Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems
For researchers working on spectral problems for Sturm-Liouville equations, this method offers a highly accurate numerical approach to compute eigendata.
The paper presents a method for solving Sturm-Liouville spectral problems by combining Liouville transformation with approximation of Delsarte transmutation operators, enabling computation of both lower and higher eigendata with extreme accuracy.
A method for solving spectral problems for the Sturm-Liouville equation $(pv^{\prime})^{\prime}-qv+λrv=0$ 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 $\frac{1}{r}\left(\frac{d}{dx}p\frac{d}{dx}-q\right)$.