Treatment of a three-dimensional central potential with cubic singularity

Abstract

We compute the bound states for a special type of singular central potential that generalizes the hyperbolic Eckart potential by adding a cubic singular term at the origin while keeping the short range exponential decay far away from the origin. Such strong singular potentials are of practical importance in atomic, nuclear and molecular physics. To bring the solution of the Schrodinger equation for finite angular momentum to analytical treatment we use an analytical approximation to the centrifugal orbital part of the potential that has a similar structure to the Eckart potential. We compute the energy spectrum associated with this potential using both the tridiagonal representation approach (TRA) and the asymptotic iteration method (AIM) and make a comparative analysis of these results.

Appendix A. Relevant matrices for the TRA

In Eq. 15, few matrices have been introduced. First, the matrix X is defined as follows [31, 49]

$$\begin{aligned} X_{nm}=Q_{n}\delta _{nm}+R_{n}\delta _{n,m-1}+R_{n-1}\delta _{n,m+1} \end{aligned}$$

(18)

where

$$\begin{aligned} Q_{n}=\frac{\nu ^{2}-\mu ^{2}}{(2n+\mu +\nu )(2n+\mu +\nu +2)} \end{aligned}$$

(19)

and

$$\begin{aligned} R_{n}=\frac{2}{2n+\mu +\nu +2}\sqrt{\frac{(n+1)(n+\mu +1)(n+\nu +1)(n +\mu +\nu +1)}{(2n+\mu +\nu +1)(2n+\mu +\nu +3)}} \end{aligned}$$

(20)

Next, the matrix $\sigma ^{\pm }$ are define by [31]

$$\begin{aligned} \sigma ^{\pm }=\varLambda \cdot D^{\pm }\cdot \varLambda ^{T} \end{aligned}$$

(21)

where $\varLambda$ is a matrix containing the eigenvectors of the truncated version of the matrix X of size $(N+1)\times (N+1)$ and $D^{\pm }_{nm}=\frac{1}{1\mp x_{n}}\delta _{nm}$ where $x_{n}$ is the associated eigenvalue of the truncated matrix. Lastly, the matrix $\omega$ introduced in Eq. 16 is $\omega =-\varLambda \cdot D^{+}D^{-}\cdot \varLambda ^{T}$, where the symbol “$\cdot$” stands for matrix product operation.