论文标题
对条件解决的特征值方程的严重误解
Gross misinterpretation of a conditionally solvable eigenvalue equation
论文作者
论文摘要
我们解决了一个特征值方程，该方程出现在几篇论文中，涉及广泛的物理问题。 Frobenius方法导致了功率系列系数的三个复发关系，在合适的截断下，对于模型参数的特定值，在合适的截断下产生了精确的分析特征值和特征函数。从这些解决方案中，一些研究人员得出了各种预测，例如允许的角频率，允许的场强度等。我们还通过变异的雷利 - 里茨方法来数值求解特征值方程，并将所得的特征值与截断条件提供的特征值进行比较。通过这种方式，我们证明这些物理预测仅仅是截断条件的伪影。
We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to a three-term recurrence relation for the coefficients of the power series that, under suitable truncation, yields exact analytical eigenvalues and eigenfunctions for particular values of a model parameter. From these solutions some researchers have derived a variety of predictions like allowed angular frequencies, allowed field intensities and the like. We also solve the eigenvalue equation numerically by means of the variational Rayleigh-Ritz method and compare the resulting eigenvalues with those provided by the truncation condition. In this way we prove that those physical predictions are merely artifacts of the truncation condition.
