An inverse spectral problem for the matrix SturmLiouville operator on the halfline
 Natalia Bondarenko^{1}Email author
https://doi.org/10.1186/s1366101402753
© Bondarenko; licensee Springer 2015
Received: 6 September 2014
Accepted: 26 December 2014
Published: 30 January 2015
Abstract
The matrix SturmLiouville operator with an integrable potential on the halfline is considered. The inverse spectral problem is studied, which consists in recovering of this operator by the Weyl matrix. The author provides necessary and sufficient conditions for a meromorphic matrix function being a Weyl matrix of the nonselfadjoint matrix SturmLiouville operator. We also investigate the selfadjoint case and obtain the characterization of the spectral data as a corollary of our general result.
Keywords
MSC
1 Introduction and main results
Inverse spectral problems consist in recovering differential operators from their spectral characteristics. Such problems arise in many areas of science and engineering, i.e., quantum mechanics, geophysics, astrophysics, electronics. The most complete results were obtained in the theory of inverse spectral problems for scalar SturmLiouville operators \(y'' + q(x) y\) (see monographs [1–4] and the references therein). The greatest progress in the study of SturmLiouville operators on the halfline has been achieved by Marchenko [1]. He studied the inverse problem for the nonselfadjoint locally integrable potential by the generalized spectral function by using the method of transformation operator. We also mention that Marchenko solved the inverse scattering problem on the halfline. Later Yurko showed that the inverse problem by the generalized spectral function is equivalent to the problem by the generalized Weyl function [4]. These problems are closely related to the inverse problem for the wave equation \(u_{tt} = u_{xx}  q(x) u\). When the potential is integrable on the halfline, the generalized Weyl function turns into the ordinary Weyl function. Yurko has studied inverse problems for the SturmLiouville operator with the potential from \(L(0, \infty)\) by the Weyl function and, in the selfadjoint case, by the spectral data. He has developed a constructive algorithm for the solution of these problems and obtained necessary and sufficient conditions for the corresponding spectral characteristics. The details are presented in [4]. In this paper, we generalize his results to the matrix case.
The research on the inverse matrix SturmLiouville problems started in connection with their applications in quantum mechanics [5]. Matrix SturmLiouville equations can be also used to describe propagation of seismic [6] and electromagnetic waves [7]. Another important application is the integration of matrix nonlinear evolution equations such as matrix KdV and Boomeron equations [8]. The theory of matrix SturmLiouville problems has been actively developed during the last twenty years. Trace formulas, eigenvalue asymptotics and some other aspects of direct problems were studied in the papers [9–13]. The works [14–18] contain results of the most resent investigations of inverse problems for matrix SturmLiouville operators on a finite interval.
For the matrix SturmLiouville operator on the halfline, Agranovich and Marchenko [5] have made an extensive research on the inverse scattering problem, using the transformation operator method [1, 2]. Freiling and Yurko [19] have started the investigation of the inverse spectral problem for the nonselfadjoint matrix SturmLiouville operator. They have proved the uniqueness theorem and provided a constructive algorithm for the solution of the inverse problem by the socalled Weyl matrix (the generalization of the scalar Weyl function [1, 4]). Their approach is based on the method of spectral mappings (see [4, 20]), whose main ingredient is the contour integration in the complex plane of the spectral parameter λ. We mention that a related inverse problem for the matrix wave equation was investigated in [21].
In this paper, we study the inverse problem for the matrix SturmLiouville operator on the halfline by the Weyl matrix. We present the necessary and sufficient conditions for the solvability of the inverse problem in the general nonselfadjoint situation. As a particular case, we consider the selfadjoint problem, and get the necessary and sufficient conditions on the spectral data of the selfadjoint operator. Our method is based on the approach of [19].
Here, \(Y (x) = [y_{k}(x)]_{k = \overline{1, m}}\) is a column vector, λ is the spectral parameter, \(Q(x) = [Q_{jk}(x)]_{j, k = 1}^{m}\) is an \(m \times m\) matrix function with entries from \(L(0, \infty)\), and \(h = [h_{jk}]_{j, k = 1}^{m}\), where \(h_{jk}\) are complex numbers.
Let \(\lambda= \rho^{2}\), \(\rho= \sigma+ i \tau\), and let for definiteness \(\tau:= \operatorname{Im} \rho\ge0\). Denote by \(\Phi(x, \lambda) = [\Phi_{jk}(x, \lambda)]_{j, k = 1}^{m}\) the matrix solution of equation (1), satisfying boundary conditions \(U(\Phi) = I_{m}\) (\(I_{m}\) is the \(m \times m\) unit matrix), \(\Phi(x, \lambda) = O(\exp(i \rho x))\), \(x \to\infty\), \(\rho\in \Omega:= \{ \rho\colon\operatorname{Im} \rho\ge0, \rho\ne0\}\). Denote \(M(\lambda) = \Phi(0, \lambda)\). We call the matrix functions \(\Phi(x, \lambda)\) and \(M(\lambda)\) the Weyl solution and the Weyl matrix of L, respectively. Further we show that the singularities of \(\Phi(x, \lambda)\) and \(M(\lambda)\) coincide with the spectrum of the problem L. The Weyl functions and their generalizations often appear in applications and in pure mathematical problems for various classes of differential operators. In this paper, we use the Weyl matrix as the main spectral characteristic and study the following problem.
Inverse problem 1
Given the Weyl matrix \(M(\lambda)\), construct the potential Q and the coefficient h.
The paper is organized as follows. In Section 2, we present the most important properties of the Weyl matrix and briefly describe the solution of Inverse problem 1 given in [19]. By the method of spectral mappings, the nonlinear inverse problem is transformed to the linear equation in a Banach space of continuous matrix functions. In Section 3, we use this solution to obtain our main result, necessary and sufficient conditions for the solvability of Inverse problem 1. In the general nonselfadjoint situation, one has to require the solvability of the main equation in the necessary and sufficient conditions. Of course, it not always easy to check this requirement, but one cannot avoid it even for the scalar SturmLiouville operator (examples are provided in [4]). Therefore we are particularly interested in the special cases, when the solvability of the main equation can be easily checked. First of all, there is the selfadjoint case, studied in Sections 4 and 5. We introduce the spectral data and get their characterization. We also consider finite perturbations of the spectrum in Section 6. In this case, the main equation turns into a linear algebraic system, and one can easily verify its solvability.
2 Preliminaries
In this section, we provide the properties of the Weyl matrix and the algorithm for the solution of Inverse problem 1 by the method of spectral mappings. We give the results without proofs, one can read [5, 19] for more details.
We use the notation \(\mathcal{A}(\mathcal{I}; \mathbb{C}^{m \times m})\) for a class of the matrix functions \(F(x) = [f_{jk}(x)]_{k = \overline{1, m}}\) with entries \(f_{jk}(x)\) belonging to the class \(\mathcal{A}(\mathcal{I})\) of scalar functions. The symbol ℐ stands for an interval or a segment. For example, the potential Q belongs to the class \(L((0, \infty); \mathbb {C}^{m \times m})\).
Denote by Π the λplane with a cut \(\lambda\ge0\), and \(\Pi_{1}= \overline{\Pi} \backslash\{ 0 \}\); note that here Π and \(\Pi_{1}\) must be considered as subsets of the Riemann surface of the square root function. Then, under the map \(\rho\to\rho^{2} = \lambda\), \(\Pi_{1}\) corresponds to the domain \(\Omega= \{\rho\colon\operatorname{Im} \rho \ge0, \rho\ne0 \}\).
 (i_{1}):

