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

The matrix Sturm-Liouville operator with an integrable potential on the half-line is considered. We study the inverse spectral problem, which consists in recovering of this operator by the Weyl matrix. The main result of the paper is the necessary and sufficient conditions on the Weyl matrix of the non-self-adjoint matrix Sturm-Liouville operator. We also investigate the self-adjoint case and obtain the characterization of the spectral data as a corollary of our general result.


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 Sturm-Liouville operators −y ′′ + q(x)y (see monographs [1][2][3][4] and references therein). The greatest progress in the study of Sturm-Liouville operators on the half-line has been achieved by V.A. Marchenko [1]. He has studied the inverse problem for the non-self-adjoint locally integrable potential by the generalized spectral function, using the method of transformation operator. We also mention, that V.A. Marchenko has solved the inverse scattering problem on the half-line. Later V.A. Yurko has shown, 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. V.A. Yurko has studied inverse problems for the Sturm-Liouville operator with the potential from L(0, ∞) by the Weyl function and, in the self-adjoint case, by the spectral data. He 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 also be 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 have been actively developed during the last twenty years. Trace formulas, eigenvalue asymptotics and some other aspects of direct problems were studied in papers [9][10][11][12][13]. The works [14][15][16][17][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, Z.S. Agranovich and V.A. Marchenko [5] have made an extensive reseach on the inverse scattering problem, using the transformation operator method [1,2]. G. Freiling and V.A. 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].
Proceed to the formulation of the problem. Consider the boundary value problem L = L(Q(x), h) for the matrix Sturm-Liouville equation Here is an m × m matrix function with entries from L(0, ∞), and h = [h jk ] m j,k=1 , where h jk are complex numbers.
Let λ = ρ 2 , ρ = σ + iτ , and let for definiteness τ : We call the matrix functions Φ(x, λ) and M(λ) the Weyl solution and the Weyl matrix of L, respectively. Further we show, that the singularities of Φ(x, λ) and M(λ) 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(λ), 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 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 can not 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.

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.
Start with the introduction of the notation. We consider the space of complex column m-vectors C m with the norm the space of complex m × m matrices C m×m with the corresponding induced norm The symbols I m and 0 m are used for the unit m × m matrix and the zero m × m matrix, respectively. The symbol † denotes the conjugate transpose.
We use the notation A(I; C m×m ) for a class of the matrix functions F (x) = [f jk (x)] k=1,m with entries f jk (x) belonging to the class A(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, ∞); C m×m ).
(i 4 ) For ρ ∈ R\{0} the columns of the matrix functions e(x, ρ) and e(x, −ρ) 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 even more general situation of the matrix pencil. In principle, the the proof is not significantly different from the similar proof in the scalar case (see [4,Section 2]).
Along with L we consider the problem L * = L * (Q(x), h) in the form where Z is a row vector. Denote Z, Y := Z ′ Y − ZY ′ . If Y (x, λ) and Z(x, λ) satisfy equations (1) and (6), respectively, then so the expression Z(x, λ), Y (x, λ) does not depend on x.
One can easily show that the Weyl solution and the Weyl matrix admit the following rep- Clearly, singularities of the Weyl matrix M(λ) coincide with the zeros of ∆(ρ).
Lemma 1 ( [19,22]). The Weyl matrix is analytic in Π outside the countable bounded set of poles Λ ′ , and continuous in Π 1 outside the bounded set Λ. For |ρ| → ∞, ρ ∈ Ω, Let For each fixed x ≥ 0, these matrix functions are entire in λ-plane. Further we also need the following relation Symmetrically one can introduce the matrix solutions Φ * (x, λ), S * (x, λ) and ϕ * (x, λ) of equation (6), and the Weyl matrix M * (λ) := Φ * (0, λ) of the problem L * . Then By virtue of (8), the expression Φ * (x, λ), Φ(x, λ) does not depend on x. Since by the boundary conditions Now proceed to the constructive solution of Inverse Problem 1. Let the Weyl matrix M(λ) of the boundary value problem L = L(Q, h) be given. Choose an arbitrary model problem L = L(Q,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 symbolγ with tilde denotes the analogous object related toL. We consider also the problemL * = L * (Q,h). Denote Suppose that the following condition is fulfilled: (13) is L 2 -function. Therefore one can take any problemL with a potential from L((0, ∞); C m×m ) ∩ L 2 ((0, ∞); C m×m ) andh = h, in order to satisfy (16). Introduce auxiliary functions Let γ ′ be a bounded closed contour in λ-plane, encircling the set of singularities Λ ∪Λ ∪ {0}, let γ ′′ be the two-sided cut along the ray {λ : λ > 0, λ / ∈ int γ ′ }, and γ = γ ′ ∪ γ ′′ be a contour with the counter-clockwise circuit (see Fig. 1). By contour integration over the contour γ, G. Freiling and V. Yurko [19] have obtained the following result.
which is called the main equation of Inverse Problem 1. This equation is uniquely solvable (with respect to ϕ(x, λ)) in the Banach space B of continuous bounded on γ matrix functions Corollary 1. The analogous relation is valid for Φ(x, λ): where Proof. Following the proof of Theorem 4.1 from [19], we define a block-matrix of spectral mappings P (x, λ) = [P jk (x, λ)] j,k=1,2 by the relation In particular, Substituting formulas (4.4) from [19]: where δ jk is the Kronecker delta, we get Note that the matrix functions Φ(x, λ) andΦ(x, λ) do not have singularities in J γ . By virtue of the relations (3.12) from [19], Substitute these relations into (20) and group the terms: If one expand Φ(x, µ) andΦ * (x, µ), using (14) and (15), the terms with S(x, µ) andS * (x, µ) will be analytic inside the contour and vanish by Cauchy's theorem. Therefore we get SinceM * (µ) ≡M (µ), we arrive at (19).
Solving the main equation (18), one gets the matrix function ϕ(x, λ) 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. Let Then similarly to [4, Section 2.2], one can obtain the relations Using the formulas (21), (22), one can construct Q and h by the solution of the main equation (18), and solve Inverse Problem 1.

