Abstract We consider expressions $$\mathcal {L}_k[y]$$ that are ordered products of $$k$$ , $$k\in \mathbb {N} $$ , linear symmetric quasidifferential expressions of the second order with matrix coefficients. Asymptotic formulas as $$x\to +\infty $$ are derived for one of the fundamental solution systems of the equation $$\mathcal {L}_k[y]=\lambda y $$ , $$\lambda \in \mathbb {C} $$ , $$\lambda \ne 0$$ , under some conditions on the behavior at infinity of its coefficients. The result is applied to the study of spectral properties of operators generated by the expression $$\mathcal {L}_k[y] $$ , and, in particular, to second-order matrix differential operators of the Sturm–Liouville type.

中文翻译：

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具有非光滑系数的矩阵微分方程的谱性质

摘要 我们考虑表达式 $$\mathcal {L}_k[y]$$ 是 $$k$$ 的有序积，$$k\in \mathbb {N} $$ ，二阶线性对称拟微分表达式矩阵系数。为方程 $$\mathcal {L}_k[y]=\lambda y $$ , $$\lambda \in \ 的基本解系统之一推导出渐近公式 $$x\to +\infty $$ mathbb {C} $$ , $$\lambda \ne 0$$ ，在其系数无穷大的某些条件下。结果应用于研究由表达式 $$\mathcal {L}_k[y] $$ 生成的算子的谱特性，特别是 Sturm–Liouville 类型的二阶矩阵微分算子。