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

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 $(i)M(x)=\lambda g^{\mathrm{\prime }}(x)$ and $(ii)M(x)=c{\left(g^{\mathrm{\prime }}(x)\right)}^{\nu }$, $\nu =\frac{2\eta }{2\eta +1},$ with $\eta =0,1,2\cdots$. 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 $(\nu )$ 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.