Stokes phenomenon for the prolate spheroidal wave equation

Fauvet, Frédéric; Ramis, Jean-Pierre; Richard-Jung, Françoise; Thomann, Johannes

doi:10.1016/j.apnum.2010.05.010

Cited by 11 publications

(17 citation statements)

References 10 publications

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“…The following result can be found in ( [10], p. 95): Let X be a unitary space, let T (x) = T + x T (1) a linear operator on X and let T be normal. Then the power series for the eigenvalues λ(x) are convergent if "the magnitude of the perturbation" ||x T (1) || is smaller than half the isolation distance of the eigenvalue λ of T .…”

Section: The Eigenvalues As Roots Of An Infinite Determinantmentioning

confidence: 99%

“…where a = ||T (1) || and d is the isolation distance of the eigenvalues of T . We apply this result here by searching the eigenvalues of…”

Section: The Eigenvalues As Roots Of An Infinite Determinantmentioning

confidence: 99%

“…In a preceding paper [1], we made the following observations. For τ ≥ 0 fixed and μ ∈ R, the following properties are equivalent:…”

Section: Introductionmentioning

confidence: 97%

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New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

et al. 2016

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In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of polynomials: the coefficients of the formal power series appearing in the solutions near 0, 1, or ∞ (in the variables x,x−1, or 1/x, respectively). The result, which seems to be true for all values of the parameter τ, according to our numerical experiments, is here proved for small values of the parameter τ.

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Section: The Eigenvalues As Roots Of An Infinite Determinantmentioning

confidence: 99%

“…where a = ||T (1) || and d is the isolation distance of the eigenvalues of T . We apply this result here by searching the eigenvalues of…”

Section: The Eigenvalues As Roots Of An Infinite Determinantmentioning

confidence: 99%

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New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

et al. 2016

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“…The results of §8.3 and §8.4 could be the first step to a numerical algorithm of approximation of the Stokes and monodromy matrices. See [16,15,18,22,35] for results of numerical approximation of the Stokes matrices and [28,27] for results of numerical approximation of the monodromy matrices.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Confluence of meromorphic solutions of q-difference equations

Dreyfus¹

2015

Ann. inst. Fourier

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In this paper, we consider a q-analogue of the Borel-Laplace summation where q > 1 is a real parameter. In particular, we show that the Borel-Laplace summation of a divergent power series solution of a linear differential equation can be uniformly approximated on a convenient sector, by a meromorphic solution of a corresponding family of linear q-difference equations. We perform the computations for the basic hypergeometric series. Following Sauloy, we prove how a basis of solutions of a linear differential equation can be uniformly approximated on a convenient domain by a basis of solutions of a corresponding family of linear q-difference equations. This leads us to the approximations of Stokes matrices and monodromy matrices of the linear differential equation by matrices with entries that are invariants by the multiplication by q. * Throughout the paper, we will use the word "confluence" to describe the q-degeneracy when q → 1.

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“…Fauvet et al [9] also studied an eigenvalue problem for a special non-Fuchsian secondorder differential equation called the prolate spheroidal wave equation [27] on a finite interval, using the differential Galois theory. They analyzed the Stokes phenomenon and clarified a relation between solutions of the eigenvalue problem and the differential Galois group.…”

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confidence: 99%

Galoisian approach for a Sturm-Liouville problem on the infinite interval

Blázquez-Sanz

Yagasaki

2012

Methods and Applications of Analysis

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Abstract. We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval (−∞, ∞). Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with three examples for a stationary Schrödinger equation having a generalized Hulthén potential; a linear stability equation for a traveling front in the Allen-Cahn equation; and an eigenvalue problem related to the Lamé equation.

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Stokes phenomenon for the prolate spheroidal wave equation

Cited by 11 publications

References 10 publications

New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

Confluence of meromorphic solutions of q-difference equations

Galoisian approach for a Sturm-Liouville problem on the infinite interval

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