Application of the Method of Decomposition into Exponential Series by the Spectral Parameter in Eigenvalue Problems

Abstract

In the paper [6], a method for decomposition into power series by the spectral parameter was proposed, which turned out to be effective for the numerical determination of the eigenvalues of the Sturm-Liouville operator. The problem of computing the eigenvalues of the Sturm-Liouville operator reduces to finding the zeros of the so-called characteristic determinant . The characteristic determinant of the Sturm-Liouville operator represents an entire function of the spectral parameter. Thus, the characteristic determinant  is decomposed into a power series by the spectral parameter  with an infinite radius of convergence. In the paper [6], a simple method for finding the Taylor coefficients was provided. It turned out that the recurrence formulas for determining the Taylor coefficients give a simple and powerful method for numerically computing the eigenvalues. However, this approach is effective for calculating relatively small eigenvalues. For very large eigenvalues, the method proposed in [6] is not exactly ineffective, but for finding such eigenvalues, it is advisable to use exponential series by the spectral parameter. The exponential series we propose are effective for calculating sufficiently large eigenvalues in terms of magnitude. Exponential Series by the Spectral Parameter for the Sturm-Liouville Equation on a Segment