论文标题
trans变运算符和用于扰动贝塞尔方程解决方案的新表示形式
Transmutation operators and a new representation for solutions of perturbed Bessel equations
论文作者
论文摘要
TransMoin操作员积分内核的新表示以及定期解决$ -U^{\ prime \ prime \ prime}+\ left（\ frac {\ ell（\ ell+1）} {x^{2}}}}+q（x）+q（x）\ right）的bessel方程。积分内核表示为傅立叶雅各比系列。该解决方案表示为Neumann系列的Bessel函数，相对于$ω$均匀收敛。对于该系列的系数，可方便地用于数值计算复发公式。新表示形式从Arxiv：1609.06679和Arxiv：1712.01363提高了$ω$和$ \ ell $的大价值，对于$ \ ell $的非全能值。 结果是基于从经典传播（转换）操作者理论的几个思想的应用，解决方案的渐近公式，结果将傅立叶变换的衰减速率与功能的平滑度联系起来，Paley-Wiener定理以及建设性近似理论的结果。 我们表明，在其他可能的应用中获得的分析表示提供了一种简单有效的数值方法，能够以非方面的精度计算大型特征。
New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form $-u^{\prime\prime}+\left(\frac{\ell(\ell+1)}{x^{2}}+q(x)\right)u=ω^{2}u$ are obtained. The integral kernel is represented as a Fourier-Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $ω$. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones from arXiv:1609.06679 and arXiv:1712.01363 for large values of $ω$ and $\ell$ and for non-integer values of $\ell$. The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley-Wiener theorem and some results from constructive approximation theory. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.
