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Recovering Differential Operators with Nonseparated Boundary Conditions in the Central Symmetric Case

Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered.


Introduction.
We study inverse spectral problems for the Sturm-Liouville operator ℓy := y ′′ + q(x)y, x ∈ (0, π), on the finite interval (0, π) with non-separated boundary conditions. Inverse problems consist in recovering coefficients of differential operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, geophysics, electronics and other branches of natural sciences and engineering. Inverse problems also play an important role in solving nonlinear evolution equations in mathematical physics. Inverse problems for differential operators with separated boundary conditions have been studied fairly completely by many authors (see the monographs [1][2][3][4][5] and the references therein). Inverse problems for Sturm-Liouville operators with non-separated boundary conditions, which are more difficult for the investigation, were treated in [6][7][8][9][10][11][12][13][14][15][16][17] and other works. In particular, the periodic boundary value problem was considered in [6,7,9,14]. Stankevich [6] suggested a statement of the inverse problem and proved the corresponding uniqueness theorem. Marchenko and Ostrovskii [7] gave the characterization of the spectrum for the periodic boundary value problem in terms of a special conformal mapping. Conditions considered in [7] are difficult to verify. Another method, used in [9], allowed to obtain necessary and sufficient conditions for the solvability of the inverse problem for the periodic case that are easier to verify. Similar results were obtained in [9] for another type of boundary conditions, namely y ′ (0) − ay(0) + by(π) = y ′ (π) + dy(π) − by(0) = 0.
In this paper we study the case when the potential q is symmetric with respect to the middle of the interval, i.e. q(x) = q(π − x) a.e. on (0, π). The symmetric case requires nontrivial modifications in the method and allows us to specify less spectral information than in the general case. Some results for the symmetric case were obtained in [10] and [17]. In the present paper for the symmetric case we construct the solution of the inverse spectral problem and give the characterization of the spectrum for various non-separated boundary conditions. For convenience of readers in Section 2 we describe briefly the known results for the general (non-symmetric) case.
Inverse problem 1. Given the spectral data {γ n , α n } n≥1 , construct the potential q. This inverse problem is related to the case of separated boundary conditions. It is known that the specification of the spectral data {γ n , α n } n≥1 uniquely determines the potential q. The global solution of Inverse problem 1 can be constructed by the transformation operator method or by the method of spectral mappings (see [1][2][3][4][5] for details). In particular, these methods allow one to describe necessary and sufficient conditions for the solvability of Inverse problem 1 which are presented in the next theorem.
Theorem 1. For real numbers {γ n , α n } n≥1 to be the spectral data for a certain BVP L 0 with a real potential q(x) ∈ L 2 (0, π), it is necessary and sufficient that (9) and (12) hold.
Lemma 2. Fix n ≥ 1. Relation δ(γ n ) = 0 holds iff γ n lies at one of the endpoints of the gap a n .
Denote by J the set of sequences Ω = {ω n } n≥1 such that ω n = 0 if γ n lies at one of the endpoints of the gap a n , and ω n = ±1, otherwise. Clearly, if Ω is the Ω -sequence for L, then Ω ∈ J. The next theorem [9] establishes necessary and sufficient conditions for the solvability of Inverse problem 2.
The next theorem [9] shows that one of the endpoints of each gap can be chosen arbitrary taking only asymptotics into account.
Theorem 3 [9]. Let real numbers θ n of the form θ n = n 2 + α + κ n , {κ n } ∈ l 2 , θ n < θ n+1 , be given. Then there exists a real function q(x) ∈ L 2 (0, π) (not unique!) such that for this potential the number θ n lies at one of the endpoints of the gap a n for all n ≥ 1.
Let us consider the inverse problem for the BVP L 0 . In the central symmetric case q(x) ∈ L ′ 2 (0, π) we do not need to specify the weight numbers {α n } n≥1 ; it is sufficient to specify only the spectrum γ.
It is known [1][2][3][4][5] that for the central symmetric case the specification of the spectrum γ = {γ n } n≥1 of the BVP L 0 uniquely determines the potential q. In order to construct q, one can calculate d(λ) via (10) and the weight numbers α n = (−1) nḋ (γ n ), and then find q by solving Inverse problem 1. Moreover, the characterization of the spectrum of L 0 is given by the following assertion.
