AN INVERSE PROBLEM FOR THE STURM-LIOUVILLE PENCIL WITH ARBITRARY ENTIRE FUNCTIONS IN THE BOUNDARY CONDITION

. The Sturm-Liouville pencil is studied with arbitrary entire functions of the spectral parameter, contained in one of the boundary conditions. We solve the inverse problem, that consists in recovering the pencil coeﬃcients from a part of the spectrum satisfying some conditions. Our main results are 1) uniqueness, 2) constructive solution, 3) local solvability and stability of the inverse problem. Our method is based on the reduction to the Sturm-Liouville problem without the spectral parameter in the boundary conditions. We use a special vector-functional Riesz-basis for that reduction.


(Communicated by Fioralba Cakoni)
Abstract. The Sturm-Liouville pencil is studied with arbitrary entire functions of the spectral parameter, contained in one of the boundary conditions. We solve the inverse problem, that consists in recovering the pencil coefficients from a part of the spectrum satisfying some conditions. Our main results are 1) uniqueness, 2) constructive solution, 3) local solvability and stability of the inverse problem. Our method is based on the reduction to the Sturm-Liouville problem without the spectral parameter in the boundary conditions. We use a special vector-functional Riesz-basis for that reduction.
The paper concerns an inverse spectral problem for L. Inverse problems of spectral analysis consist in recovering differential operators from their spectral characteristics. In applications, those spectral characteristics are usually related with observed data, and operator coefficients being recovered are related with unknown properties of a medium.
The most complete results of inverse problem theory have been obtained for the Sturm-Liouville equations in the form (1.1)-(1.3) with the constant coefficients f 1 and f 2 (see, e.g., the monographs [17,27,28,35]). Pencils of differential operators with eigenparameter-dependent boundary conditions are more difficult for investigation, since such problems are rarely self-adjoint. Inverse problems for pencils, having linear dependence on the spectral parameter in the boundary conditions, were studied in [9,10,21,44,46] and many other papers. The case of boundary conditions, containing rational Herglotz-Nevanlinna functions or arbitrary polynomials of λ, also has been considered (see [3,4,13,15,18,19,22,33,45]).
In the present paper, we study the differential pencil (1.1)-(1.3) with arbitrary entire functions in the boundary condition. As far as we know, inverse problems for this class of pencils have not been investigated before. The behavior of the spectrum of the problem L substantially depends on the functions f j (λ), j = 1, 2. However, we assume that a subsequence of the eigenvalues is given, that satisfy some specific properties. This subsequence is used for recovering the potential q(x) and the coefficient h. Such inverse problem statement generalizes various inverse spectral problems being actively studied in recent years, in particular: • the Hochstadt-Lieberman problem [10, 20, 24-26, 29, 34, 36]; • the inverse transmission eigenvalue problem [6,[30][31][32]; • partial inverse problems for Sturm-Liouville operators with discontinuities [23,38,41,42]; • partial inverse problems for quantum graphs [5,7,8,40,43].
The listed problems are applied in mechanics, geophysics, nanotechnology, acoustics and other branches of science and engineering. We discuss applications of our results to these inverse problems in Section 5 in more details.
The eigenvalues of the problem L coincide with the zeros of the characteristic function Let {λ n } n∈N be a sequence of the eigenvalues of L, satisfying the following conditions.
Note that the sequence {λ n } n∈N may include only a part of all the eigenvalues. Suppose that λ n is a zero of ∆(λ) of multiplicity at least m n ∈ N, n ∈ N. The assumption λ n = 0, n ∈ N, is imposed for simplicity. The case λ n = 0 requires minor technical modifications.
Define ω := h + 1 2 π 0 q(x) dx. We study the following inverse problem. Inverse Problem 1.1. Given eigenvalues {λ n } n∈N , satisfying (A 1 ) and (A 2 ), the corresponding numbers {m n } n∈N and ω, find the potential q(x) and the coefficient h.
The functions f 1 (λ) and f 2 (λ) are supposed to be known a priori. In applications, the constant ω often can be found from the asymptotics of the eigenvalues {λ n } n∈N .