For \(x \to\infty\), \(\nu= 0, 1\), and each fixed \(\delta> 0\),uniformly in \(\Omega_{\delta} := \{ \operatorname{Im} \rho\ge0, \rho \ge\delta\}\).$$ e^{(\nu)}(x, \rho) = (i \rho)^{\nu} \exp(i \rho x) \bigl(I_{m} + o(1)\bigr), $$(4)
 (i_{2}):

For \(\rho\to\infty\), \(\rho\in\Omega\), \(\nu= 0, 1\),uniformly for \(x \ge0\).$$ e^{(\nu)}(x, \rho) = (i \rho)^{\nu} \exp(i \rho x) \biggl( 1 + \frac{\omega (x)}{i \rho} + o\bigl(\rho^{1}\bigr) \biggr),\quad \omega(x) :=  \frac{1}{2} \int_{x}^{\infty} Q(t) \,dt, $$(5)
 (i_{3}):

For each fixed \(x \ge0\) and \(\nu= 0, 1\), the matrix functions \(e^{(\nu)}(x, \rho)\) are analytic for \(\operatorname{Im} \rho> 0\) and continuous for \(\rho\in\Omega\).
 (i_{4}):

For \(\rho\in\mathbb{R} \backslash\{ 0\}\) the columns of the matrix functions \(e(x, \rho)\) and \(e(x, \rho)\) form a fundamental system of solutions for equation (1).
The construction of the Jost solution in the matrix case was given in the Appendix of [22] for an even more general situation of the matrix pencil. In principle, the proof is not significantly different from the similar proof in the scalar case (see [4, Section 2]).
Denote \(u(\rho) := U(e(x, \rho)) = e'(0, \rho)  h e(0, \rho)\), \(\Delta(\rho) = \det u(\rho)\). By property (i_{3}) of the Jost solution, the functions \(u(\rho)\) and \(\Delta(\rho)\) are analytic for \(\operatorname{Im} \rho> 0\) and continuous for \(\rho\in\Omega\).
Clearly, singularities of the Weyl matrix \(M(\lambda)\) coincide with the zeros of \(\Delta(\rho)\).
Lemma 1
Now we proceed to the constructive solution of Inverse problem 1. Let the Weyl matrix \(M(\lambda)\) of the boundary value problem \(L = L(Q, h)\) be given. Choose an arbitrary model problem \(\tilde{L} = L(\tilde{Q}, \tilde{h})\) in the same form as L, but with other coefficients. We agree that if a certain symbol γ denotes an object related to L, then the corresponding symbol \(\tilde{\gamma}\) with tilde denotes the analogous object related to \(\tilde{L}\). We consider also the problem \(\tilde{L}^{*} = L^{*}(\tilde{Q}, \tilde{h})\).
Theorem 1
Corollary 1
Proof
Solving the main equation (18), one gets the matrix function \(\varphi(x, \lambda)\) and can follow the algorithm from [19] to recover the original problem L. But further we need an alternative way to construct the potential Q and the coefficient h.
3 Necessary and sufficient conditions
In this section, we give the necessary and sufficient conditions in a very general form, with requirement of the solvability of the main equation.
 (i_{1}):

\(M(\lambda)\) is analytic in Π outside the countable bounded set of poles \(\Lambda'\), and continuous in \(\Pi_{1}\) outside the bounded set Λ;
 (i_{2}):

\(M(\lambda)\) enjoys the asymptotic representation$$ M(\lambda) = \frac{1}{i \rho} \biggl( I_{m} + \frac{h}{i \rho} + o\bigl(\rho^{1}\bigr) \biggr),\quad \rho \to \infty, \rho\in\Omega. $$(23)
Theorem 2
Similarly one can study the classes of potentials Q with higher degree of smoothness, then the potential of the model problem \(\tilde{Q}\) and ε should belong to the same classes.
Proof
