Spectral parameter power series representation for solutions of linear system of two first order differential equations
Instituto de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia（安蒂奥基亚大学精确与自然科学学院数学研究所）；Departamento de Matemáticas, CINVESTAV del IPN, Unidad Querétaro（墨西哥国立理工学院CINVESTAV奎雷塔罗校区数学系）
本文给出了一个一般解的谱参数幂级数（SPPS）表示形式，适用于一维狄拉克系统，该系统包含任意矩阵系数在谱参数处。方程形式为$ B \frac{dY}{dx} + P(x)Y = λR(x)Y $，其中$ Y=(y_1,y_2)^T $是未知向量函数，$ λ $是谱参数，$ B = \begin{pmatrix}0 & 1 \\\ -1 & 0\end{pmatrix} $，$ P $是对称的$ 2\times 2 $矩阵，$ R $是任意的$ 2\times 2 $矩阵，其元素是可积的复值函数。这些级数的系数函数通过递归迭代一个简单的积分过程获得，从一个非消失解开始，对于特定的$ λ= λ_0 $。证明了这种解的存在性。对于一般的线性系统，由两个一阶微分方程组成，$ P(x)\frac{dY}{dx}+Q(x)Y = λR(x)Y $，其中$ P $、$ Q $、$ R $是$ 2\times 2 $矩阵，其元素是可积的复值函数，$ P $在每个$ x $处可逆，展示了一种将其转换为上述类型的方法。给出了SPPS表示法在求解初值问题和谱问题中的通用应用方案以及数值示例。
英文摘要：
A representation in the form of spectral parameter power series (SPPS) is given for a general solution of a one dimension Dirac system containing arbitrary matrix coefficient at the spectral parameter, \[ B \frac{dY}{dx} + P(x)Y = λR(x)Y,\] where $Y=(y_1,y_2)^T$ is the unknown vector-function, $λ$ is the spectral parameter, $B = \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$, and $P$ is a symmetric $2\times 2$ matrix, $R$ is an arbitrary $2\times 2$ matrix whose entries are integrable complex-valued functions. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a non-vanishing solution for one particular $λ= λ_0$. The existence of such solution is shown.
For a general linear system of two first order differential equations \[
P(x)\frac{dY}{dx}+Q(x)Y = λR(x)Y,\ x\in [a,b], \] where $P$, $Q$, $R$ are $2\times 2$ matrices whose entries are integrable complex-valued functions, $P$ being invertible for every $x$, a transformation reducing it to a type considered above is shown.
The general scheme of application of the SPPS representation to the solution of initial value and spectral problems as well as numerical illustrations are provided.
Power series with Taylor coefficients of sum-product type and algebraic differential equations /SP-series and the interplay between their resurgence and differential properties
Analogs of generalized resolvents and eigenfunction expansions of relations generated by pair of differential operator expressions one of which depends on spectral parameter in nonlinear manner
Analogs of generalized resolvents of relations generated by pair of differential operator expressions one of which depends on spectral parameter in nonlinear manner
