Corrigendum to “Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconfluent Heun Equation” [Ann. Phys. 347 (2014) 130–140]

Caruso, Francisco; Martins, J.; Oguri, V.; Silveira, Felipe

doi:10.1016/j.aop.2016.12.027

Cited by 11 publications

(14 citation statements)

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“…The problem of two electrons in an external oscillator potential [24] is studied in three dimensions [25], and it is shown that the above radial equation is quasi-exactly solvable [26,27,28], which means that it is possible to find exact simple solutions for some, but not all, eigenfunctions, corresponding to a certain infinite set of discrete oscillator frequencies. In a recent paper [29,30], adopting the Ansatz V (r) = 1/r, it was shown that in 2D it is possible to determine exactly and in a closed form a finite portion of the positive energy spectrum and the associated eigenfunctions for the Schrödinger equation describing the relative motion of a two-electron system, by putting Eq. ( 3) into the form of a Biconfluent Heun Equations (BHE) [31].…”

Section: Planar Physics and The Ln(r) Potentialmentioning

confidence: 99%

How the inter-electronic potential Ansätze affect the bound state solutions of a planar two-electron quantum dot model

Caruso

Oguri

Silveira

2019

Physica E: Low-dimensional Systems and Nanostructures

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The model of a two-electron quantum dot, confined to move in a two dimensional flat space, in the presence of an external harmonic oscillator potential, is revisited for a specific purpose. Indeed, eigenvalues and eigenstates of the bound state solutions are obtained for any oscillation frequency considering both the 1/r and ln r Ansätze for inter-electronic Coulombic-like potentials in 2D. Then, it is pointed out that the significative difference between measurable quantities predicted from these two potentials can shed some light on the problem of space dimensionality as well as on the physical nature of the potential itself.

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Section: Planar Physics and The Ln(r) Potentialmentioning

confidence: 99%

How the inter-electronic potential Ansätze affect the bound state solutions of a planar two-electron quantum dot model

Caruso

Oguri

Silveira

2019

Physica E: Low-dimensional Systems and Nanostructures

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“…There has been increased interest in the bi-confluent Heun equation [6][7][8] recently [10,21,23,[47][48][49][50][51], as a second-order differential equation into which the stationary Schrödinger equation can be transformed. In this Section we combine two approaches that have been applied previously to obtain potentials solvable in terms of certain special functions of mathematical physics, and specify them for the case of the bi-confluent Heun equation.…”

Section: Potentials Solvable In Terms Of the Bi-confluent Heun Equationmentioning

confidence: 99%

Exact solutions of the sextic oscillator from the bi-confluent Heun equation

Lévai

Ishkhanyan

2019

Mod. Phys. Lett. A

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The sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments.The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Hermite functions turn into Hermite polynomials with shifted arguments, and, at the same time, a polynomial expression is obtained for one of the parameters, the roots of which supply the energy eigenvalues. With the δ = 0 choice the quartic potential term is cancelled, leading to the reduced sextic oscillator. It was found that the expressions for the energy eigenvalues and the corresponding wave functions of this potential agree with those obtained from the quasi-exactly solvable formalism. Possible generalizations of the method are also presented.

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“…This steams from the fact that these expressions do not represent the energies of the system in its present form. Actually, the condition (76) allows us to establish a quantum condition that links the energy and others physical quantities, including η C [79,85,86]. As a result, it is possible to express the energy in terms of all the physical parameters involved in the problem, namely, η C , η L , M, and ω.…”

Section: The Analysis Of Both the Spin And The Pseudo-spin Symmetriesmentioning

confidence: 99%

On the 2D Dirac oscillator in the presence of vector and scalar potentials in the cosmic string spacetime in the context of spin and pseudospin symmetries

et al. 2019

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The Dirac equation with both scalar and vector couplings describing the dynamics of a two-dimensional Dirac oscillator in the cosmic string spacetime is considered. We derive the Dirac-Pauli equation and solve it in the limit of the spin and the pseudo-spin symmetries. We analyze the presence of cylindrical symmetric scalar potentials which allows us to provide analytic solutions for the resultant field equation. By using an appropriate ansatz, we find that the radial equation is a biconfluent Heun-like differential equation. The solution of this equation provides us with more than one expression for the energy eigenvalues of the oscillator. We investigate these energies and find that there is a quantum condition between them. We study this condition in detail and find that it requires the fixation of one of the physical parameters involved in the problem. Expressions for the energy of the oscillator are obtained for some values of the quantum number n. Some particular cases which lead to known physical systems are also addressed.

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Corrigendum to “Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconfluent Heun Equation” [Ann. Phys. 347 (2014) 130–140]

Cited by 11 publications

References 0 publications

How the inter-electronic potential Ansätze affect the bound state solutions of a planar two-electron quantum dot model

How the inter-electronic potential Ansätze affect the bound state solutions of a planar two-electron quantum dot model

Exact solutions of the sextic oscillator from the bi-confluent Heun equation

On the 2D Dirac oscillator in the presence of vector and scalar potentials in the cosmic string spacetime in the context of spin and pseudospin symmetries

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