On a parameter-based asymptotics of solutions to differential equations with coefficients from \(L_{2}(a, b)\) (Q951687)

Consider the system of \(n\) ordinary differential equations \[ y' - \lambda{\sum_0^n\lambda^{-j}A_j(x)y = 0\;,\;-\infty<a\leq x\leq b<+\infty} \] with \(A_j(x)\) and \(A'_0(x)\) are complex valued matrices of class \(L^2(a,b)\); the eigenvalues \(\varphi_i(x)\) of \(A_0(x)\) are different for all \(x\) and their arguments and the arguments of their differences are independent of \(x\). Assume that any matrix \(M(x)\) such that \[ M^{-1}(x)A_0(x)M(x) = D(x)\equiv \text{diag}\{\varphi_1(x),\ldots,\varphi_n(x)\} \] can be written as \(M(x)=m(x)d(x)\) with \(d(x)\) any non-degenerate diagonal matrix. Under these assumptions there exists a fundamental matrix of solutions of the system such that \[ {y(x,\lambda)=(M(x,\lambda)+E(x,\lambda))\exp\left(\lambda \int_a^xD(\xi)d\xi\right)}, \] where \(M(x)\) is as above and chosen such that the main diagonal \(M^{-1}(A_1(x)M(x)-M'(x))\) is zero and \(E(x,\lambda)\in L^2(l)\), where \(l\in S\cap(| \lambda| \gg 1)\) is any ray whose support contains the origin. If \(l\) is strictly inside \(S\) then \(| E(x,\lambda)| \leq C/\sqrt{\lambda}\). Here, \(S\) is one of the sectors centered at the origin such that inside of it \[ \Re e\;\lambda\varphi_1(x)\leq\ldots\leq\Re e\lambda\varphi_n(x)\;,\;\lambda\in S \]