Transmutations and spectral parameter power series in eigenvalue problems
For mathematicians and physicists working on Sturm-Liouville problems, this provides a unified framework linking transmutation operators and SPPS, but the results are incremental as they extend known theory.
The paper reviews recent advances in Sturm-Liouville theory using transmutation operators and spectral parameter power series (SPPS), showing that SPPS yields analytic dispersion equations for spectral problems and that recursive integral systems are complete in L2 and are images of power series under transmutation. Numerical illustrations are provided.
We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations corresponding to a variety of spectral problems related to Sturm-Liouville equations in an analytic form is an attractive feature of the SPPS method. It is based on a computation of certain systems of recursive integrals. Considered as families of functions these systems are complete in the $L_{2}$-space and result to be the images of the nonnegative integer powers of the independent variable under the action of a corresponding transmutation operator. This recently revealed property of the Delsarte transmutations opens the way to apply the transmutation operator even when its integral kernel is unknown and gives the possibility to obtain further interesting properties concerning the Darboux transformed Schrödinger operators. We introduce the systems of recursive integrals and the SPPS approach, explain some of its applications to spectral problems with numerical illustrations, give the definition and basic properties of transmutation operators, introduce a parametrized family of transmutation operators, study their mapping properties and construct the transmutation operators for Darboux transformed Schrödinger operators.