By necessity, conditions 1 and 3 are obvious, while condition 2 is contained in Theorem 1. So it remains to prove that the potential Q and the coefficient h, constructed by formulas (22), form a problem L with the Weyl matrix, coinciding with the given \(M(\lambda)\).
Analogously one can prove the relation \(\ell\Phi(x, \lambda) = \lambda\Phi(x, \lambda)\) for the matrix function \(\Phi(x, \lambda)\) constructed via (19).
4 Selfadjoint case: properties of the spectral data
In this section, we assume that the boundary value problem L is selfadjoint: \(Q(x) = Q^{\dagger}(x)\) a.e. on \((0, \infty)\), \(h = h^{\dagger}\). We show that its spectrum has the following Properties (i_{1})(i_{6}). Similar facts for the Dirichlet boundary condition were proved in [5].
Property (i_{1})
The problem L does not have spectral singularities: \(\Lambda'' = \varnothing\).
Proof
Property (i_{2})
All the nonzero eigenvalues are real and negative: \(\lambda_{k} = \rho_{k}^{2} < 0\), \(\rho_{k} = i \tau_{k}\), \(\tau_{k} > 0\).
Indeed, the eigenvalues of L are real because of the selfadjointness. In view of [19, Theorem 2.4], they cannot be positive.
Property (i_{3})
The poles of the matrix function \((u(\rho ))^{1}\) in the upper halfplane are simple. (They coincide with \(i \tau_{k}\).)
Proof
The inverse \((u(\rho))^{1}\) has a simple pole at \(\rho= \rho_{0}\) if and only if the relationswhere a and b are constant vectors, yield \(a = 0\).$$ u(\rho_{0}) a = 0, \qquad u(\rho_{0}) b + \frac{d}{d\rho} u(\rho_{0}) a = 0, $$(30)
Property (i_{4})
The number of eigenvalues is finite.
Proof
Prove the assertion by contradiction. Suppose that there is an infinite sequence \(\{ \lambda_{k} \}_{k = 1}^{\infty}\) of negative eigenvalues, \(\rho_{k} = \sqrt{\lambda}_{k}\), and \(\{ Y_{k}(x) \}_{k = 1}^{\infty}\) is an orthogonal sequence of corresponding vector eigenfunctions. Note that there can be multiple eigenvalues, their multiplicities are finite and equal to \(m  \operatorname{rank} u(\rho_{k})\). Multiple eigenvalues occur in the sequence \(\{ \lambda_{k}\}_{k = 1}^{\infty}\) multiple times with different eigenfunctions \(Y_{k}(x)\). The eigenfunctions can be represented in the form \(Y_{k}(x) = e(x, \rho_{k}) N_{k}\), \(\ N_{k} \ = 1\).
Clearly, \(\mathcal{I}_{2} \ge0\). Using arguments similar to the proof of [4, Theorem 2.3.4], one can show that \(\mathcal{I}_{3}\) tends to zero as \(k, n \to\infty\). Thus, for sufficiently large k and n, \(\mathcal{I}_{1} + \mathcal {I}_{2} + \mathcal{I}_{3} > 0\), which contradicts (32). Hence, the number of negative eigenvalues is finite. □
Property (i_{5})
\(\lambda= 0\) is not an eigenvalue of L.
Proof
Property (i_{6})
\(\rho(u(\rho))^{1} = O(1)\) and \(M(\lambda) = O(\rho^{1})\) as \(\rho\to0\), \(\rho\in\Omega\).
Proof
Since under condition (31) the Jost solution exists for \(\rho= 0\), we have \(e(0, \rho) = O(1)\) as \(\rho\to0\). Taking (12) and \(g(\rho) = O(1)\) into account, we arrive at \(M(\lambda) = O(\rho^{1})\), \(\rho\to0\). □
We combine the properties of the Weyl matrix in the next theorem.
Theorem 3
Proof
The remaining assertions of the theorem do not need a proof. □
5 Selfadjoint case: the inverse problem
Now we are going to apply the general results of Section 3 to the selfadjoint case.
 (i_{1}):

