Biconfluent Heun equation in quantum chemistry: Harmonium and related systems

Karwowski, Jacek; Witek, Henryk A.

doi:10.1007/s00214-014-1494-5

Cited by 43 publications

(47 citation statements)

References 39 publications

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“…Differential equations of the form similar to Equation (15) arise in many areas of both physics and quantum chemistry, ranging from the study of the harmonium to the confinement potentials [38][39][40][41][42][43][44][45][46]. Various exact and approximate methods are known for solving such an equation with central potentials [47][48][49][50][51][52][53][54][55].…”

Section: Four Methods Leading To Quantizationmentioning

confidence: 99%

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Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case

Landry

Hammad

2021

Universe

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We investigate the gravitational effect on Landau levels. We show that the familiar infinite Landau degeneracy of the energy levels of a quantum particle moving inside a uniform and constant magnetic field is removed by the interaction of the particle with a gravitational field. Two independent approaches are used to solve the relevant Schrödinger equation within the Newtonian approximation. It is found that both approaches yield qualitatively similar results within their respective approximations. With the goal of clarifying some results found in the literature concerning the use of a third independent approach for extracting the quantization condition based on a similar differential equation, we show that such an approach cannot yield a general and yet consistent result. We point out to the more accurate, but impractical, way to use such an approach; a way which does in principle yield a consistent quantization condition. We discuss how our results could be used to contribute in a novel way to the existing methods for testing gravity at the tabletop experiments level as well as at the astrophysical observational level by deriving the corrections brought by Yukawa-like and power-law deviations from the inverse-square law. The full relativistic regime is also examined in detail.

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Section: Four Methods Leading To Quantizationmentioning

confidence: 99%

“…The special function H(a, b, c, d; z) is called the biconfluent Heun function and it is given explicitly by the following infinite series [45,60],…”

Section: Using the Biconfluent Heun Equation: The Polynomial Approachmentioning

confidence: 99%

Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case

Landry

Hammad

2021

Universe

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“…For more such properties of k-gamma and related functions, we refer the reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and related bibliography therein. The subsequent Section 2.2 is taken from author work [22].…”

Section: K-gamma Functionsmentioning

confidence: 99%

“…Set et al have used the analogue to the Riemann-Liouville singular kernel at k-calculus in [7]. For more comprehensive and detailed studies of related works, I refer the interested reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and associated references therein. The literature review for k-gamma functions motivates us to state "on the one hand k-gamma functions excited to mathematician for the analysis of mathematical concepts in a new way and on the other applications of these functions in diverse problems are fundamental".…”

Section: Introductionmentioning

confidence: 99%

A New Representation of the k-Gamma Functions

Tassaddiq

2019

Mathematics

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The products of the form z ( z + l ) ( z + 2 l ) … ( z + ( k − 1 ) l ) are of interest for a wide-ranging audience. In particular, they frequently arise in diverse situations, such as computation of Feynman integrals, combinatory of creation, annihilation operators and in fractional calculus. These expressions can be successfully applied for stated applications by using a mathematical notion of k-gamma functions. In this paper, we develop a new series representation of k-gamma functions in terms of delta functions. It led to a novel extension of the applicability of k-gamma functions that introduced them as distributions defined for a specific set of functions.

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“…To be more specific, consider, for instance, the details of derivation of the expansion (9). It is readily shown, e.g., by dividing Eq.…”

Section: Expansions In Terms Of the Incomplete Beta Functionsmentioning

confidence: 99%

Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

et al. 2016

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Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four-and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.

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Biconfluent Heun equation in quantum chemistry: Harmonium and related systems

Abstract: Schrödinger equation for harmonium and related models may be transformed to the biconfluent Heun equation. The solubility of this equation and its applications in quantum chemistry are briefly discussed.

Cited by 43 publications

References 39 publications

Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case

Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case

A New Representation of the k-Gamma Functions

Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

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