Solving New Potentials in Terms of Exceptional Orthogonal Polynomials and Their Supersymmetric Partners

Abstract

Point canonical transformation has been used to find out new exactly solvable potentials in the position-dependent mass framework. We solve 1-D Schrödinger equation in this framework by considering two different fairly generic position-dependent masses \((i) M(x)=\lambda g'(x)\) and \((ii) M(x) = c \left( {g'(x)} \right) ^\nu \), \(\nu =\frac{2\eta }{2\eta +1},\) 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 position-dependent mass.