A partial inverse problem for the Sturm-Liouville operator on the lasso-graph

The Sturm-Liouville operator with singular potentials on the lasso graph is considered. We suppose that the potential is known a priori on the boundary edge, and recover the potential on the loop from a part of the spectrum and some additional data. We prove the uniqueness theorem and provide a constructive algorithm for the solution of this partial inverse problem.

1. Introduction. The paper concerns the spectral theory of differential operators on graphs, also called quantum graphs. Such operators consist of geometrical graphs, eqipped with differential expressions on the edges and appropriate matching conditions in the vertices. A nice elementary introduction to the theory of quantum graphs is provided in [1,14]. Quantum graphs model wave propagation through a domain being a thin neighborhood of a graph. Such models arise in organic chemistry, mesoscopic physics, nanotechnology, theory of waveguides and other branches of science [15]. In [23], differential operators on graps are used for analysis of mechanical models, consisting of strings and rods, electrical circuits, hydro and heating networks, etc.
In this paper, we study an inverse spectral problem for a quantum graph. Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems arise in various applications, when it is required to determine properties of the medium (described by coefficients of some differential operator), by using observable data (e.g., spectral data). Basic results of inverse problems theory for differential operators on intervals can be found in the monographs [7,17,18,24]. The recent paper [31] contains a good overview of inverse spectral problems for differential operators on graphs. The majority of results in this direction concern the second-order Sturm-Liouville operators −y j + q j y j . In particular, in [32] a constructive algorithm was developed for recovering the Sturm-Liouville potentials q j on a graph of an arbitrary structure.
However, reconstruction of the operator coefficients on the whole graph usually requires a large amount of given spectral data, e.g. multiple spectra, corresponding to various conditions in boundary vertices. We are interested in the question, how to decrease an amount of given data, and therefore we focus on the so-called partial inverse problems. We suppose that the potential is known a priori on a part of the graph, and reconstruct the potential on the remaining part, using some partial information on the spectrum. Such problems generalize the well-known Hochstadt-Lieberman problem on a finite interval [9]. Partial inverse problems have been studied in [3][4][5]22,[27][28][29] for star-shaped graphs. However, the methods, developed in those works, can not be straightforward applied to more general graphs, because the latter ones have more complex behavior of the spectrum.
In the present paper, we obtain the first results on partial inverse problems for a graph with a loop. We formulate a partial inverse problem for the Sturm-Liouville operator on a lasso graph (see Figure 1), prove the uniqueness theorem and provide a constructive algorithm for solution of this problem. The algorithhm consists in the reduction of the partial inverse problem on the graph to a complete inverse problem on a finite interval. We develop the technique of [3][4][5], based on the Riesz basis property of some systems of vector functions. Note that complete inverse problems for differential operators on lasso graphs were studied in [16,19,21]. We hope that in the future, our results will be generalized for graphs with a more complicated structure.
Another feature of the paper is that the operator with singular potentials q j ∈ W −1 2 is studied (see rigorous definitions in Section 2). Inverse problems for Sturm-Liouville operators with potentials from this class on a finite interval were investigated in [11][12][13]. However, there are only a few results for such operators on graphs (see [5,6]). The methods of the present paper work for W −1 2 , as well as for L 2 , so we choose the more general class. Nevertheless, in order to recover the potential on the loop, we need to solve the periodic inverse Sturm-Liouville problem, which has been studied only for regular potentials from L 2 (see [20,26]). We generalize the periodic inverse problem solution for singular potentials in Section 3. This auxiliary step can be treated as a separate result.
The paper is organized as follows. In Section 2, we state the boundary value problem on the lasso graph and study asymptotic properties of its eigenvalues. Section 3 is devoted to the periodic inverse Sturm-Liouville problem, which is further used as an auxiliary step for recovering the potential on the loop. In Section 4, we formulate the partial inverse problem, provide our main results and proofs.
2. Asymptotic formulas for eigenvalues. Consider the lasso graph G, represented in Figure 1. The edge e 1 is a boundary edge of length l 1 = m ∈ N, the edge e 2 is a loop of length l 2 = 1. Introduce a parameter x j for each edge e j , j = 1, 2, x j ∈ [0, l j ]. The value x 1 = 0 corresponds to the boundary vertex, and x 1 = m corresponds to the internal vertex. For the loop e 2 , both ends x 2 = 0 and x 2 = 1 correspond to the internal vertex.
Let y = [y j (x j )] j=1,2 be a vector function on the graph G. Consider Sturm-Liouville expressions θ j y j := −y j + q j (x j )y j , j = 1, 2, on the edges of G, where q j , j = 1, 2, are real-valued functions from W −1 2 (0, l j ). This means that q j = σ j , σ j ∈ L 2 (0, l j ), where the derivative is understood in the j = y j − σ j y j , j = 1, 2. Then the differential expressions j can be understood in the following sense: We study the boundary value problem L for the Sturm-Liouville equations on the graph G: with the matching conditions 2 (1) = 0 in the internal vertex, and the Dirichlet boundary condition y 1 (0) = 0 in the boundary vertex. Note that, if the functions σ j , j = 1, 2, are continuous, the matching conditions (2) take the form . Consequently, if p = 0, we get the standard matching conditions, which express the Kirchhoff's law in electrical circuits, the balance of tension in elastic string networks, etc. (see [15,23]).
For each fixed j = 1, 2, let C j (x j , λ) and S j (x j , λ) be the solutions of the corresponding equation (1) under the initial conditions Further we use the following notations. Let B 2,a be the class of Paley-Wiener functions of exponential type not greater than a, belonging to L 2 (R). The symbols κ k,odd (ρ) and κ k,even (ρ) denote various odd and even functions from B 2,k , respectively. Note that where K, N ∈ L 2 (0, k). The notation {ξ n } stands for various sequences in l 2 .
The boundary value problem L has a purely discrete spectrum, consisting of real eigenvalues. The eigenvalues of L coincide with the zeros of the characteristic function with respect to their multiplicities. The asymptotic behavior of the eigenvalues is described by the following lemma.
3. Periodic inverse Sturm-Liouville problem. Inverse spectral problems on graphs with cycles usually generalize the periodic inverse problem on a finite interval. We describe the periodic problem on a loop e 2 in this section, because we need it for statement and solution of our partial inverse problem. Define In view of (3), the zeros {ν n } n∈N and {µ n } n∈Z of the entire functions h(λ) and d(λ) satisfy the following asymptotic formulas: √ µ n = |2πn| + ξ n , n ∈ Z.
Using the approach of [7, Theorem 1.1.4] and Hadamard's factorization theorem, one can obtain the following formulas: By virtue of (8) and (9), the infinite products in (10) and (11) converge uniformly on compact sets.
Inverse Problem 1. Given the sequences {ν n } n∈N , {µ n } n∈Z and the sequence of signs Ω, construct the function σ 2 .
Analogs of Inverse Problem 1 for the case of a regular potential q 2 ∈ L 2 (0, 1) have been studied in [20,26] (see also paper [30], where the solution of the periodic problem has been applied to the inverse problem on a graph). However, the known results can be easily generalized for the case q 2 ∈ W −1 2 (0, 1). Indeed, it is easy to check that Consequently, we have 2 (1, λ)h(λ)). (14) H(ν n ) = ω n d(ν n )(d(ν n ) + 4).
Using (10), we obtain the following formula for the derivative of h(λ): It is proved in [11], that the spectral data {ν n , β n } n∈N uniquely specify the function σ 2 , and an algorithm for the reconstruction is provided, based on the Gelfand-Levitan-Marchenko (GLM) equation: Here (18) F (x, y) = ϕ(x+y)−ϕ(x−y), ϕ(s) = k∈N cos πks − 1 β k cos ν k s ∈ L 2 (0, 2), and K σ2 (x, y) is the transformation operator kernel: Without loss of generality, we assume that ν n > 0. One can achieve this condition by a shift λ → λ + C. One can find K σ2 (x, y) from the GLM equation (17) and then calculate the function σ 2 by the formula Thus, Inverse Problem 1 has a unique solution, which can be found by the following algorithm. Algorithm 1. Let the sequences {ν n } n∈N , {µ n } n∈Z and the signes Ω be given.
Impose the following assumptions: Assumption (A 1 ) is used for simplicity, the case of multiple eigenvalues require some technical modifications (see discussion in [3]). Assumption (A 2 ) can be achieved by a shift of the spectrum. Assumption (A 3 ) is the only principal one. One can easily check, that assumption (A 3 ) follows from the condition ω n = 0, n ∈ N. Under assumptions (A 1 )-(A 3 ), we study the following partial inverse problem.
Inverse Problem 2. Given the function σ 1 , the subspectrum Λ and the signs Ω, find the function σ 2 .
Relying on Lemma 2, we shall prove the uniqueness theorem for the solution of Inverse Problem 2. Along with the boundary value problem L, consider the problem L of the same form, but with different functionsσ j ∈ L 2 (0, l j ), j = 1, 2. We agree that if a certain symbol γ denotes an object related to L, the corresponding symbol γ denotes an analogous object related toL.

