Reconstruction formula for differential systems with a singularity

Our studies concern some aspects of scattering theory of the singular differential systems $ y'-x^{-1}Ay-q(x)y=\rho By, \ x>0 $ with $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, where $A,B$ are constant and $\rho$ is a spectral parameter. We concentrate on the important special case when $q(\cdot)$ is smooth and $q(0)=0$ and derive a formula that express such $q(\cdot)$ in the form of some special contour integral, where the kernel can be written in terms of the Weyl - type solutions of the considered differential system. Formulas of such a type play an important role in constructive solution of inverse scattering problems: use of such formulas, where the terms in their right-hand sides are previously found from the so-called main equation, provides a final step of the solution procedure. In order to obtain the above-mentioned reconstruction formula we establish first the asymptotical expansions for the Weyl - type solutions as $\rho\to\infty$ with $o\left(\rho^{-1}\right)$ rate remainder estimate.


Introduction
Our studies concern some aspects of scattering theory of the differential systems with n × n matrices A, B, q(x), x ∈ (0, ∞), where A, B are constant and ρ is a spectral parameter. Differential equations with coefficients having non-integrable singularities at the end or inside the interval often appear in various areas of natural sciences and engineering. For n = 2, there exists an extensive literature devoted to different aspects of spectral theory of the radial Dirac operators, see, for instance [1], [2], [3], [4], [5].
Systems of the form (1) with n > 2 and arbitrary complex eigenvalues of the matrix B appear to be considerably more difficult for investigation even in the "regular" case A = 0 [6]. Some difficulties of principal matter also appear due to the presence of the singularity. Whereas the "regular" case A = 0 has been studied fairly completely to date [6], [7], [8], for system (1) with A = 0 there are no similar general results.
In this paper, we consider the important special case when q(·) is smooth and q(0) = 0 and, provided also that the discrete spectrum is empty, derive a formula that express such q(·) in the form of some special contour integral, where the kernel can be written in terms of the Weyl -type solutions of system (1). Formulas of such a type play an important role in constructive solution of inverse scattering problems: use of such formulas, where the terms in their right-hand sides are previously found from the so-called main equation (see, for instance, [9], [10]), provides a final step of the solution procedure. In order to obtain the above-mentioned reconstruction formula we establish first the asymptotical expansions for the Weyl -type solutions as ρ → ∞ with o (ρ −1 ) rate remainder estimate.

Preliminary remarks
Consider first the following unperturbed system: and its particular case corresponding to the value ρ = 1 of the spectral parameter but to complex (in general) values of x. Assumption 1. Matrix A is off-diagonal. The eigenvalues {µ j } n j=1 of the matrix A are distinct and such that µ j − µ k / ∈ Z for j = k, moreover, Reµ 1 < Reµ 2 < · · · < Reµ n , Reµ k = 0, k = 1, n.
• the symbol V (m) , where V is n × n matrix, denotes the operator acting in ∧ m C n so that for any vectors u 1 , . . . , u m the following identity holds: • if h ∈ ∧ n C n then |h| is a number such that h = |h|e 1 ∧ e 2 ∧ · · · ∧ e n ; • for h ∈ ∧ m C n we set:

