem of recovering the non-self-adjoint Sturm-Liouville operator with the Dirichlet boundary conditions from GSD. However, the authors of [15] considered only such perturbations of GSD that preserve eigenvalue multiplicities. In the present paper, arbitrary perturbations that can change eigenvalue multiplicities are studied. We obtain special conditions on a GSD perturbation, which allow GSD to change their structure, but the perturbation of the potential remains small in L 2 -norm. In our sequel studies [21,22], the results of this paper are applied to investigate the non-self-adjoint Sturm-Liouville problem with arbitrary entire functions in the boundary condition.
The paper is organized as follows. In Sect. 2, our main Theorems 2.

Main results
We start with some preliminaries. Let Φ(x, λ) be the solution of Eq. (1.1) satisfying the is called the Weyl function of the problem L. Weyl functions are natural spectral characteristics for various self-adjoint and non-self-adjoint operators (see [5,8]). It is easy to show that Φ(x, λ) for each fixed x ∈ [0, π] and M(λ) are meromorphic functions in the λ-plane having the poles at λ = λ n , n ≥ 0. In [13], the following representation has been obtained: The coefficients {M n } ∞ n=0 can be uniquely determined by the generalized weight numbers {α n } ∞ n=0 and vice versa from the linear system ν k=0 α n+ν-k M n+m n -k-1 = δ ν,0 , n ∈ S, ν = 0, m n -1.
In particular, M n = α -1 n , if m n = 1. Thus, Inverse problem 1.1 by the GSD is equivalent to the following one.

Inverse problem 2.1 Given the data
Further we study Inverse problem 2.1 instead of Inverse problem 1.1.
Along with the problem L, we consider complex numbersG = {λ n ,M n } ∞ n=0 . We will show that, if the dataG are "sufficiently close" to G = {λ n , M n } ∞ n=0 in some sense (a rigorous formulation is given in Theorem 2.2), thenG will correspond to some BVPL = L(q(x),h,H) of the same form as L, but with different coefficients. We agree that, if a certain symbol γ is related to L, then the symbolγ with tilde is the analogous object constructed by the dataG.
Then there exists δ 0 > 0 (depending on L) such that, for any δ ∈ (0, δ 0 ] and any complex numbersG = {λ n ,M n } ∞ n=0 satisfying the conditions there exist a complex-valued functionq ∈ L 2 (0, π) and complex numbersh,H being the solution of Inverse problem 2.1 forG. Moreover, Here and below, the same symbol C is used for various positive constants depending on L and δ 0 and independent of δ,G, etc.
Recall that the function M N (λ) is fixed and all its poles lie inside γ N . Condition (2.2) for sufficiently small δ implies that all the poles ofM N (λ) also lie inside γ N . Moreover, the following estimate holds: However, the values λ n andλ n can have different multiplicities. Namely, multiple values λ n can split into smaller groups, so S ⊆S,m n = m n = 1 for all n > N .
We also obtain local solvability and stability conditions on the discrete data, not involving the continuous functionM N . Such conditions are provided in the following theorem.

Main equation
The goal of this section is to derive the main equation in a Banach space, which plays a crucial role in the proofs of the main results. Our approach is based on the method of spectral mappings (see [14]). Since a part of the proofs repeat the standard technique of [8, Sect. 1.6] and [13], we omit the details and focus on the differences of our methods from the classical ones.
Let us consider two BVPs L = L(q(x), h, H) andL = (q(x),h,H) with different coefficients. Fix N = N(L) and the contour γ N . Assume that the eigenvalues {λ n } N n=0 lie inside γ N and {λ n } ∞ n=N+1 lie outside γ N . Define For K ∈ N, consider the region Υ K := {λ ∈ C :p < Re λ < (K + 1 2 ) 2 , | Im λ| < p} and its boundary υ K := ∂Υ K with the counter-clockwise circuit. The constant p is chosen so that Re λ n > -p, Reλ n > -p, | Im λ n | < p, | Imλ n | < p for all n ≥ 0. Using the contour integration (see [8, p. 53] for details), we obtain the relation whereM :=M -M. Applying the residue theorem and observing that the function (M(λ) -M N (λ)) is analytic inside γ N , we obtain the relation We use relation (3.1) for deriving the main equation of Inverse problem 2.1 in a special Banach space. Denote by B C the Banach space of functions continuous on γ N with the norm Define the Banach space Here and below the lower indices C and D mean a "continuous" and a "discrete" part, respectively.
For every x ∈ [0, π], define the element The elementψ(x) is defined analogously by usingφ instead of ϕ. For the solution ϕ(x, λ), the following standard asymptotics is valid: where ρ = √ λ, Re ρ ≥ 0. Using (1.4) and (3.2), we obtain the estimates where the constant C does not depend on x and n. Analogous relations are valid forφ(x, λ).
For each fixed x ∈ [0, π], we define the linear bounded operator R(x): B → B as follows: . Taking λ ∈ γ N , λ =λ n and λ = λ n , n > N , in (3.1), we obtain the so-called main equation in the Banach space B: (3.5) Here, I is the identity operator in B. Now suppose that the problem L and the dataG = {λ n ,M n } ∞ n=0 satisfy the conditions of Theorem 2.2. We choose δ 0 to be so small that the values {λ n } N n=0 definitely lie inside γ N and the values {λ n } n>N definitely lie outside γ N . It is not known whether the dataG correspond to any problemL or not. Let ψ(x) and R(x) be constructed by L andG via the formulas above. Then the following assertion holds.

Lemma 3.1
For each fixed x ∈ [0, π], the following estimate is valid: where the constant C does not depend on x, δ and on the choice ofG satisfying the conditions of Theorem 2.2.
Proof In order to prove (3.6), it is sufficient to obtain similar estimates for The standard estimates (see [8, Lemma 1.6.2]) imply Combining the latter relation with (3.4), (2.3) and the obvious estimates we get One can similarly study the components R DC (x) and R DD (x) and finally arrive at the assertion of the lemma.

Proofs
The aim of this section is to prove Theorems 2.
Define the functioñ It is easy to check that Consequently, Lemma 4.1 implies Similar estimates also hold for λ N+k replaced byλ N+k . Introduce the functions

Case of Dirichlet boundary conditions
In this section, we formulate the results similar to Theorems 2.2 and 2.3 for the case of the Dirichlet boundary conditions. Since the proofs for different types of boundary conditions are quite similar, we provide only formulations in this section. Consider the boundary problem L 0 = L 0 (q(x)) for equation (1.1) with the complexvalued potential q ∈ L 2 (0, π) and the Dirichlet boundary conditions y(0) = y(π) = 0. (5.1) Denote by {λ n } ∞ n=1 the eigenvalues of L counted with their multiplicities and numbered so that |λ n | ≤ |λ n+1 |, n ≥ 1. Equal eigenvalues are consecutive.
Let Φ(x, λ) be the solution of equation ( The following inverse problem is analogous to Inverse problem 2.1.