In this paper, we prove the uniqueness theorem, local solvability and stability for Inverse Problem 1.1, and also provide a constructive algorithm for solution of this problem. Our technique develops the ideas of [5,6,42]. In order to solve the inverse problem, we reduce it to the classical Sturm-Liouville problem with boundary conditions independent of λ. For the reduction, a special vector-functional Riesz basis is used. The proof of the Riesz-basicity relies on the theory of entire functions and nonharmonic analysis of exponential sequences in the form {exp(iθ n t)}. Note that our methods require no self-adjointness. We work with the complex-valued potential q(x) and possibly multiple eigenvalues. To deal with the inverse problem, having eigenparameter independent boundary conditions, we apply the Borg-type theorem, recently proved in [11] for the complex-valued potential.
The paper is organized as follows. In Section 2, the uniqueness theorem for Inverse Problem 1.1 is proved. We introduce a special exponential sequence, related to the given eigenvalues {λ n } n∈N and their multiplicities {m n } n∈N . The proof of the uniqueness theorem is based on the completeness of that sequence and on the theory of entire functions. Section 3 is devoted to a constructive solution of Inverse Problem 1.1. We require the exponential sequence, introduced in Section 2, to be a Riesz basis. A crucial step of our algorithm for solving the inverse problem is recovering a vector-function from its coordinates by that Riesz basis. In Section 4, local solvability and stability are proved for Inverse Problem 1.1 under some additional requirements on the subspectrum {λ n } n∈N and the functions f 1 (λ) and f 2 (λ). In Section 5, we discuss applications of our results to various problems of the modern inverse spectral theory.
Along with the problem L, we consider the boundary value problemL = L(q(x),h) of the same form as L, but with different coefficientsq(x) andh. The entire functions f 1 (λ) and f 2 (λ) forL are the same as for L. Let us agree that, if a symbol γ denotes an object related to L, the symbolγ with tilde denotes the analogous object related toL. Now we are ready to formulate the uniqueness theorem for Inverse Problem 1.1.
Theorem 2.1. Suppose that the data {λ n , m n } n∈N satisfy the assumptions (A 1 )-(A 3 ), and that λ n =λ n , m n =m n for n ∈ N, and ω =ω. Then q(x) =q(x) for a.a.
x ∈ (0, π) and h =h. Thus, the data {λ n } n∈N , {m n } n∈N and ω uniquely specify the potential q(x) and the boundary condition coefficient h.
In order to prove Theorem 2.1, we need some standard results. For any real a > 0, we denote by B 2 a the Paley-Wiener class of entire functions of exponential type not greater than a, belonging to L 2 (R). The following proposition easily follows from [37,Theorem 1.1]. For convenience, we formulate it directly for the sequence E , defined above.
The function q(x) and the constant h can be constructively recovered from M (λ) by the method of spectral mappings (see [17]).
3. Constructive solution. In this section, we develop an algorithm for constructive solution of Inverse Problem 1.
1. An important role in our solution is played by the vector-functional sequence {v n }, defined in (3.5). We show that this sequence is complete in an appropriate Hilbert space and, moreover, that it is an unconditional basis. The main equations (3.6) are derived, which allow us to reduce Inverse Problem 1.1 to the classical inverse problem without the spectral parameter in the boundary conditions. Finally, we obtain the constructive Algorithm 3.4 for solving Inverse Problem 1.1.
For simplicity, assume that m n = 1, n ∈ N. The multiplicities of the eigenvalues {λ n } n∈N may be greater than 1, but we will not use this fact in our reconstruction procedure. In order to work with multiple eigenvalues, one can use the ideas of the paper [6] and of [5,Remark 2].
Note that the functions κ j (ρ), j = 1, 2, from Lemma 2.3 have the form Substituting (3.1) together with the relations of Lemma 2.3 into (1.4) and taking λ = ρ 2 n , we get Let the eigenvalues {λ n } n∈N and the number ω be given. Note that all the values in (3.2) can be easily constructed by the given data, except the functions K(t) and N (t). Consider (3.2) as a system of equations with respect to these functions. It is convenient to rewrite (3.2) in the form . Introduce the complex Hilbert space H := L 2 (−π, π) ⊕ L 2 (−π, π) of two-element vector-functions with elements from L 2 (−π, π). The scalar product and the norm in H are defined as follows: Define the functions Clearly, u ∈ H and v n ∈ H for n ∈ Z 0 . We rewrite the relation (3.3) in the following form: We call the relations (3.6) the main equations of Inverse Problem 1.1, since they play a crucial role in our solution.
