Abstract

We consider a \(2 \times 2\) system of ordinary differential equations

$$y'-By=\lambda Ay, \qquad y=y(x), \quad x \in [0, 1],$$

where \(A=\operatorname{diag}\{a_1(x), a_2(x)\}\), \(B=\{b_{kj}(x)\}_{k, j=1}\), and all functions occurring in the matrices are complex-valued and integrable. In the case

$$a_1,a_2, b_{21},b_{12} \in W^n_1[0,1], \qquad b_{11}, b_{22} \in W^{n-1}_1[0,1],$$

we obtain \(n+1\) terms of the asymptotic expansion in powers of \(\lambda^{-1}\), \(\lambda \to \infty\), of the fundamental matrix of solutions of this equation. These asymptotic expansions are valid in the half-planes \(\Pi_{\kappa}=\{\lambda \in \mathbb{C} \mid \operatorname{Re}{\lambda} \ge -\kappa \}\), \(\kappa \in \mathbb{R}\), and \(-\Pi_{\kappa}\) if \(a_1(x)-a_2(x) > 0\). They hold in the sectors \(S=\{\lambda \in \mathbb{C} \mid \lvert\operatorname{arg}\lambda\rvert \le \pi/2-\phi-\varepsilon\}\), \(\varepsilon > 0\), and \(-S\) under the condition that \(\lvert\operatorname{arg}\{a_1(x)-a_2(x)\}\rvert<\phi<\pi /2\). The main novelty of the work is that we assume minimal conditions for the smoothness of the functions and in addition we obtain explicit formulae for matrices involved in asymptotic expansions. The results are also new for the Dirac system.

中文翻译：

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$$2 \times 2$$ 常微分方程组解的谱参数渐近

摘要

我们考虑一个\(2 \times 2\)常微分方程组

$$y'-By=\lambda Ay, \qquad y=y(x), \quad x \in [0, 1],$$

其中\(A=\operatorname{diag}\{a_1(x), a_2(x)\}\) , \(B=\{b_{kj}(x)\}_{k, j=1}\ )，并且矩阵中出现的所有函数都是复值且可积的。在这种情况下

$$a_1,a_2, b_{21},b_{12} \in W^n_1[0,1], \qquad b_{11}, b_{22} \in W^{n-1}_1[0, 1],$$

我们获得该方程解的基本矩阵的\(\lambda^{-1}\)、\(\lambda \to \infty\)幂的 \( n+1 \) 项渐近展开式。这些渐近展开式在半平面中有效\(\Pi_{\kappa}=\{\lambda \in \mathbb{C} \mid \operatorname{Re}{\lambda} \ge -\kappa \}\)、\(\kappa \in \mathbb{R}\)和\(-\Pi_{\kappa}\)如果\(a_1(x)-a_2(x) > 0\)。它们保存在扇区\(S=\{\lambda \in \mathbb{C} \mid \lvert\operatorname{arg}\lambda\rvert \le \pi/2-\phi-\varepsilon\}\) 中，\(\varepsilon > 0\)和\(-S\)条件是\(\lvert\operatorname{arg}\{a_1(x)-a_2(x)\}\rvert<\phi<\pi /2\)。这项工作的主要新颖之处在于，我们假设函数平滑的最小条件，此外，我们还获得了渐近展开所涉及的矩阵的显式公式。对于狄拉克系统来说，这个结果也是新的。