论文标题
$ 2 \ times 2 $普通微分方程的频谱渐近学
Spectral asymptotics for solutions of $2\times 2$ system of ordinary differential equations of the first order
论文作者
论文摘要
本文的目的是找到$ 2 \ times 2 $常规微分方程的解决方案的表示形式$ \ mathbf {y^\ prime} - b（x）\ mathbf {y} =λa（x）\ mathbf {y} a_2（x）\} $，$ b（x）=（b_ {ij}（x））$，$ a_1（x）> 0，\ a_2（x）<0 $和所有功能$ a_ {i {i}，b_ {ij {ij} $属于Sobolev Space $ W^n_1 $ n_1 $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n unt。我们证明，上述系统的解决方案的基本矩阵存在$$ y（x，λ）= m（x）= m（x）\ left（i + \ frac {r^1（x）}λ + \ dots + dots + \ dots + \ frac {r^n（x）其中$ o（1）\ to 0 $在[0,1] $ in [0,1] $ in [0,1] $中均匀地为频谱参数$λ\ to \ infty $在半平面$ \ re \ re \，λ>-κ$或$ \ re \ re \，λ<κ$，$κ$的情况下，$κ$是任何固定的实际数量。主要的新颖性是，我们在此表示中为所有矩阵$ m，e $和$ r^m $提供明确的公式。
The aim of the paper is to find representation for solutions of $2\times 2$ system of ordinary differential equations $$ \mathbf{y^\prime} - B(x)\mathbf{y} = λA(x)\mathbf{y}, \quad \ x \in [0, 1], $$ where $A(x) = diag\{a_1(x), a_2(x)\}$, $B(x) = (b_{ij}(x))$, $a_1(x) > 0, \ a_2(x) < 0$ and all the functions $a_{i}, b_{ij}$ belong to the Sobolev spaces $W^n_1[0,1]$ for given integer $n\geqslant 0$. We prove that there exists a fundamental matrix of solutions for the above system, which have representation $$ Y(x, λ) = M(x)\left(I + \frac{R^1(x)}λ + \dots + \frac{R^n(x)}{λ^n} + o(1)λ^{-n}\right)E(x, λ), $$ where $o(1) \to 0$ uniformly for $x\in [0,1]$ as the spectral parameter $λ\to \infty$ in the half plane $\Re\,λ>-κ$ or $\Re\,λ<κ$, where $κ$ is any fixed real number. The main novelty is that we give explicit formulae for all matrices $M,E$ and $R^m$ in this representation.
