Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconfluent Heun equation

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

doi:10.1016/j.aop.2014.04.023

Cited by 37 publications

(30 citation statements)

References 16 publications

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“…In Fig. 2 some theoretical polynomial eigenfunctions u(r) calculated with the results of the references [8,9] are compared to the respective numerical eigenfunctions. Fig.…”

Section: First Numerical Resultsmentioning

confidence: 99%

Numerical Solutions for a Two-dimensional Quantum Dot Model

2019

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In this paper, a quantum dot mathematical model based on a twodimensional Schrödinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. The known polynomial solutions are confronted with new numerical calculations based on the Numerov method. A good qualitative agreement between them emerges. The numerical method being more general gives rise to new solutions. In particular, we are now able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound state for such planar system, in the case = 0, is predicted and its respective eigenvalue is determined. Keywords Quantum dot model · Numerov numerical method · two-electron system · Schrödinger equation PACS PACS 81.07.Ta · 78.67.Hc · 36.10.-k 1 IntroductionModern technics in nanometer-scale semiconductor manufacturing enable the creation of quantum confinement of only a few electrons. These few-body systems are often called quantum dots [1]. They can be described by a model F. Caruso

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“…In Fig. 2 some theoretical polynomial eigenfunctions u(r) calculated with the results of the references [8,9] are compared to the respective numerical eigenfunctions. Fig.…”

Section: First Numerical Resultsmentioning

confidence: 99%

Numerical Solutions for a Two-dimensional Quantum Dot Model

2019

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“…Several tests of the procedure have been mentioned along our presentation. Additional tests of the whole method may be made in its application to the known cases of BHE with polynomial solutions [3].…”

Section: Final Commentsmentioning

confidence: 99%

“…In the present work we are concerned with applications of the biconfluent Heun equation (BHE), whose relevance has been recognized in a variety of physical systems, like confined interacting electrons in a magnetic field [3,10,11], plane wave diffraction by elongated objects [13], fermions in graphene [4,5], and quantum Newtonian cosmology [17].…”

Section: Introductionmentioning

confidence: 99%

Global solutions of the biconfluent Heun equation

Ferreira

Sesma

2015

Numer Algor

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An algorithm is proposed for obtaining global solutions of the biconfluent Heun equation, which appears when dealing with a variety of physical problems. The procedure, which provides algebraic expressions of the solutions in the form of convergent series or asymptotic expansions, lies on the determination of the connection factors relating the solutions about the regular singular point at the origin and the irregular one at infinity. The algorithm is illustrated by examples.

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“…On one hand, the existence of simple polynomial solutions of this equation, discovered in connection with studies on electron correlation, motivated numerous works in the community of quantum chemists [18][19][20][21][22][23][24][25][26][27][28]. However, this equation is known in mathematics since more than a century as the biconfluent Heun equation and its properties were studied from both purely mathematical perspective [29,30,[35][36][37][38] and in the context of its applications in different areas of physics [39][40][41]. Very recently a brief review on its physical applications has been published by Hortaçsu [42].…”

Section: Harmoniummentioning

confidence: 99%