Theorem 2. The system of vector functions V is a Riesz basis in H.
Proof. Using (5) and (22), we get a nj = cos ρ nj m + κ n , b nj = 2 sin ρ nj m + κ n , (n, j) ∈ I.
Note that (6) implies cos α k m = 0, sin α k m = 0. Let us show that the system V 0 := {v 0 nj } (n,j)∈I is a Riesz basis in H. It follows from the results of [5, Appendix A], that the systems {sin(2πn + α k )t} n∈Z and {cos(2πn + α k )t} n∈Z are Riesz bases in L 2 (0, 1). Consider the linear operator A : H → H, defined as follows.
where g(u)(t) = n∈Z c n (u) sin |2πn + α k |t, u(t) = n∈Z c n (u) cos |2πn + α k |t, i.e. c n (u) are the coordinates of the function u ∈ L 2 (0, 1) with respect to the Riesz basis {cos |2πn + α k |t} n∈Z . It follows from the Riesz-basis property, that there exist positive constants C 1 and C 2 such that Consequently, the operator A and its inverse: are bounded in H. Note that the operator A transforms the sequence V 0 into a Riesz basis in H: (Av 0 nk )(t) = 2 sin α k m 0 cos(2πn + α k )t , n ∈ Z, (Av 0 n0 )(t) = sin πnt 0 , n ∈ N.
Hence the system V 0 is also a Riesz basis.
Since the system V is complete by Lemma 2 and l 2 -close to the Riesz basis V 0 , we conclude that V is a Riesz basis in H.
Recovering the vector function f from its coordinates with respect to the Riesz basis, one can solve Inverse Problem 2 by the following algorithm.
Algorithm 2. Let the function σ 1 , the eigenvalues Λ and the signs Ω be given.
One can investigate local solvability and stability of Inverse Problem 2, using the approach of [2,3].