Rationally Extended Harmonic Oscillator potential, Isospectral Family and the Uncertainity Relations

We consider the rationally extended harmonic oscillator potential which is isospectral to the conventional one and whose solutions are associated with the exceptional, $X_m$- Hermite polynomials and discuss its various important properties for different even codimension of $m$. The uncertainty relations are obtained for different $m$ and it is shown that for the ground state, the uncertainity increases as $m$ increases. A one parameter $(\lambda )$ family of exactly solvable isospectral potential corresponding to this extended harmonic oscillator potential is obtained. Special cases corresponding to the $\lambda =0$ and $\lambda =-1$, which give the Pursey and the Abhram-Moses potentials respectively, are discussed. The uncertainty relations for the entire isospectral family of potentials for different $m$ and $\lambda$ are also calculated.