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

2019 ◽  
Vol 34 (18) ◽  
pp. 1950134 ◽  
Author(s):  
Géza Lévai ◽  
Artur M. Ishkhanyan
Keyword(s):  
Recurrence Relation ◽  
Hermite Polynomials ◽  
Hermite Functions ◽  
Heun Equation ◽  
Potential Term ◽  
Energy Eigenvalues ◽  
Polynomial Expression ◽  
Expansion Coefficients ◽  
Confluent Heun Equation ◽  
Three Term Recurrence Relation

In this paper, 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 (QES) 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 [Formula: see text] choice the quartic potential term is canceled, 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 QES formalism. Possible generalizations of the method are also presented.

Effect of the Electric and Magnetic Fields with Aharonov–Bohm Flux Field in Quantum Dots

2021 ◽  
pp. 2150013
Author(s):  
M. Eshghi ◽  
I. Ahmadi Azar ◽  
S. Soudi
Keyword(s):  
Free Energy ◽  
Thermodynamic Quantities ◽  
Series Method ◽  
Body System ◽  
Persistent Currents ◽  
Energy Eigenvalues ◽  
Electric And Magnetic Fields ◽  
Expansion Coefficients ◽  
Aharonov Bohm ◽  
Three Term Recurrence Relation

This paper has solved the nonrelativistic equation with the external uniform electric potential and magnetic and Aharonov–Bohm (AB) fields in a dot. We have obtained the three-term recurrence relation for the expansion coefficients using the series method. In continuing, we have found two different conditions. Then, using the obtained conditions, we have calculated the energy eigenvalues and eigenfunction. We have then obtained the main thermodynamic quantities such as the free energy, mean energy, entropy, specific heat, magnetization and persistent currents for our system. Also, we extended the calculations to an interaction-free [Formula: see text]-body system. The obtained analytic results are compared with other results, and some of the obtained results are discussed, too.

scholarly journals Numerical construction of the generalized Hermite polynomials

2003 ◽  
pp. 49-63
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Stable Method ◽  
Numerical Construction ◽  
Three Term Recurrence Relation

In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x ? z|? exp(?x2) on R, where z?R and ? > ? 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.

scholarly journals An ubiquitous three-term recurrence relation

2021 ◽  
Vol 62 (3) ◽  
pp. 032106
Author(s):  
Paolo Amore ◽  
Francisco M. Fernández
Keyword(s):  
Recurrence Relation ◽  
Three Term Recurrence Relation

scholarly journals On hypergeometric generalized negative binomial distribution

2002 ◽  
Vol 29 (12) ◽  
pp. 727-736 ◽  
Author(s):  
M. E. Ghitany ◽  
S. A. Al-Awadhi ◽  
S. L. Kalla
Keyword(s):  
Recurrence Relation ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Field Data ◽  
Goodness Of Fit ◽  
Negative Binomial ◽  
Generalized Negative Binomial Distribution ◽  
The Right ◽  
Three Term Recurrence Relation

It is shown that the hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right, and leptokurtic. Also, a three-term recurrence relation for computing probabilities from the considered distribution is given. Application of the distribution to entomological field data is given and its goodness-of-fit is demonstrated.

scholarly journals On the three term recurrence relation and BDF methods

2020 ◽  
Author(s):  
Larissa Ferreira Marques ◽  
Vanessa Botta ◽  
Messias Meneguette
Keyword(s):  
Recurrence Relation ◽  
Bdf Methods ◽  
Three Term Recurrence Relation

scholarly journals ELEMENTARY OSCILLATION CRITERIA FOR A THREE TERM RECURRENCE RELATION WITH OSCILLATORY COEFFICIENT SEQUENCE

1998 ◽  
Vol 29 (3) ◽  
pp. 227-232
Author(s):  
GUANG ZHANG ◽  
SUI-SUN CHENG
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Qualitative Properties ◽  
Oscillation Criteria ◽  
Three Term Recurrence Relation

Qualitative properties of recurrence relations with coefficients taking on both positive and negative values are difficult to obtain since mathematical tools are scarce. In this note we start from scratch and obtain a number of oscillation criteria for one such relation : $x_{n+1}-x_n+p_nx_{n-r}\le 0$.

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