Unified Theory of Recurrence Formulas. I

Inui, Teturo

doi:10.1143/ptp/3.2.168

Cited by 10 publications

(3 citation statements)

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“…Eqs. (24), (A5)] [19][20][21][22][23][24][25][26][27], (ii) all Fuchsian 2nd order ODE [28] and (iii) non-Fuchsian 2nd order ODE of the present article [such as Eq. (37) below, which has an irregular singular point at infinity (see…”

Section: Gradation Slicingmentioning

confidence: 99%

“…The subset of ODE (2) where A(z), B(z), C(z) are polynomial of degree at most 4, 3, 2, respectively, covers (i) all QES models within the context of sl 2 (see equations ( 24) and (A.5)) [19][20][21][22][23][24][25][26][27], (ii) all Fuchsian 2nd order ODE [28] and (iii) non-Fuchsian 2nd order ODE of the present article (such as equation (37) below, which has an irregular singular point at infinity (see appendix C)).…”

Section: Gradation Slicingmentioning

confidence: 99%

“…The general necessary condition (28) for the existence of a polynomial solution of nth degree of theorem 4 for 2j = n can be then recast in terms of C * ,…”

Section: Sl 2 Algebraizationmentioning

confidence: 99%

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A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

Moroz¹

2018

J. Phys. A: Math. Theor.

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General concept of a gradation slicing is used to analyze polynomial solutions of ordinary differential equations (ODE) with polynomial coefficients, Lψ = 0, where L = l p l (z)d l z , p l (z) are polynomials, z is a one-dimensional coordinate, and d z = d/dz. It is not required that ODE is ei-Fuchsian or (ii) leads to a usual Sturm-Liouville eigenvalue problem. General necessary and sufficient conditions for the existence of a polynomial solution are formulated involving constraint relations. The necessary condition for a polynomial solution of nth degree to exist forces energy to a nth baseline. Once the constraint relations on the nth baseline can be solved, a polynomial solution is in principle possible even in the absence of any underlying algebraic structure. The usefulness of theory is demonstrated on the examples of various Rabi models. For those models, a baseline is known as a Juddian baseline (e.g. in the case of the Rabi model the curve described by the nth energy level of a displaced harmonic oscillator with varying coupling g). The corresponding constraint relations are shown to (i) reproduce known constraint polynomials for the usual anddriven Rabi models and (ii) generate hitherto unknown constraint polynomials for the two-mode, two-photon, and generalized Rabi models, implying that the eigenvalues of corresponding polynomial eigenfunctions can be determined algebraically. Interestingly, the ODE of the above Rabi models are shown to be characterized, at least for some parameter range, by the same unique set of grading parameters.

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Section: Gradation Slicingmentioning

confidence: 99%

Section: Gradation Slicingmentioning

confidence: 99%

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A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

Moroz¹

2018

J. Phys. A: Math. Theor.

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The factorization method for the Askey–Wilson polynomials

Bangerezako

1999

Journal of Computational and Applied Mathematics

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Codiagonal Perturbations

Green

Triffet

1969

Journal of Mathematical Physics

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Matrix methods are developed for calculating the eigenvalues and eigenvectors of a large class of quantum-mechanical operators which may be regarded as perturbed forms of special-function operators. Specific representations are obtained for the latter, including all of the important cases treated by Infeld and Hull. To these are added representations for terms sufficient to generate forms corresponding to the Mathieu equation, the Lamé equation, and others. A rapidly convergent computational scheme applicable to asymmetric matrices, which retains its stability even when the perturbing terms become large, is described; and its use is illustrated by application to the operator p(1 − q2)p − α2q2, corresponding to the Legendre-like form (d/dx)(1 − x2)(d/dx) + α2x2. Though group-theoretic considerations are stressed, appropriate correlations with differential and integral equations are presented throughout.

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Unified Theory of Recurrence Formulas. I

Cited by 10 publications

References 0 publications

A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

The factorization method for the Askey–Wilson polynomials

Codiagonal Perturbations

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