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.
Denote by W the class of the matrix functions M(λ), satisfying the conditions of Lemma 1, namely (i 1 ) M(λ) is analytic in Π outside the countable bounded set of poles Λ ′ , and continuous in Π 1 outside the bounded set Λ; (i 2 ) M(λ) enjoys the asymptotic representation Theorem 2. For the matrix function M(λ) ∈ W to be the Weyl matrix of some boundary value problem L of the form (1), (2), it is necessary and sufficient to satisfy the following conditions.
Similarly one can study the classes of potentials Q with higher degree of smotheness, then the potential of the model problemQ 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(λ).
. Then, using the same arguments, as in the scalar case [4], one can show, that the matrix function η(x, λ) belongs to the Banach space B for each fixed x ≥ 0. Consider the operatorR(x) : B → B, acting in the following way: Here and below in similar situations, we write an operator to the right of an operand, because the action of the operator involves noncommutative matrix multiplication in the such order. For each fixed x ≥ 0, the operatorR(x) is compact, therefore it follows from the unique solvability of the main equation (18), that the corresponding homogeneous equation (25) is also uniquely solvable. Hence η(x, λ) ≡ 0, and (24) is proved.
Step 2. In the general case, when (16) holds, the proof of the equality (24) is more complicated, so we only outline the main ideas. Introduce contours γ N = γ ∩ {|λ| ≤ N 2 }, and consider operatorsR The sequence {R N (x)} converges toR(x) in the operator norm. In view of the unique solvability of the main equation, the operator (I +R(x)) is invertible for each fixed x ≥ 0. So for sufficiently large values of N, the operators (I +R N (x)) are also invertible, and the equations can repeat the arguments of Step 1 for the matrix functions ϕ N (x, λ), and prove the relations −ϕ ′′ The sequence {ϕ N (x, λ)} converges to ϕ(x, λ) uniformly with respect to x and λ on compact sets, and the sequence {Q N (x)} converges to Q(x) in L-norm on every bounded interval. These facts yield (24). Analogously one can prove the relation ℓΦ(x, λ) = λΦ(x, λ) for the matrix function Φ(x, λ), constructed via (19).

