Infinite families of (non)-Hermitian Hamiltonians associated with exceptional $X_m$ Jacobi polynomials

Abstract: Using an appropriate change of variable, the Schr\"odinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi type $X_m$ exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric Scarf potentials and finite number of Hermitian and infinite number of non-Hermitian PT-symmetric hyperbolic Scarf potentials. The bound state solutions of all these potentials are associated with the aforesaid exceptional orthogonal polynomials. These infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic Scarf potentials by the addition of some rational terms characterized by the presence of classical Jacobi polynomials. All the members of a particular family of these 'rationally extended polynomial-dependent' potentials have the same energy spectrum and possess translational shape invariant symmetry. The obtained non-Hermitian trigonometric Scarf potentials are shown to be quasi-Hermitian in nature ensuring the reality of the associated energy spectra.