Let us investigate the properties of the vector-functional sequence {v n } n∈Z0 .
Lemma 3.1. The relation v n = c n u n holds for n ∈ Z 0 , where c n are nonzero constants and Proof. Using (1.4), (3.5) and (3.7), we derive The assumption (A 2 ) implies v n = 0, n ∈ Z 0 . Since the values ϕ(π, ρ 2 n ) and ϕ (π, ρ 2 n ) cannot both equal zero, we also have u n = 0, n ∈ Z 0 . These arguments lead to the assertion of the lemma.
Proof. In view of Lemma 3.1, it is sufficient to prove the completeness of the sequence {u n } n∈Z0 . Consider an element z = z 1 z 2 ∈ H, such that (z, u n ) H = 0 for all n ∈ Z 0 . In the elementwise form, we have Consequently, the function has zeros {ρ n } n∈Z0 . Note that Z(ρ) can have a singularity at ρ = 0. However, the functionZ (ρ) := ρ(ρ − ρ 1 ) −1 Z(ρ) is entire and belongs to B 2 2π . Applying Proposition 2.2 toZ(ρ) and using the completeness of the sequence Further we need some additional assumptions on the given part of the spectrum.
A reader can find more information about Riesz bases in [17, Section 1.8.5], [1] and [14]. Note that (A 3 ) follows from (A 4 ). Proof. In order to prove the theorem, it is sufficient to show that {u n } n∈Z0 is a Riesz basis in H. Using Lemma 2.3 and the relation (3.7), we obtain the asymptotics where (3.9) u 0 n (t) := cos(ρ n π) −i sin(ρ n π) exp(iρ n t), n ∈ Z 0 , and the O-estimate is uniform with respect to t ∈ [−π, π]. Taking (A 5 ) into account, we conclude that the sequences {u n } n∈Z0 and {u 0 n } n∈Z0 are quadratically close: Let us prove that {u 0 n } n∈Z0 is a Riesz basis in H. It is sufficient to show, that this sequence is complete and there exist positive constants M 1 and M 2 , such that for any sequence {b n } n∈Z0 of complex numbers the two-side inequality holds (see [14, Theorem 3.6.6]): We emphasize that the constants M 1 and M 2 do not depend on the sequence {b n } n∈Z0 . One can prove completeness of the sequence {u 0 n } n∈Z0 in H, by using arguments similar to the proof of Lemma 3.2.
Further we prove the inequality (3.10). Note that n∈Z0 b n u 0 Using (3.9), we calculate (u 0 n , u 0 k ) H = (cos(ρ n π) cos(ρ k π) + sin(ρ n π) sin(ρ k π)) Note that, for the functions e n := exp(iρ n t), we have Hence (u 0 n , u 0 k ) H = 2(e n , e k ) L2(−2π,2π) , n, k ∈ Z 0 . Consequently, the two-side inequality (3.10) holds for {u 0 n } n∈Z0 if and only if the similar inequality holds for the sequence E = {e n } n∈Z0 with the norm in L 2 (−2π, 2π). By virtue of (A 4 ), the sequence E is a Riesz basis in L 2 (−2π, 2π), so the two-side inequality for E is valid. Therefore {u 0 n } n∈Z0 is also a Riesz basis. Thus, the sequence {u n } n∈Z0 is complete by Lemma 3.2 and quadratically close to the Riesz basis {u 0 n } n∈Z0 . By virtue of [17, Proposition 1.8.5], the sequence {u n } n∈Z0 is also a Riesz basis. Taking Lemma 3.1 into account, we conclude that {v n } n∈Z0 is an unconditional basis in H.
Using the basisness of the sequence {v n } n∈Z0 , now we can constructively solve Inverse Problem 1.1 as follows. We have to reconstruct the potential q(x) and the coefficient h of the boundary condition.

4.
