Effects of Topological Defects and AB Fields on the Thermal Properties, Persistent Currents and Energy Spectra with an Exponential-Type Pseudoharmonic Potential

Abstract

The Schrodinger equation with exponential-type pseudo-harmonic oscillator in the global monopole space-time is solved using extended Nikiforov-Uvarov (ENU) method. The energy spectrum of the Klein-Gordon oscillator for the exponential-type pseudo-harmonic oscillator is obtained in closed form, and the corresponding wave function is determined using the biconfluent Heun differential equation. Special cases are deduced and some graphical results are shown to illustrate the variation of the energy levels with topological defect, screening parameter, magnetic field flux, and quantum numbers. In addition, the thermodynamic function expressions for the exponential-type pseudo-harmonic oscillator are obtained in closed form, and their variations with temperature are discussed extensively for various values of the potential and Klein-Gordon oscillator parameters. Our results agree with those obtained in the literatures.

Appendix A Extended Nikiforov Uvarov Method

As the name implies, the ENU method is an extension of the standard NU method. This method was developed and proposed by Karayer et al. [23] and her co-workers. The ENU takes the form [23,24,25],

$${\psi}^{{\prime\prime} }(z)+\frac{{\overset{\sim }{\tau}}_e(z)}{\sigma_e(z)}{\psi}^{\prime }(z)+\frac{{\overset{\sim }{\sigma}}_e(z)}{\sigma_e^2(z)}\psi (z)=0$$

(52)

where ${\overset{\sim }{\tau}}_e$ is a polynomial of at most second degree, σ_e is a polynomial of at most third degree, ${\overset{\sim }{\sigma}}_e$ is a polynomial of at most fourth degree, ψ is the wave function and the subscript (e) denotes the extended in order to distinguish it from the NU whose polynomials are of degree one for $\overset{\sim }{\tau }$ and two for σ and $\overset{\sim }{\sigma }$. In order to solve Eq. (52), we used the following ansatz for the wave function,

$$\psi (s)={\phi}_e(s){\chi}_n(s)$$

(53)

The second part of the wave function becomes hypergeometric-type equation of the type

$${\sigma}_e(s){\chi}^{{\prime\prime} }(s)+{\tau}_e(s){\chi}^{\prime }(s)+h(s)\chi (s)=0$$

(54)

where

$${\tau}_e(z)={\overset{\sim }{\tau}}_e(z)+2{\pi}_e(z)$$

(55)

$$h(z)=G(z)+{\pi}_e^{\prime }(z)$$

(56)

and the π_e(z) and G(z) polynomials are defined as follows,

$${\pi}_e(z)=\frac{\sigma_e^{\prime }(z)-{\overset{\sim }{\tau}}_e(z)}{2}\pm \sqrt{{\left(\frac{\sigma_e^{\prime }(z)-{\overset{\sim }{\tau}}_e(z)}{2}\right)}^2-{\overset{\sim }{\sigma}}_e(z)+G(z){\sigma}_e(z)}$$

(57)

$$G(z)= Pz+Q$$

(58)

where P and Q are constants that will be determine in Eq. (57) based on the condition that the expression under the square root sign must be square of a polynomial of at most degree two. Eq. (54) has a particular polynomial solution of the form χ(z) = χ_n(z) then Eq. (55) becomes,

$${\sigma}_e(z){\chi}^{{\prime\prime} }(z)+{\tau}_e(z){\chi}^{\prime }(z)+{h}_n(z)\chi (z)=0$$

(59)

where,

$${h}_n(z)=-\frac{n}{2}{\tau}^{\prime }(z)-\frac{n\left(n-1\right)}{6}{\sigma}^{{\prime\prime} }(z)+{C}_n,$$

(60)

where C_n is an integration constant. The eigenvalue equation for the extended Nikiforov Uvarov takes the form.

$$h(s)={h}_n(s)=-\frac{1}{2}n{\tau}^{\prime }(s)-\frac{n\left(n-1\right)}{6}{\sigma}^{{\prime\prime} }(s)\ \left(n=0,1,2,\dots \right)$$

(61)