On a Series Representation for Integral Kernels of Transmutation Operators for Perturbed Bessel Equations

Abstract

A representation for the kernel of the transmutation operator relating the perturbed Bessel equation with the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of the regular solution of the perturbed Bessel equation is given presenting a remarkable feature of uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of application to the solution of Dirichlet spectral problems is presented.

Transmutation operators and complete systems of solutions for the radial Schrödinger equation

“…The constant c p does not depend on r, and the last inequality is valid for all r ∈ [0, b], from which we obtain (19), and thus the uniform convergence.…”

“…In Section 6, we present an explicit procedure to construct the formal powers by solving a sequence of perturbed Bessel equations. In Section 7, a relation between the transmutation operator and another transmutation operator for the perturbed Bessel operator (studied in a number of publications, eg, previous studies 16–21 ) is established.…”

A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications

“…It admits a spectral parameter independent estimate for the approximation of the solution by partial sums of the series, which in practice allows one to compute huge numbers of eigenvalues and eigenfunctions with a controllable accuracy [9], [8], [13]. Similar results were obtained for perturbed Bessel equations [12], [10].…”

Weighted generalized functions and hyperbolic Riesz B‐potential

“…[37][38][39] For the approach to study transmutations operators connected with Bessel operator, see other studies. [40][41][42][43] In (6), we define the Riesz hyperbolic B-potentials I P±i0, as…”