Theorem 5. For real numbers {γ n } n≥1 to be the spectrum of a BVP L 0 with a real potential q(x) ∈ L ′ 2 (0, π), it is necessary and sufficient that (9) holds. Proof. The necessity is obvious. We will prove the sufficiency. Let real numbers {γ n } n≥1 satisfying (9) be given. We construct d(λ) via (10) and the numbers {α n } n≥1 by α n = (−1) nḋ (γ n ). Our plan is to use Theorem 1. For this purpose we should obtain the asymptotics for the numbers α n . This seems to be difficult because the function d(λ) is by construction the infinite product. But fortunately, for calculating the asymptotics of α n one can also use Theorem 1, as an auxiliary assertion. Indeed, by virtue of Theorem 1 there exists a potential q(x) ∈ L 2 (0, π) (not unique) such that γ = {γ n } n≥1 is the spectrum ofL 0 := L 0 (q) with this potential. Then d(λ) is the characteristic function ofL 0 , and consequently, (13) holds. Therefore, (12) is valid. Then, by Theorem 1 there exists a unique potential q(x) ∈ L 2 (0, π) such that {γ n , α n } n≥1 are the spectral data of L 0 (q). Since β n = (−1) n−1 , n ≥ 1, it follows from Theorem 4 that q(x) ∈ L ′ 2 (0, π). Theorem 6 [9]. q(x) ∈ L ′ 2 (0, π) iff γ n lies at one of the endpoints of the gap a n for all n ≥ 1.
We will write a n ∈ I 0 , if the length of the gap a n is equal to zero, and a n ∈ I 1 , otherwise. Let us now consider the inverse problem for the periodic BVP L. In the general case in Inverse problem 2 we have to specify Λ, γ and Ω. In the central symmetric case we do not need γ. On the other hand, the sequence Ω = {ω n } n≥1 does not bring any information because in the central symmetric case ω n = 0 for all n ≥ 1. Unfortunately, in contrast to the separated boundary conditions, for the periodic case the specification of the spectrum Λ does not uniquely determine the potential q, and we need additional information. For this purpose we introduce the sequence E = {ε n } n≥1 , where ε n = 0 if a n ∈ I 0 , ε n = 1, if a n ∈ I 1 and γ n lies at the right endpoint of a n , ε n = −1, if a n ∈ I 1 and γ n lies at the left endpoint of a n . The sequence E = {ε n } n≥1 is called the E -sequence for the potential q(x) ∈ L ′ 2 (0, π). The inverse problem for the periodic BVP L in the central symmetric case is formulated as follows.
Inverse problem 4. Given Λ and E, construct q. Theorem 7 [9]. Let q(x) ∈ L ′ 2 (0, π). Then the specification of Λ and E uniquely determines the potential q. The solution of Inverse problem 1 can be found by the following algorithm.
3) Find zeros Λ + = {λ + n } n≥1 of p + (λ). 4) Construct γ = {γ n } n≥1 as follows: γ n lies at the right endpoint of a n if ε n = 1; γ n lies at the left endpoint of a n if ε n = −1, and γ n = a n if ε = 0. 5) Using {γ n } construct the potential q(x) ∈ L ′ 2 (0, π) by solving Inverse problem 3. Denote by J 1 the set of sequences E = {ε n } n≥1 such that ε n = 0 if a n ∈ I 0 , and ε n = ±1 if a n ∈ I 1 . Clearly, if E is the E -sequence for q, then E ∈ J 1 . The next theorem [9] establishes necessary and sufficient conditions for the solvability of Inverse problem 4.
In particular, this means that the sequence Λ = {λ n } n≥0 coincides with the spectrum of the BVP L for the potential q. Since γ n lies at one of the endpoints of the gap a n for all n ≥ 1, it follows from Theorem 6 that q(x) ∈ L ′ 2 (0, π). Now it is clear that E is the E -sequence for q.
Denote by J ′ the set of sequences η = {η n } n≥0 such that (i) η n = ±1 if the corresponding zeros of the functions r(λ) and r(λ) − 4b are simple, and η n = 0 otherwise; (ii) there exists N (depending on the sequence) such that η n = 1 for all n > N.
Clearly, if η is the η -sequence for B, then η ∈ J ′ . The next theorem gives us necessary and sufficient conditions for the solvability of Inverse problem 5.
We note that in [17] stability of the solution of the inverse problem for the BVP B is established.