Factorization of some confluent Heun’s differential equations

Hounkonnou, Mahouton Norbert; Ronveaux, A.; Sodoga, Komi

doi:10.1016/j.amc.2006.11.170

Cited by 22 publications

(16 citation statements)

References 5 publications

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“…Due to its wide appearance in theoretical physics, mathematical properties of the biconfluent Heun equation have been studied by many authors (see, e.g., [1][2][3][13][14][15][16][17][18][19][20][21][22][23][24][25][26]). In particular, the power-series solutions near the regular singularity at the origin and in the neighborhood of the irregular singularity at the infinity [17,18], the continued fraction technique [19] and the Hill determinant approach [20] for a class of confinement potentials have been discussed in detail.…”

Section: Introductionmentioning

confidence: 99%

“…In [21,22] relations between the linear equations of the (deformed) Heun class and the six Painlevé nonlinear equations have been established via an anti-quantization procedure. In [23] the factorization of the confluent Heun equations is re-examined. In [24] k -summability is used to obtain new integral formulas for the solutions near the infinity and in [25] integral representations for a fundamental system of solutions to the bi-confluent Heun equation are derived using the properties of the Meijer G -functions.…”

Section: Introductionmentioning

confidence: 99%

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Solutions of the bi-confluent Heun equation in terms of the Hermite functions

Ishkhanyan

2017

Annals of Physics

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We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finitesum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schrödinger equation is written in terms of the bi-confluent Heun functions and further identify a particular conditionally integrable potential for which the involved bi-confluent Heun function admits a four-term finite-sum expansion in terms of the Hermite functions. This is an infinite well defined on a half-axis. We present the explicit solution of the one-dimensional Schrödinger equation for this potential and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and construct an accurate approximation for the bound-state energy levels.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solutions of the bi-confluent Heun equation in terms of the Hermite functions

Ishkhanyan

2017

Annals of Physics

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“…(22) and (23) consists in performing a series of transformations that map Eq. 22into a symmetric canonical form of a double confluent Heun's equation [27,[43][44][45].…”

Section: The Generalized Dipole Equationmentioning

confidence: 99%

“…Although not obvious, Eq. (31) has an intrinsic symmetry that allows mapping it into a very symmetrical form, called the symmetrical-canonical double confluent Heun equation [27,43,44]. We will discuss in Section 4.2 multiple advantages of expressing the dipole equation in this form, especially related to the existence of simple analytic solutions.…”

Section: This Valley Creates a Barrier Of Heightmentioning

confidence: 99%

Vortex Structures inside Spherical Mesoscopic Superconductor Plus Magnetic Dipole

Ludu

2018

Advances in Mathematical Physics

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We investigate the existence of multivortex states in a superconducting mesoscopic sphere with a magnetic dipole placed at the center. We obtain analytic solutions for the order parameter Ψ(r→) inside the sphere through the linearized Ginzburg-Landau (GL) model, coupled with mixed boundary conditions, and under regularity conditions and decoupling coordinates approximation. The solutions of the linear GL equation are obtained in terms of Heun double confluent functions, in dipole coordinates symmetry. The analyticity of the solutions and the associated eigenproblem are discussed thoroughly. We minimize the free energy for the fully nonlinear GL system by using linear combinations of linear analytic solutions, and we provide the conditions of occurring multivortex states. The results are not restricted to the particular spherical geometry, since the present formalism can be extended for large samples, up to infinite superconducting space plus magnetic dipole.

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“…we map the dipole equation equation (10) into a symmetric canonical doubleconfluent Heun equation in y(z) [6,[13][14][15]26].…”

Section: The Linear Equationmentioning

confidence: 99%