Asymptotics of Solutions of Some Integral Equations Connected with Differential Systems with a Singularity

Our studies concern some aspects of scattering theory of the singular differential systems y − xAy − q(x)y = ρBy, x > 0 with n × n matrices A,B, q(x), x ∈ (0,∞), where A,B are constant and ρ is a spectral parameter. We concentrate on investigation of certain Volterra integral equations with respect to tensor-valued functions. The solutions of these integral equations play a central role in construction of the so-called Weyl-type solutions for the original differential system. Actually, the integral equations provide a method for investigation of the analytical and asymptotical properties of the Weyl-type solutions while the classical methods fail because of the presence of the singularity. In the paper, we consider the important special case when q is smooth and q(0) = 0 and obtain the classical-type asymptotical expansions for the solutions of the considered integral equations as ρ → ∞ with o (


INTRODUCTION
Our studies concern some aspects of scattering theory of the differential systems y ′ − x −1 Ay − q(x)y = ρBy, x > 0 (1) 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 the system (1) with A = 0 there are no similar general results.
The important role in scattering theory is played by a certain distinguished basis of generalized eigenfunctions for (1) (the so-called Weyl-type solutions, see, for instance [9]). In the presence of the singularity construction and investigation of this basis encounters some difficulties which do not appear in the "regular" case A = 0. In particular, one can not use the auxiliary Cauchy problems with the initial conditions ❝ ■❣♥❛t✐❡✈ ▼✳ (✉✳✱ ✷✵✷✵ ➮çâ✳ Ñàðàò✳ óí✲òà✳ ❮îâ✳ ñåð✳ Ñåð✳ ❒àòåìàòèêà✳ ❒åõàíèêà✳ ➮íôîðìàòèêà✳ ✷✵✷✵✳ Ò✳ ✷✵✱ âûï✳ ✶ at x = 0. The approach presented in [10] (see also [11] and references therein) for the scalar differential operators ℓy = y (n) + n−2 j=0 ν j x n−j + q j (x) y (j) (2) is based on using some special solutions of the equation ℓy = λy that also satisfy certain Volterra integral equations. This approach assumes some additional decay condition for the coefficients q j (x) as x → 0, moreover, the required decay rate depends on eigenvalues of the matrix A. In this paper, we do not impose any additional restrictions of such a type. Instead, we use a modification of the approach first presented in [12] for the higher-order differential operators with regular coefficients on the whole line and recently adapted for differential systems of the form (1) on the semi-axis in [9]. In brief outline the approach can be described as follows. We consider some auxiliary systems with respect to the functions with values in the exterior algebra ∧C n . Our study of these auxiliary systems centers on two families of their solutions that also satisfy some asymptotical conditions as x → 0 and x → ∞ respectively, and can be constructed as solutions of certain Volterra integral equations. As in [12] we call these distinguished tensor solutions the fundamental tensors. The main difference from the above-mentioned method used in [10] is that we use the integral equations to construct the fundamental tensors rather than the solutions for the original system. Since each of the fundamental tensors has minimal growth (as x → 0 or x → ∞) among solutions of the same auxiliary system, this step does not require any decay of q(x) as x → 0.
Construction and properties of the fundamental tensors were considered in details in our paper [9] provided that q(·) is absolutely continuous and both q, q ′ are integrable on the semi-axis (0, ∞). In this paper, we consider the important special case q(0) = 0 and obtain the classical-type asymptotical expansions for the fundamental tensors as ρ → ∞ with o (ρ −1 ) rate remainder estimate.

