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: $e^{-αp} x e^{αp} = x+ i α$, to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.

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