Spin–orbit interaction for the double ring-shaped oscillator

Chen, Changyuan; Lu, Fa-Lin; Sun, Dongsheng; You, Yuan; Dong, Shi‐Hai

doi:10.1016/j.aop.2016.04.012

Cited by 25 publications

(7 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where 72)is an example of differential equations scattered over a vast range of applications in applied mathematics and theoretical physics [16][17][18][19]. The coefficients {C i } m i=0 of the polynomial solutions Q m (r) = ∑ m i=0 C i r i satisfy the following recurrence relation (53):…”

Section: Two-dimensional Dirac Equationmentioning

confidence: 99%

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

et al. 2020

View full text Add to dashboard Cite

The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, n−1, and n−2 respectively. We show that for n≥3 the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of n=4. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.

show abstract

Section: Two-dimensional Dirac Equationmentioning

confidence: 99%

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

et al. 2020

View full text Add to dashboard Cite

show abstract

“…(5.15) Equation (5.15) is an example of differential equations scattered over a vast range of applications in applied mathematics and theoretical physics [15][16][17][18]. The coefficients {C i } m i=0 of the polynomial solutions Q m (r) = m i=0 C i r i satisfy the following recurrence relation (4.5): For the zero-degree polynomial solution Q 0 (r) = 1 where C 0 = 1, m = 0 and therefore = 0, 1, 2, it follows that: = 0, 1 :…”

Section: Two-dimensional Dirac Equationmentioning

confidence: 99%

On Polynomial Solutions of Linear Differential Equations with Applications

Bryenton1,

Cameron,

Kirk

et al. 2020

Preprint

View full text Add to dashboard Cite

The analysis of many physical phenomena can be reduced to the study of solutions of differential equations with polynomial coefficients. In the present work, we establish the necessary and sufficient conditions for the existence of polynomial solutions to the linear differential equationfor arbitrary n ≥ 2. We show by example that for n ≥ 3, the necessary condition is not enough to ensure the existence of the polynomial solutions. Using Scheffé's criteria, we show that from this differential equation there are n-generic equations solvable by a two-term recurrence formula. The closed-form solutions of these generic equations are given in terms of the generalized hypergeometric functions. For the arbitrary n differential equations, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly. The algorithm is used to establish the polynomial solutions in the case of n = 4. To demonstrate the simplicity and applicability of this approach, it is used to study the solutions of Heun and Dirac equations.

show abstract

“…Concerning this, there are lots of reports on theoretical studies for characteristics of interacting molecules based on the Schrödinger solutions [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. For instance, quantum features of molecular interactions which are subjected to ring-shaped potentials [22][23][24][25] or double ring-shaped potentials [26][27][28][29][30][31] in quantum chemistry were extensively investigated. The mechanism of interaction between deformed nuclei has been analyzed [32,33].…”

Section: Introductionmentioning

confidence: 99%

Quantum features of molecular interactions associated with time-dependent non-central potentials

2017

View full text Add to dashboard Cite

The research for quantum features of molecular Coulomb interactions subjected to a time-dependent non-central potential has many applications. The wave functions of this system can be obtained by using an invariant operator, which is necessary for investigating a time-dependent Hamiltonian system. Regarding time dependence of the system, one can confirm that the formula of this operator is in general somewhat complicated. Hence, in order to solve its eigenvalue equation, special mathematical techniques beyond separation of variables method, such as the unitary transformation method, the Nikiforov-Uvarov method, and the asymptotic iteration method, should be employed. The double ring-shaped generalized non-central potential of which evolution explicitly depends on time is introduced as a particular case. The complete quantum solutions of the system can be identified from the eigenstates of the invariant operator. These solutions are useful for analyzing dynamical properties of the system.

show abstract

Spin–orbit interaction for the double ring-shaped oscillator

Cited by 25 publications

References 25 publications

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

On Polynomial Solutions of Linear Differential Equations with Applications

Quantum features of molecular interactions associated with time-dependent non-central potentials

Contact Info

Product

Resources

About