ASSUMPTIONS AND NOTATIONS. FORMULATIONS OF THE RESULTS
We are to discuss first the 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. det c(x) ≡ 1 and allĉ k (·) are entire functions,ĉ k (0) = h k , h k is an eigenvector of the matrix A corresponding to the eigenvalue µ k . We define C k (x, ρ) := c k (ρx), x ∈ (0, ∞), ✶✽ ❮àó÷íûé îòäåë ρ ∈ C. We note that the matrix C(x, ρ) is a solution of the unperturbed system (3) (with respect to x for the given spectral parameter ρ). Let Σ be the following union of lines through the origin in C: By virtue of Assumption 2 for any z ∈ C \ Σ there exists the ordering R 1 , . . . , R n of the numbers b 1 , . . . , b n such that Re(R 1 z) < Re(R 2 z) · · · < Re(R n z). Let S be a sector {z = r exp(iγ), r ∈ (0, ∞), γ ∈ (γ 1 , γ 2 )} lying in C \ Σ. Then [13] the system (4) has the fundamental matrix e(x) = (e 1 (x), . . . , e n (x)) which is analytic in S , continuous in S \ {0} and admits the asymptotics: Everywhere below we assume that the following additional condition is satisfied. Condition 1. For all k = 2, n the numbers are not equal to 0. Under Condition 1 the system (4) has the fundamental matrix ψ 0 (x) = (ψ 0 1 (x), . . . , ψ 0 n (x)) which is analytic in S , continuous in S \ {0} and admits the asymptotics: We define Ψ 0 (x, ρ) := ψ 0 (ρx). As above, we note that the matrices E(x, ρ), Ψ 0 (x, ρ) solve (3).
Everywhere below the symbol S denotes some (arbitrary) open sector with the vertex at the origin lying in C \ Σ.
For each fixed ρ ∈ S \ {0} =: S ′ we consider the following Volterra integral equations (k = 1, n): where and G m (x, t, ρ) is an operator acting in ∧ m C n as follows: Here and below σ α := |h α ∧ h α ′ |. For any ρ ∈ S ′ equations (5) and (6) were shown to have the unique solutions T k (x, ρ) and F k (x, ρ) respectively such that (see [9] for details): We call the functions F k (x, ρ), T k (x, ρ) the fundamental tensors. Note that the fundamental tensors solve the auxiliary systems with m = k and m = n − k + 1.

✷✵ ❮àó÷íûé îòäåë
We note that the tensors {E α (x, ρ)} α∈Am form the fundamental system of solutions for the system (10) in the "unperturbed" case. Therefore, the following representation holds: with x-independent coefficients T 0 kα . Taking into account the special construction of the fundamental matrices C(x, ρ), E(x, ρ) one can conclude that the coefficients T 0 kα do not depend on ρ as well.
The G m (x, t, ρ) terms in equations (5), (6) are actually the Green operator functions for the nonhomogeneous systems: In order to construct them one can use variuos fundamental systems of solutions of the unperturbed system (3). In particular the following representations hold: Here and below χ α := |f α ∧ f α ′ |.

PROOF OF THEOREM 1
We consider in details the function T k (x, ρ), for the function F k (x, ρ) similar arguments are valid.
In order to make a more detailed study we represent the operator K (ρ) in the form K (ρ) = K 0 (ρ) + K 1 (ρ), where: Here and below the symbols θ ± (·) denote the Heaviside step functions:

Lemma 1. Under the conditions of Theorem 1 one has the estimate
Proof. We split the operator as follows: By virtue of (15) we have: Proceeding in a similar way and taking into account that (K 2 . Using the representation (9) for G n−k+1 (x, t, ρ), the asymptotics which is uniform in |ρx| 1 and taking into account that Re(ρ(x − t)(R α − ← − R k )) 0 for any 0 t x, ρ ∈ S ′ , α ∈ A n−k+1 we obtain the estimate: with some absolute constant M . Since under the conditions of Theorem 1 t −1 q(t) ∈ L 1 (0, ∞) the estimate above yields and therefore K

Lemma 2. Under the conditions of Theorem 1 one has the estimate
Proof. We have: Thus, we can rewrite: where: We notice again that Re(ρ( Moreover, under the conditions of Theorem 1 Q αβ (·) are absolutely continuous and Q αβ (t) ≡ 0 if α = β. This yields the estimate which is uniform in 0 τ x, ρ ∈ S ′ . The estimate implies the required assertion.