AI 中文总结
研究具有可变系数的径向Sturm-Liouville算子，通过变量分离得到相关问题，推导出谱方程。应用刘维尔-格林变换得到渐近公式，建立相关性质，用于构造近似解和误差估计，数值计算验证了渐近量化公式及谱近似的准确性。
AI 中文摘要
我们研究一个径向对称的初边值问题，其变量分离导致一个具有明确确定系数和正权函数的正则自伴Sturm-Liouville问题。所得的Sturm-Liouville算子由特殊的几何诱导系数对定义，产生非标准加权谱结构。相关径向算子在相应加权希尔伯特空间中有离散实谱和完备正交特征函数系。推导出控制特征值的精确超越谱方程。为分析高频情况，将刘维尔-格林变换直接应用于径向Sturm-Liouville方程，得到特征值、特征函数及相应刘维尔-格林拟模的显式渐近公式。建立了它们的加权正交性和渐近完备性。所得渐近谱数据用于构造原边值问题的近似解并获得谱重构的误差估计。精确Sturm-Liouville谱的数值计算与刘维尔-格林预测高度吻合，验证了渐近量化公式并确认了所提谱近似的准确性。
英文摘要
We investigate a radially symmetric initial-boundary value problem whose separation of variables leads to a regular self-adjoint Sturm-Liouville problem with explicitly determined coefficients and a positive weight function. The resulting Sturm-Liouville operator is defined by a special geometry-induced coefficient pair that gives rise to a nonstandard weighted spectral structure. The associated radial operator possesses a discrete real spectrum and a complete orthonormal system of eigenfunctions in the corresponding weighted Hilbert space, yielding an exact spectral representation of the evolution problem. An exact transcendental spectral equation governing the eigenvalues is derived. To analyze the high-frequency regime, the Liouville-Green transformation is applied directly to the radial Sturm-Liouville equation. This yields explicit asymptotic formulas for the eigenvalues and eigenfunctions together with corresponding Liouville-Green quasimodes. Their weighted orthogonality properties and asymptotic completeness are established within the spectral framework of the exact operator. The resulting asymptotic spectral data are used to construct approximate solutions of the original boundary-value problem and to obtain error estimates for the spectral reconstruction. Numerical computations of the exact Sturm-Liouville spectrum show excellent agreement with the Liouville-Green predictions, thereby validating the asymptotic quantization formula and confirming the accuracy of the proposed spectral approximation.
Comments30 pages, 1 figure, 1 table
