Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials

Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: $e^{-\alpha p} x e^{\alpha p} = x+ i \alpha$, to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.