嬗变算子与双曲伪解析函数的构造
Construction of transmutation operators and hyperbolic pseudoanalytic functions
摘要：
给出了Delsarte变换算子积分核的函数级数表示及其项的精确公式。它是基于双曲伪解析函数理论的应用和最近的结果变形算子的映射性质。联系$A=-\frac{d^2}{dx^2}+q_1(x)$和$B=-\frac{d^2}{dx^2}$的嬗变算子的核$K_1$是满足特殊形式的Vekua型双曲型方程的双复值双曲伪解析函数的复分量之一。伪解析函数的另一个组成部分是与$C=-\frac{d^2}{dx^2}+q_2(x)$和$B$相关的嬗变算子的核，其中$q_2$通过达布变换从$q_1$获得。我们证明了这个特殊的双曲型Vekua方程的展开定理和Runge型定理，并利用双曲伪解析函数理论的几个已知结果以及最近发现的变形算子的映射性质，得到了它们的核的新表示。给出了几个例子。此外，基于所给出的结果，提出了嬗变核的数值计算方法和谱问题的数值解方法。
英文摘要：
A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with the exact formulas for the terms of the series. It is based on the application of hyperbolic pseudoanalytic function theory and recent results on mapping properties of the transmutation operators. The kernel $K_1$ of the transmutation operator relating $A=-\frac{d^2}{dx^2}+q_1(x)$ and $B=-\frac{d^2}{dx^2}$ results to be one of the complex components of a bicomplex-valued hyperbolic pseudoanalytic function satisfying a Vekua-type hyperbolic equation of a special form. The other component of the pseudoanalytic function is the kernel of the transmutation operator relating $C=-\frac{d^2}{dx^2}+q_2(x)$ and $B$ where $q_2$ is obtained from $q_1$ by a Darboux transformation. We prove the expansion theorem and a Runge-type theorem for this special hyperbolic Vekua equation and using several known results from hyperbolic pseudoanalytic function theory together with the recently discovered mapping properties of the transmutation operators obtain the new representation for their kernels. Several examples are given. Moreover, based on the presented results approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed.
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