\(\lambda_{k}\) are distinct negative numbers,
 (i_{2}):

\(\alpha_{k}\) are nonzero Hermitian matrices, \(\alpha_{k} \ge0\),
 (i_{3}):

the \(m \times m\) matrix function \(\rho V(\lambda)\) is continuous and bounded as \(\lambda> 0\), \(V(\lambda) > 0\) and \(M(\lambda) = O(\rho^{1})\) as \(\rho\to0\), where \(M(\lambda)\) is defined by (37),
 (i_{4}):

there exists a model problem \(\tilde{L}\) such that (16) holds.
Note that the spectral data of any selfadjoint boundary value problem \(L(Q, h)\) belong to Sp.
Lemma 2
Let data \(( \{ V(\lambda) \}_{\lambda> 0}, \{ \lambda_{k}, \alpha_{k} \}_{k = 1}^{P} )\) belong to Sp. Then, for each fixed \(x \ge0\), system (38)(39) is uniquely solvable. In other words, the operator \((I + \tilde{R}(x))\) is invertible.
Proof
Theorem 4
For data \(S := ( \{ V(\lambda)\}_{\lambda> 0}, \{ \lambda_{k}, \alpha_{k} \}_{k = 1}^{P} )\) to be the spectral data of some selfadjoint boundary value problem \(L(Q, h)\), \(Q = Q^{\dagger}\), \(h = h^{\dagger}\), satisfying (31), it is necessary and sufficient to belong to the class Sp and to have such a property that \((1 + x) \varepsilon(x) \in L((0, \infty ); \mathbb {C}^{m \times m})\), where \(\varepsilon(x)\) is constructed via (44) by the unique solution of system (38)(39) \(\varphi(x, \lambda)\).
6 Perturbation of the discrete spectrum
Theorem 5
For the matrix function \(M(\lambda)\) in the form (45) to be the Weyl matrix of a certain boundary value problem L, it is necessary and sufficient that the determinant of system (46) differs from zero, and \(\varepsilon(x) \in L((0, \infty); \mathbb{C}^{m \times m})\), where \(\varepsilon(x)\) is defined in (47).
There is an example, provided in [4, Section 2.3.2], showing that even in the simple case of a finite perturbation, the condition \(\varepsilon(x) \in L((0, \infty); \mathbb{C}^{m \times m})\) is essential and cannot be omitted. So it is crucial in Theorems 2, 4 and 5.
Declarations
Acknowledgements
This work was supported by Grant 1.1436.2014K of the Russian Ministry of Education and Science and by Grants 130100134 and 140131042 of the Russian Foundation for Basic Research.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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