New solutions of Isochronous potentials in terms of exceptional orthogonal polynomials in heterostructures

Abstract

Point canonical transformation (PCT) has been used to find out new exactly solvable potentials in the position-dependent mass (PDM) framework. We solve $1$-D Schrödinger equation in the PDM framework by considering two different fairly generic position-dependent masses $ (i) M(x)=λg'(x)$ and $(ii) M(x) = c \left( {g'(x)} \right)^ν$, $ν=\frac{2η}{2η+1},$ with $η= 0,1,2\cdots $. In the first case, we find new exactly solvable potentials that depend on an integer parameter $m$, and the corresponding solutions are written in terms of $X_m$-Laguerre polynomials. In the latter case, we obtain a new one parameter $(ν)$ family of isochronous solvable potentials whose bound states are written in terms of $X_m$-Laguerre polynomials. Further, we show that the new potentials are shape invariant by using the supersymmetric approach in the framework of PDM.

Figures (5)

• Figure 1: Plot of the potential $V_{eff}$ given in Eq. (\ref{['Vs']}) for different $m$ values, square of first three bound state wavefunctions ${|\psi_0|}^2$ (purple line), ${|\psi_1|}^2$ (orange line), ${|\psi_2|}^2$ (green line) corresponding to different $m$ values, for mass function $M(x)$ (red line) given in Eq. (\ref{['mxgxM']}). We have consider here $b=1$, $\alpha=2$
• Figure 2: Plot of the potential $V_{eff}$ given in Eq. (\ref{['Vsm']}) for $m=1$ and different $\nu$ values, square of first three bound state wavefunctions ${|\psi_0|}^2$ (purple line), ${|\psi_1|}^2$ (orange line), ${|\psi_2|}^2$ (green line) corresponding to different $\nu$ values, for mass function $M(x)$ (red line). We have consider here $\alpha=2$.
• Figure 3: Plot of the potential $V_{eff}$ given in Eq. (\ref{['Vsm']}) for $m=2$ and different $\nu$ values, square of first three bound state wavefunctions ${|\psi_0|}^2$ (purple line), ${|\psi_1|}^2$ (orange line), ${|\psi_2|}^2$ (green line) corresponding to different $\nu$ values, for mass function $M(x)$ (red line). We have consider here $\alpha=2$
• Figure 4: Plot of the potential $V_{eff}$ given in Eq. (\ref{['Vsm']}) for $m=3$ and different $\nu$ values, square of first three bound state wavefunctions ${|\psi_0|}^2$ (purple line), ${|\psi_1|}^2$ (orange line), ${|\psi_2|}^2$ (green line) corresponding to different $\nu$ values, for mass function $M(x)$ (red line). We have considered here $\alpha=2$
• Figure 5: Plot for the probability density of a two-dimensional potential for $m=1, \nu=\frac{2}{3}$ and different combinations of $n_1$ and $n_2$ values. The wavefunctions are normalized.