Transmutations and Spectral Parameter Power Series in Eigenvalue Problems

Abstract

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 L2-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. (2010). Primary 34B24, 34L16, 65L15, 81Q05, 81Q60; Secondary 34L25, 34L40. Mathematics Subject Classification∞ k=0 there exists a transmutation operator T such that T[x k ] = ϕ k , i.e., the functions ϕ k are the images of the usual powers of the independent variable. Moreover, it was shown how this operator can be constructed and how it is related to the "canonical" transformation operator considered, e.g., in [51, Chapter 1]. This result together with the practical formulas for calculating the functions ϕ k makes it possible to apply the transmutation technique even when the integral kernel of the operator is unknown. Indeed, now it is easy to apply the transmutation operator to any function approximated by a polynomial.Deeper understanding of the mapping properties of the transmutation operators led us in [46] to the explicit construction of the transmutation operator for a Darboux transformed Schrödinger operator by a known transmutation operator for the original Schrödinger operator as well as to several

A method for computation of scattering amplitudes and Green functions of whole axis problems

“…Remark 5. The inequalities (19) and (20) are of particular importance when using representations (14) and (15) because they guarantee a uniform ( -independent) error estimate for an approximation of solutions, which was illustrated by numerical experiments in Kravchenko et al 6 Remark 6. For numerical implementation of the NSBF representations, the formulas (16) and (23) Remark 7.…”

“…where l k,n is the coefficient of x k in the Legendre polynomial of order n. The series in (14) and (15) converge uniformly with respect to x on any segment and converge uniformly with respect to on any compact subset of the complex plane of the variable . Moreover, for the functions…”

“…Notice that the magnitude f(d) − h 1 (d) is invariant with respect to the choice of the particular solution f(x) satisfying the condition f(0) = 1; see Proposition 4.7 from Kravchenko and Torba. 14 In particular, for f 0 , (34) takes the form (30), from which we see that a( ) has a simple pole in the origin iff ′ 0 (d) ≠ 0, that is, iff zero is not a Neumann eigenvalue of the operator H on…”

Wave polynomials, transmutations and Cauchy’s problem for the Klein–Gordon equation

“…The corresponding second-order ordinary differential equation (15), f ′′ −c 2 f = 0 admits a nonvanishing solution f (x) = e cx , f (0) = 1. Based on this solution we construct n functions ϕ k defined by (14) and (11)-(13) with x 0 = 0.…”

“…In [5] it was shown that given a system of functions {ϕ k } ∞ k=0 defined by (14) where f is any particular solution of (15) …”