An inverse spectral problem for the matrix Sturm-Liouville operator on the half-line

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 Sturm-Liouville operators \(-y'' + q(x) y\) (see monographs [1–4] and the references therein). The greatest progress in the study of Sturm-Liouville operators on the half-line has been achieved by Marchenko [1]. He studied the inverse problem for the non-self-adjoint 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 half-line. 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 half-line, the generalized Weyl function turns into the ordinary Weyl function. Yurko has studied inverse problems for the Sturm-Liouville operator with the potential from \(L(0, \infty)\) by the Weyl function and, in the self-adjoint 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 Sturm-Liouville problems started in connection with their applications in quantum mechanics [5]. Matrix Sturm-Liouville 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 Sturm-Liouville 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 Sturm-Liouville operators on a finite interval.

For the matrix Sturm-Liouville operator on the half-line, 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 non-self-adjoint matrix Sturm-Liouville operator. They have proved the uniqueness theorem and provided a constructive algorithm for the solution of the inverse problem by the so-called 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 Sturm-Liouville operator on the half-line by the Weyl matrix. We present the necessary and sufficient conditions for the solvability of the inverse problem in the general non-self-adjoint situation. As a particular case, we consider the self-adjoint problem, and get the necessary and sufficient conditions on the spectral data of the self-adjoint 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.

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 non-self-adjoint 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 Sturm-Liouville 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 self-adjoint 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.

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 λ-plane 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 \(\Omega= \{\rho\colon\operatorname{Im} \rho \ge0, \rho\ne0 \}\).

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 (i₃) 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 \(\Delta(\rho)\).

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 \(\tilde{L}^{*} = L^{*}(\tilde{Q}, \tilde{h})\).

Let \(\gamma'\) be a bounded closed contour in λ-plane encircling the set of singularities \(\Lambda\cup\tilde{\Lambda}\cup\{ 0 \}\), let \(\gamma''\) be the two-sided cut along the ray \(\{ \lambda\colon \lambda> 0, \lambda\notin\operatorname{int} \gamma' \}\), and let \(\gamma= \gamma' \cup\gamma''\) be a contour with the counter-clockwise circuit (see Figure 1). By contour integration over the contour γ, Freiling and Yurko [19] have obtained the following result.

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.

In this section, we give the necessary and sufficient conditions in a very general form, with requirement of the solvability of the main equation.

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.

In this section, we assume that the boundary value problem L is self-adjoint: \(Q(x) = Q^{\dagger}(x)\) a.e. on \((0, \infty)\), \(h = h^{\dagger}\). We show that its spectrum has the following Properties (i₁)-(i₆). Similar facts for the Dirichlet boundary condition were proved in [5].

Indeed, the eigenvalues of L are real because of the self-adjointness. In view of [19, Theorem 2.4], they cannot be positive.

We combine the properties of the Weyl matrix in the next theorem.

Now we are going to apply the general results of Section 3 to the self-adjoint case.

Note that the spectral data of any self-adjoint boundary value problem \(L(Q, h)\) belong to Sp.

Proof

Fix \(x \ge0\). The operator \(\tilde{R}(x)\) is compact, so it is sufficient to prove the unique solvability of the homogeneous system (38)-(39). Let \((\{ \beta(x, \lambda) \}_{\lambda> 0}, \{ \beta_{ni}(x) \} _{(n, i) \in \mathcal{M}} ) \in B_{S}\) be a solution of the homogeneous system

$$\begin{aligned}& \beta(x, \lambda) + \int_{0}^{\infty} \beta(x, \mu) \hat{V}(\mu) \tilde{D}(x, \lambda , \mu) \,d\mu \\& \quad {}+\sum_{k = 1}^{P} \beta_{k0}(x) \alpha_{k0} \tilde{D}(x, \lambda, \lambda_{k0}) - \sum_{k = 1}^{\tilde{P}} \beta_{k1}(x) \alpha_{k1} \tilde{D}(x, \lambda, \lambda _{k1}) = 0_{m}, \quad \lambda> 0, \\& \beta_{ni}(x) + \int_{0}^{\infty} \beta(x, \mu) \hat{V}(\mu) \tilde{D}(x, \lambda _{ni}, \mu) \,d\mu \\& \qquad {}+\sum_{k = 1}^{P} \beta_{k0}(x) \alpha_{k0} \tilde{D}(x, \lambda_{ni}, \lambda_{k0}) - \sum_{k = 1}^{\tilde{P}} \beta_{k1}(x) \alpha_{k1} \tilde{D}(x, \lambda_{ni}, \lambda_{k1}) \\& \quad = 0_{m},\quad (n, i) \in\mathcal{M}. \end{aligned}$$

(40)

Note that formula (40) gives an analytic continuation of the matrix function \(\beta(x, \lambda)\) to the whole λ-plane. Clearly, \(\beta(x, \lambda_{ni}) = \beta_{ni}(x)\).

