A Uniform Expansion for the Eigenfunction of a Singular Second-Order Differential Operator

Fitouhi, Ahmed; Hamza, Mongi

doi:10.1137/0521088

Cited by 37 publications

(13 citation statements)

References 4 publications

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“…In an interesting research reported in [11] and [16] there appears a representation of solutions of Sturm-Liouville equations in the form of Neumann series of Bessel functions different to the representation obtained in the present work. The representation from [11] and [16] does not possess the uniformity with respect to ω to the difference from our representation, and the convergence of the series which is guaranteed on a certain interval of x for holomorphic q only is achieved due to the exponential decay of j n (z) when n → ∞. Apart from that previous work, to our best knowledge, the Neumann series of Bessel functions have not been used to represent solutions of a general linear differential equation.…”

Section: Introductioncontrasting

confidence: 75%

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

Kravchenko

Navarro

Torba

2017

Applied Mathematics and Computation

151

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A new representation of solutions to the equation −y ′′ + q(x)y = ω 2 y is obtained. For every x the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter ω. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to ω which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.

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Section: Introductioncontrasting

confidence: 75%

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

Kravchenko

Navarro

Torba

2017

Applied Mathematics and Computation

151

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“…Consider the asymptotic expansion (4.18) with m = 1. According to [11] the remainder R 1 (ω, x) satisfies the integral equation…”

Section: Representation Of the Derivative Of The Regular Solutionmentioning

confidence: 99%

“…The convergence rate of the Fourier-Legendre series (and, consequently, of the series (1.3)) depends on the smoothness of the integral kernel R. Only few basic properties of the kernel R can be obtained using the results from [33] and [9]. We implement a different approach based on the asymptotic formulas from [15] and [11], a Paley-Wiener theorem and the constructive approximation theory [10]. As a result, we present close to optimal convergence rate estimates depending on the parameter l and the smoothness of the potential q.…”

Section: Introductionmentioning

confidence: 99%

A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations

Kravchenko

Torba

Castillo-Pérez

2017

Applicable Analysis

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A new representation for a regular solution of the perturbed Bessel equation of the form Lu = −u ′′ + l(l+1)x 2 + q(x) u = ω 2 u is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to ω. For the coefficients of the series explicit direct formulas are obtained in terms of the systems of recursive integrals arising in the spectral parameter power series (SPPS) method, as well as convenient for numerical computation recurrent integration formulas.The result is based on application of several ideas from the classical transmutation (transformation) operator theory, recently discovered mapping properties of the transmutation operators involved and a Fourier-Legendre series expansion of the transmutation kernel. For convergence rate estimates, asymptotic formulas, a Paley-Wiener theorem and some results from constructive approximation theory were used.We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy. * Research was supported by CONACYT, Mexico via the projects 166141 and 222478. R. Castillo would like to thank the support of CONACYT and of the SIBE and EDI programs of the IPN as well as that of the project SIP 20160525.

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“…We assume in this section that χ is holomorphic in a disc D(0, 2b) = {z ∈ C, |z| < 2b}, b > 0. Therefore, from [13], we have:…”

Section: Definitionmentioning

confidence: 99%

New Results on the DLp-Type Spaces Associated with a Singular Second Order Differential Operator

Jelassi¹

2015

IJOPCA

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A Uniform Expansion for the Eigenfunction of a Singular Second-Order Differential Operator

Cited by 37 publications

References 4 publications

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations

New Results on the DLp-Type Spaces Associated with a Singular Second Order Differential Operator

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