Local solvability and stability. In this section, local solvability and stability of Inverse Problem 1.1 are investigated. We consider a small perturbation of the subspectrum {λ n } n∈N of the problem L. It is shown that, by applying Algorithm 3.4 to the perturbed sequence {λ n } n∈N , one can construct another problem L = L(q(x),h). The numbers {λ n } n∈N will be among the eigenvalues ofL, and the coefficientsq andh will be sufficiently close to q and h, respectively, in an appropriate sense. Our results are rigorously formulated in Theorem 4.1 below. The results of such type for the standard Sturm-Liouville inverse problems are provided in the book [17] (see Theorems 1.6.4, 1.6.5, 1.8.1). In order to study local solvability and stability, we impose the additional assumption on the functions f j (λ), j = 1, 2.
The assumption (A 6 ) is natural for applications. The estimate (4.2) together with (3.5) imply For j = 0, 1, denote by L j = L j (q(x), h) the boundary value problem for equation (1.1) with the boundary conditions (1.2) and y (j) (π) = 0. Clearly, the problem L j has the characteristic function η j (λ) := ϕ (j) (π, λ). It is well-known (see, e.g., [17]), that the functions η j (λ), j = 0, 1, have countable sets of zeros. Denote by {θ nj } n≥0 the zeros of η j (λ) counted with their multiplicities. The numbers {θ nj } n≥0 are the eigenvalues of L j . Choose among {θ nj } n≥0, j=0,1 the eigenvalue with the largest multiplicity and denote this multiplicity by p. Note that p < ∞, since the eigenvalues of L j are simple for large n. In the special case, when q(x) and h are real, we have p = 1.
The next theorem is the main result of this section.   Fix the problem L and its subspectrum {λ n } n∈N , satisfying the conditions of Theorem 4.1. In order to prove the theorem, we will apply Algorithm 3.4 to the data {{λ n } n∈N , ω}. If a symbol γ denotes an object related to the problem L, further we will denote byγ with tilde the similar object, constructed by {λ n } n∈N and ω. The same symbol C will be used for various positive constants, depending only on L.
Further we intend to apply the following proposition, which is a combination of [17, Proposition 1.
is also a Riesz basis in B. Furthermore, suppose that there is τ ∈ B, and τ n = (τ, ξ n ) B for all n. Then for any sequence {τ n }, such that there exists a uniqueτ ∈ B, such thatτ n = (τ ,ξ n ) B for all n. Moreover, where the constant C depends only on {ξ n } and τ .
Using Proposition 4.3 together with Lemma 4.2 and the main equations (3.6), we arrive at the following assertion.
is a solution of the system (ũ − ,ṽ n ) =w n , n ∈ Z 0 . But, by virtue of Lemma 4.4, the solution of this system is unique in H. Henceũ =ũ − , and the lemma is proved.
Define the functions  It follows from Lemma 4.5, thatη j (λ) are entire functions of λ. One can easily show that, for j = 0, 1,η j (λ) have countable sets of zeros {θ nj } n≥0 . Lemma 4.6. There exists ε > 0, such that for any complex-valued functionsK and N from L 2 (−π, π), satisfying the estimate the zeros of the functions η j (λ) andη j (λ), j = 0, 1, can be numbered according to their multiplicities, so that Proof. One can prove this lemma similarly to [6,Lemma 3] and [42,Lemma 5], relying on the relations (3.11), (3.12), (4.10), (4.11). The only difference is, that the functions K(t) and N (t) are complex-valued, so the functions η j (λ), j = 0, 1, may have multiple zeros. However, there is only a finite number of multiple zeros, so it is sufficient to prove the following assertion.
Proposition 4.7 establishes local solvability and stability of the inverse Sturm-Lioville problem by the two spectra {θ nj } n≥0 , j = 0, 1. For a real-valued potential q(x) and h ∈ R, Proposition 4.7 is the classical result by Borg (see [17,Theorem 1.8.1]). In the complex case, the similar proposition has been recently proved by Buterin and Kuznetsova (see [11]) for the Dirichlet boundary condition y(0) = 0 at the left. The case of the Robin boundary condition (1.2) has no principal differences. Note that, in the complex case, some of the eigenvalues {θ nj } n≥0 can be multiple, and they can split under a small perturbation. This case is included into Proposition 4.7.

5.
Applications. In this section, we transform several important problems of spectral theory to the form similar to (1.1)-(1.3). In some applications, the Dirichlet boundary condition y(0) = 0 is imposed instead of the Robin condition (1.2). However, there is no significant difference between these two cases.