Using the standard estimate \(\| \tilde{D}(x, \lambda, \lambda_{kj}) \| \le\frac {C}{|\rho|} \exp(|\tau|x)\) (see [4, 19]) together with (16), one can show that

$$ \bigl\Vert \beta(x, \lambda) \bigr\Vert \le\frac{C}{|\rho|} \exp\bigl(\vert \tau \vert x\bigr). $$

(41)

Define the function

$$\begin{aligned} \Gamma(x, \lambda) =& -\int_{0}^{\infty} \beta(x, \mu) \hat{V}(\mu) \frac{\langle \tilde{\varphi}^{*}(x, \mu), \tilde{\Phi}(x, \lambda) \rangle}{\lambda - \mu} \,d\mu \\ &{}- \sum_{k = 1}^{P} \beta_{k0}(x) \alpha_{k0} \frac{\langle\tilde{\varphi} ^{*}_{k0}(x), \tilde{\Phi}(x, \lambda) \rangle}{\lambda- \lambda _{k0}} + \sum_{k = 1}^{\tilde{P}} \beta_{k1}(x) \alpha_{k1} \frac{\langle \tilde{\varphi} ^{*}_{k1}(x), \tilde{\Phi}(x, \lambda) \rangle}{\lambda- \lambda_{k1}}. \end{aligned}$$

(42)

Using the relations \(\tilde{\Phi}(x, \lambda) = \tilde{S}(x, \lambda) + \tilde{\varphi} (x, \lambda) \tilde{M}(\lambda)\) and (40), one can easily derive the following formula:

$$\begin{aligned} \Gamma(x, \lambda) =& \beta(x, \lambda) \tilde{M}(\lambda) - \int _{0}^{\infty} \beta(x, \mu ) \hat{V}(\mu) \frac{\langle\tilde{\varphi}^{*}(x, \mu), \tilde{S}(x, \lambda) \rangle}{\lambda- \mu } \,d\mu \\ &{}-\sum_{k = 1}^{P} \beta_{k0}(x) \alpha_{k0} \frac{\langle\tilde{\varphi} ^{*}_{k0}(x), \tilde{S}(x, \lambda) \rangle}{\lambda- \lambda_{k0}} + \sum_{k = 1}^{\tilde{P}} \beta_{k1}(x) \alpha_{k1} \frac{\langle \tilde{\varphi} ^{*}_{k1}(x), \tilde{S}(x, \lambda) \rangle}{\lambda- \lambda_{k1}}. \end{aligned}$$

Since \(\langle\tilde{\varphi}^{*}(x, \mu), \tilde{S}(x, \lambda) \rangle_{x = 0} = -I_{m}\), we have

$$\frac{\tilde{\varphi}^{*}(x, \mu), \tilde{S}(x, \lambda)}{\lambda- \mu } = -\frac {I_{m}}{\lambda- \mu} + \int_{0}^{x} \tilde{\varphi}^{*}(t, \mu) \tilde{S}(t, \lambda) \,dt. $$

Consequently, we can represent \(\Gamma(x, \lambda)\) in the following form:

$$\begin{aligned} \Gamma(x, \lambda) =& \beta(x, \lambda) \tilde{M}(\lambda) + \int _{0}^{\infty} \frac{\beta (x, \mu) \hat{V}(\mu)}{\lambda- \mu} \,d\mu \\ &{}+ \sum _{k = 1}^{P} \frac{\beta_{k0}(x) \alpha_{k0}}{\lambda- \lambda _{k0}} - \sum _{k = 1}^{\tilde{P}} \frac{\beta_{k1}(x) \alpha_{k1}}{\lambda- \lambda_{k1}} + \Gamma_{1}(x, \lambda), \end{aligned}$$

where the matrix function \(\Gamma_{1}(x, \lambda)\) is entire by λ, since \(\tilde{S}(x, \lambda)\) is entire. Taking (37) into account, we obtain

$$\Gamma(x, \lambda) = \beta(x, \lambda) M(\lambda) + \Gamma_{1}(x, \lambda) - \Gamma_{2}(x, \lambda), $$

where

$$\begin{aligned} \Gamma_{2}(x, \lambda) =& \int_{0}^{\infty} \frac{(\beta(x, \lambda) - \beta(x, \mu)) \hat{V}(\mu)}{\lambda- \mu}\,d\mu+ \sum_{k = 1}^{P} \frac{(\beta(x, \lambda) - \beta_{k0}(x)) \alpha _{k0}}{\lambda- \lambda_{k0}} \\ &{}- \sum_{k = 1}^{\tilde{P}} \frac{(\beta(x, \lambda) - \beta_{k1}(x)) \alpha _{k1}}{\lambda- \lambda_{k1}}. \end{aligned}$$

