Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems

Abstract

A method for solving spectral problems for the Sturm-Liouville equation $(p{v}^{\prime }{)}^{\prime }-qv+\lambda rv=0$ based on the approximation of the Delsarte transmutation operators combined with the Liouville transformation is presented. The problem of numerical approximation of solutions and of eigendata is reduced to approximation of a pair of functions depending on the coefficients $p$, $q$ and $r$ by a finite linear combination of certain specially constructed functions related to generalized wave polynomials introduced in arXiv:1208.5984, arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy. Several necessary results concerning the action of the Liouville transformation on formal powers arising in the method of spectral parameter power series are obtained as well as the transmutation operator for the Sturm-Liouville operator $\frac{1}{r}\left(\frac{d}{dx}p\frac{d}{dx}-q\right)$.

21 pages, 3 figures

References in corpus (5)

• Transmutations and Applications: a survey
• Wave polynomials, transmutations and Cauchy's problem for the Klein-Gordon equation
• Modified spectral parameter power series representations for solutions of Sturm-Liouville equations and their applications
• Spectral parameter power series for polynomial pencils of Sturm-Liouville operators and Zakharov-Shabat systems
• Analytic approximation of transmutation operators and related systems of functions

Cited by in corpus (4)

• Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions
• Analytic approximation of transmutation operators and related systems of functions
• Supersymmetric generalized power functions
• Generalized exponential basis for efficient solving of homogeneous diffusion free boundary problems: Russian option pricing