scholarly journals Solutions of the bi-confluent Heun equation in terms of the Hermite functions

2017 ◽  
Vol 383 ◽  
pp. 79-91 ◽  
Author(s):  
T.A. Ishkhanyan ◽  
A.M. Ishkhanyan
Keyword(s):  
Hermite Functions ◽  
Heun Equation ◽  
Confluent Heun Equation

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.

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