Obviously, the function \(\Gamma_{2}(x, \lambda)\) is entire in λ. Therefore, the function \(\Gamma(x, \lambda)\) has simple poles at the points \(\Lambda'\) and

$$ \mathop{\operatorname{Res}}_{\lambda= \lambda_{k0}} \Gamma(x, \lambda) = \beta_{k0}(x) \alpha_{k0}, \quad k = \overline{1, P}. $$

Furthermore,

$$\frac{1}{2 \pi i} \bigl( \Gamma^{-}(\lambda) - \Gamma^{+}(\lambda) \bigr) = \beta (x, \lambda) V(\lambda),\qquad \Gamma^{\pm}(\lambda) := \lim _{z \to0, \operatorname{Re} z > 0} \Gamma (\lambda\pm i z), \quad \lambda> 0. $$

Using (41) and the standard asymptotics for \(\tilde{\varphi}^{*}(x, \mu)\) and \(\tilde{\Phi}(x, \lambda)\), one arrives at the estimate

$$ \bigl\Vert \Gamma(x, \lambda) \bigr\Vert \le C | \rho|^{-2} \exp\bigl(-\vert \tau \vert x\bigr),\quad |\lambda| \to \infty. $$

(43)

Introduce the matrix function $\mathcal{B}(x,\lambda ):=\mathrm{\Gamma }(x,\lambda ){\beta }^{†}(x,\overline{\lambda })$, and consider the integral

$\mathcal{I}:=\frac{1}{2\pi i}{\int }_{{\gamma }_R^0}\mathcal{B}(x,\lambda )d\lambda$

over the contour \(\gamma_{R}^{0} := ( \gamma\cap\{ \lambda\colon \lambda\le R\} ) \cup\{ \lambda\colon|\lambda| = R \}\) (see Figure 2). For a sufficiently large radius R, $\mathcal{I}=0_m$ by the Cauchy theorem. In view of estimates (41), (43), we have $\parallel \mathcal{B}(x,\lambda )\parallel \le C{|\rho |}^{-3}$, \(|\lambda| \to \infty\). Hence

$\begin{array}{c}\underset{R\to \mathrm{\infty }}{lim}\frac{1}{2\pi i}{\int }_{|\lambda |=R}\mathcal{B}(x,\lambda )d\lambda =0_m,\hfill \\ \frac{1}{2\pi i}{\int }_{\gamma }\mathcal{B}(x,\lambda )d\lambda =0_m.\hfill \end{array}$

The last integral over γ can be calculated by the residue theorem. It equals

$\sum _{k=0}^{\mathrm{\infty }}{\beta }_{k0}(x){\alpha }_{k0}{\beta }_{k0}^{†}(x)+\frac{1}{2\pi i}{\int }_{|\lambda |=\epsilon }\mathcal{B}(x,\lambda )d\lambda +{\int }_{\epsilon }^{\mathrm{\infty }}\beta (x,\lambda )V(\lambda ){\beta }^{†}(x,\lambda )d\lambda ,$

where \(\varepsilon> 0\) is sufficiently small. Note that since \(M(\lambda) = O(\rho ^{-1})\) as \(\lambda\to0\), the second term in the sum tends to zero as \(\varepsilon\to0\). Finally, we obtain

$$\sum_{k = 0}^{\infty} \beta_{k0}(x) \alpha_{k0} \beta _{k0}^{\dagger}(x) + \int _{0}^{\infty} \beta(x, \lambda) V(\lambda) \beta^{\dagger}(x, \lambda) \,d\lambda= 0. $$

Since \(\alpha_{k0} \ge0\), \(V(\lambda) > 0\), we get \(\beta_{k0}(x) \alpha_{k0} = 0_{m}\), \(\beta(x, \lambda) V(\lambda) = 0_{m}\), and \(\beta(x, \lambda) = 0\) for \(\lambda> 0\). Since \(\beta(x, \lambda )\) is an entire function in λ, we conclude that \(\beta(x, \lambda) \equiv0_{m}\). Consequently, \(\beta_{k0}(x) = \beta(x, \lambda_{k0}) = 0_{m}\). Thus, the homogeneous system has only a trivial solution, so system (38)-(39) is uniquely solvable.

 □

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.