Asymptotics of the Weyl -type solutions
Let S ⊂ C \ Σ be an open sector with vertex at the origin. For arbitrary ρ ∈ S and k ∈ {1, . . . , n} we define the k-th Weyl -type solution Ψ k (x, ρ) as a solution of (1) normalized with the asymptotic conditions: If q(·) is off-diagonal matrix function summable on the semi-axis (0, ∞) then for arbitrary given ρ ∈ S k-th Weyl -type solution exists and is unique provided that the characteristic function: are certain tensor-valued functions (fundamental tensors) defined as solutions of certain Volterra integral equations, see [14], [16] for details.
For arbitrary fixed arguments x, ρ (where ∆ k (ρ) = 0) the value Ψ k = Ψ k (x, ρ) is the unique solution of the following linear system: This fact and also some properties of the Weyl -type solutions were established in works [14], [17], in particular, the following asymtotics for ρ → ∞ was obtained: For our purposes we need more detailed asymptocis that can be obtained provided that the potential q(·) is smooth enough and vanishes as x → 0.
We denote by P(S) the set of functions F (ρ), ρ ∈ S admitting the representation: Here the set Λ (depending on F (·) ∈ P(S)) is such that Re(λρ) < 0 for all λ ∈ Λ, ρ ∈ S. We note that the set of scalar functions belonging to P(S) is an algebra with respect to pointwise multiplication.
Suppose that all the functions Then for each fixed x > 0 and ρ → ∞, ρ ∈ S the following asymptotics holds: where Γ(x) is some diagonal matrix, E(x, ·) ∈ P(S).
For the Weyl -type solutions of the unperturbed system we have the asymptotics (following directly from their definition):Ψ whereΨ 0k (x, ρ) := exp(−ρxR k )Ψ 0k (x, ρ). Here and below we use the same symbol E(·, ·) for different functions such that E(x, ·) ∈ P(S) for each fixed x. We rewrite relations (5) in the form of the following linear system with respect to valuẽ Ψ k =Ψ k (x, ρ) of the functionΨ k (x, ρ) := exp(−ρxR k )Ψ k (x, ρ): By making the substitution:Ψ we obtain:F The obtained relations we transform into the following system of linear algebraic equations: with respect to coefficients {γ jk } of the expansion: Coefficients {m ij }, {u i } can be calculated as follows: Using (7), (8) and taking into account that: we obtain the following asymptotics for the coefficients of SLAE (10) as ρ → ∞: and for i = 1, k − 1.
Proceeding in a similar way we obtain: where δ i,k is a Kroeneker delta, Using the obtained asymptotics we obtain from (10) the auxiliary estimate γ ik (x, ρ) = O (ρ −1 ).
Then, using in (10) the substitution γ ik (x, ρ) = ρ −1γ ik (x, ρ) (where, as it was shown above, γ ik (x, ρ) = O(1)) we obtain for i = k, n: In view of (12), (15) this yields: Similarly, for i < k we have: Using (13) the obtained relation can be transformed as follows: Now, using in the right hand side of the obtained formula (13) for m ij (x, ρ) and (18) forγ jk (x, ρ) with j = k, n we conclude that formulas (18) are true for i < k as well.
In our further calculations we use particular form of the coefficients f k,α (x) and g k,α,β (x) given by Theorem 1 [16].
For i = k, n from (18), (16), (12) we get: Theorem 1 [16] yields: Recall that any arbitrary linear operator V acting in C n can be expanded onto the wedge algebra ∧C n so that the identity remains true for any set of vectors h 1 , . . . , h m , m ≤ n; moreover, for any h ∈ ∧ n C n one has V h = |V |h (here |V | denotes determinant of matrix of the operator V in the standard coordinate basis {e 1 , . . . , e n }). In what follows the symbol f denotes the above mentioned expansion of the operator corresponding to the transmutation matrix f. We should note also that the relation f is true for any n × n matrix V . Taking this into account we obtain: For the particular multi-index α = α * (k − 1) ∪ i arising at (19) and arbitrary n × n matrix V we have: Substituting the obtained relations into (19) we arrive at: Proceeding in a similar way in the case i < k, using (13), (17) we obtain: Theorem 1 [16] yields: Repeating the arguments above we obtain: In particular, one gets: If β = α ′ \ k, α = α * (k − 1) \ i, i < k, then for arbitrary n × n matrix V we have: Substituting the obtained relations into (21) we arrive at: From (22), (20), (18) we obtain: In terms of the matrix γ = (γ ik ) i,k=1,n this is equivalent to: where the matrix Γ(x) is diagonal. Finally, using (11) in the formΨ(x, ρ) = fγ(x, ρ) we obtain the required relation.
Taking into account that F + (x, ζ) − F − (x, ζ) = ζ[B,P (x, ζ)] we obtain: On the other hand, we can proceed in a similar way applying the Cauchy formula to the function P (x, ρ) − I. Thus we obtain: P (x, ρ) − I = 1 2πi Substituting this to the definition of the function F (x, ρ) we arrive at the representation: F (x, ρ) = q(x) + lim Compare it with (25) we obtain the desired relation.