5.1. Hochstadt-Lieberman problem. Consider the boundary value problem (5.1) − y + q(x)y = λy, x ∈ (0, π), y (0) − hy(0) = y (π) + Hy(π) = 0, where q ∈ L 2 (0, π) is a real-valued potential, h, H ∈ R. Suppose that the potential q(x) for x ∈ (π/2, π) and the coefficient H are known a priori. The Hochstadt-Lieberman problem consists in recovering q(x) on (0, π/2) and h from the spectrum {λ n } n≥0 of (5.1). This problem is also often called the half-inverse problem or the inverse problem by mixed data. Note that, without any preliminary information on the potential, not less than two spectra are required for its reconstruction (see, e.g., [17]). The uniqueness theorem for the half-inverse problem has be proved in the classical study by Hochstadt and Lieberman [24]. Later on, constructive methods for its solution have been developed and conditions for solvability have been obtained in [10,26,29,34,36].
The similar problems with the potential known on the interval (a, π) for arbitrary a ∈ (0, π/2) have been studied in [20,25] and other papers. Our results also can be applied to this case.
The problem (5.2) attracted much attention of both mathematicians and physicists in connection with the inverse acoustic scattering problem (see [6,12,[30][31][32]39] and references therein). In particular, McLaughlin and Polyakov [30] stated the problem of recovering the potential on the interval 0, |a−1| 2 , a = 1, by a part of the spectrum, called an almost real subspectrum. In [30], the uniqueness of solution for this problem has been established. Further the theory of the McLaughlin-Polyakov inverse problem has been developed in the studies [6,31,32]. The majority of those results are particular corollaries of the general approach of the present paper. , where λ is the spectral parameter; q ∈ L 2 (0, 1) is a real-valued function; 0 < d ≤ 1/2 is the discontinuity position; a j and h j are real numbers for j = 1, 2, a 1 > 0.
Discontinuous Sturm-Liouville problems arise in geophysical models for oscillations of the Earth and in other applications (see, e.g., [17,Section 4.4]). Suppose that the potential q(x) is known on (a, 1), a ≤ d, and our goal is to recover the potential on the remaining part of the interval by a part of the spectrum. Such partial inverse problems have been studied in [23,38,41,42].
Similarly to the Hochstadt-Lieberman problem, discontinuous partial inverse problems can be reduced to the form with entire functions in the right-hand side boundary condition. Uniqueness theorems, constructive algorithms, local solvability and stability for these problems can be easily obtained from the results of the present paper. 5.4. Partial inverse problems for quantum graphs. Consider the system of Sturm-Liouville equations (5.3) − y j (x j ) + q j (x j )y j (x j ) = λy j (x j ), x j ∈ (0, π), j = 1, m, with the matching conditions y 1 (π) = y j (π), j = 2, m, The potentials q j belong to L 2 (0, π), j = 1, m. The system (5.3)-(5.5) corresponds to the Sturm-Liouville operator on a star-shaped graph (see [5,40]).
Differential operators on geometrical graphs are also called quantum graphs. There is an extensive literature, devoted to such operators (see, e.g., [2,16,47] and references therein). Quantum graphs have applications in organic chemistry, mesoscopic physics, nanotechnology, theory of waveguides, etc.
Suppose that the potentials {q j } m j=2 are known a priori, and it is required to find q 1 , by using a part of the spectrum of the problem (5.3)-(5.5). Problems of such kind are called partial inverse problems for differential operators on graphs.
For j = 1, m, denote by S j (x j , λ) the solution of (5.3) under the initial conditions S j (0, λ) = 0, S j (0, λ) = 1. The solutions S j (x j , λ) for j = 2, m can be constructed by the known potentials {q j } m j=2 . One can easily check, that the eigenvalues of (5.3)-(5.5) coincide with the eigenvalues of the boundary value problem for equation (5.3) for j = 1 with the boundary conditions y 1 (0) = 0, f 1 (λ)y 1 (π) + f 2 (λ)y 1 (π) = 0, where f j (λ), j = 1, 2, are entire functions of λ, defined as follows: Therefore the results of [5,40] for the partial inverse problem of recovering q 1 follow from the results of the present paper. Note that the other partial inverse problems for quantum graphs, studied in [7,8,43], also can be reduced to the form similar to (1.1)-(1.3).