Global solutions of the biconfluent Heun equation

Ferreira, E.; Sesma, J.

doi:10.1007/s11075-015-0024-4

Cited by 7 publications

(15 citation statements)

References 15 publications

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“…(III. 19) makes the latter take the following canonical form, known as the bi-confluent Heun differential equation [53][54][55][56], given by a linear combination of two independent bi-confluent Heun functions as follows,…”

Section: Using the Biconfluent Heun Equation: The Polynomial Approachmentioning

confidence: 99%

Landau levels in a gravitational field: The Schwarzschild spacetime case

Landry,

Hammad

2019

Preprint

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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. First, two independent approaches are used to solve the relevant Schrödinger equation. It is found that both approaches yield qualitatively similar results within their respective approximations. Then, with the goal of clarifying some confusing 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, and yet impractical, way to use such an approach, a way which does in principle yield a consistent quantization condition.Finally, we show 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.

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Section: Using the Biconfluent Heun Equation: The Polynomial Approachmentioning

confidence: 99%

Landau levels in a gravitational field: The Schwarzschild spacetime case

Landry,

Hammad

2019

Preprint

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“…However, this method seems to be very complicated. For this reason we use a comparatively new and more simple method proposed by the authors of [15] for the biconfluent Heun equation and based on the asymptotic series for Wronskians. It allows one to obtain the value of T 4 in the form of a transcendental expression…”

Section: Numerical Calculation Of Eigenvalues: the Methods And Resultsmentioning

confidence: 99%

“…with the same boundary conditions (16). It is equivalent to the solution of the equation (15) with µ → −µ < 0, E → −E. Calculations made in Section 5 show that for µ < 0 all values E are positive.…”

Section: Eigenvalues Problemmentioning

confidence: 99%

On the one-dimensional reggeon model: eigenvalues of the Hamiltonian and the propagator

et al. 2019

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“…First, we need to introduce the new variable z = β/2 ρ. Substituting this inside Equation (34) makes the latter take the following canonical form, known as the biconfluent Heun differential equation [60][61][62][63],…”

Section: Using the Biconfluent Heun Equation: The Polynomial Approachmentioning

confidence: 99%

“…In fact, for a magnetic white dwarf of radius of about 0.015 R and a mass of about 0.4 M [76], the ratio GM/c 2 ρ is of the order of 10 −4 on the surface of the star. Our approximations done in this section are therefore suitable to fully apply our results (63) and (64) to study the behavior of the charged electron gas making the surface of such stars and confront them with observations regarding the mass-radius predictions [76]. Furthermore, as in the case of the Newtonian approximation done in Section 3.1, any significant deviation from the predicted corrections ( 63) and ( 64) within the range of our approximations would signal a possible deviation from general relativity.…”

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confidence: 93%

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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Global solutions of the biconfluent Heun equation

Cited by 7 publications

References 15 publications

Landau levels in a gravitational field: The Schwarzschild spacetime case

Landau levels in a gravitational field: The Schwarzschild spacetime case

On the one-dimensional reggeon model: eigenvalues of the Hamiltonian and the propagator

Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case

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