Self-adjoint case: properties of the spectral data
In this section, we assume that the boundary value problem L is self-adjoint: Q(x) = Q † (x) a.e. on (0, ∞), h = h † . We show that its spectrum has the following properties (i 1 )-(i 6 ). The similar facts for the Dirichlet boundary condition were proved in [5].
So we arrive at the contradiction, which proves the property.
Property (i 2 ). All the nonzero eigenvalues are real and negative: Indeed, the eigenvalues of L are real because of the self-adjointness. In view of [19, Theorem 2.4], they cannot be positive.
Property (i 3 ). The poles of the matrix function (u(ρ)) −1 in the upper half-plane are simple. (They coincide with iτ k ).
The next properties take place, if the additional condition holds: Property (i 4 ). The number of eigenvalues is finite.
Proof. Prove the assertion by contradiction. Suppose there is an infinite sequence {λ k } ∞ k=1 of negative eigenvalues, ρ k = √ λ k , and {Y k (x)} ∞ k=1 is an ortogonal sequence of corresponding vector eigenfunctions. Note that there can be multiple eigenvalues, their multiplicities are finite and equal to m − rank u(ρ k ). Multiple eigenvalues occur in the sequence {λ k } ∞ k=1 multiple times with different eigenfunctions Y k (x). The eigenfunctions can be represented in the form Using the ortogonality of the eigenfunctions, we obtain for k = n Similarly to the scalar case [4,Theorem 2.3.4], one can show that e(x, ρ k ) = exp(−τ k x)(I m + α k (x)), where α k (x) ≤ 1 8 as x ≥ A for all k ≥ 1 and for sufficiently large A. Consequently Since the vectors N k belong to the unit sphere, one can choose a convergent subsequence {N ks } ∞ s=1 . Further we consider N k and N n from such subsequence. Then for sufficiently large k and n, we have Hence Clearly, I 2 ≥ 0. Using arguments similar to the proof of [4,Theorem 2.3.4], one can show that I 3 tends to zero, as k, n → ∞. Thus, for sufficiently large k and n, I 1 + I 2 + I 3 > 0, that contradicts (32). Hence, the number of negative eigenvalues is finite.
Property (i 5 ). λ = 0 is not an eigenvalue of L.
We combine the properties of the Weyl matrix in the next theorem.
We call the collection {V (λ)} λ>0 , {λ k , α k } P k=1 the spectral data of L. Similarly to the scalar case (see [4]), the Weyl matrix can be uniquely determined by the spectral data:

Self-adjoint case: the inverse problem
Now we are going to apply the general results of Section 3 to the self-adjoint case. Let us rewrite the main equation (18) of the inverse problem in terms of the spectral data. Denote Then the main equation (18) can be transformed into the system of equations with the norm The system (38)-(39) has the form ψ(x)(I +R(x)) =ψ(x), whereR(x) : B S → B S is a linear compact operator for each fixed x ≥ 0. By necessity, we have the unique solvability of the main equation (18), so the equivalent system (38)-(39) is uniquely solvable, and the operator (I +R(x)) has a bounded inverse. Now we are going to prove, that all these facts follow from some simple properties of spectral data. We will say that data {V (λ)} λ>0 , {λ k , α k } P k=1 belongs to the class Sp, if (i 1 ) λ k are distinct negative numbers. (i 2 ) α k are nonzero Hermitian matrices, α k ≥ 0. (i 4 ) There exists a model problemL, such that (16) holds. Note that the spectral data of any self-adjoint boundary value problem L(Q, h) belong to Sp.

Perturbation of the discrete spectrum
Return to the general non-self-adjoint problem, and consider one more particular case, when the solvability of the main equation (18) can be easily checked. Let the problemL be given, andM (λ) is its Weyl matrix. Consider the matrix function where λ k ∈ C are some distinct numbers and α kν ∈ C m×m , k = 1, P , ν = 1, m k . Then V (λ) = 0 m , and by virtue of the residue theorem, the main equation (18)  whereD i,j (x, λ, µ) := ∂ i+j ∂λ i ∂µ jD (x, λ, µ). Differentiating this relation with respect to λ, we arrive at the following system of linear algebraic equations with respect to the unknown variables ∂ s ∂λ s ϕ(x, λ n ) : n = 1, P , s = 0, m n − 1. The system (46) has a unique solution if and only if its determinant is not zero. Having the solution of (46), one can construct and then find Q(x) and h via (22).
Theorem 5. For the matrix function M(λ) 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 the system (46) differs from zero, and ε(x) ∈ L((0, ∞); C m×m ), where ε(x) is defined in (47), There is the example, provided in [4, Section 2.3.2], showing that even in the simple case of a finite perturbation, the condition ε(x) ∈ L((0, ∞); C m×m ) is essential and can not be omitted. So it is crucial in Theorems 2, 4 and 5.