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 _X1$X_1$ exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of coordinate: e^−αpxe^αp=x+iα$e^{-\alpha p}x{e}^{\alpha p}=x+i\alpha$, to be both quasi- and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.