Study about Pseudo-Hermitian Matrix Exactly Solvable Hamiltonian

The non symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field that couples to a harmonic oscillator via some pseudo-hermitian interaction is considered. We highlight all of the properties of both the original Mandal and Jaynes-Cummings Hamiltonians. We show that the Mandal Hamiltonian is non hermitian and non invariant under the combined action of the parity operator P and the time-reversal operator T Even if the previous properties are not satisfied, it has been proved that the Mandal Hamiltonian is pseudo-hermitian with respect to P and with respect to also [1,2]. Thus, we show that the original Jaynes-Cummings Hamiltonian is hermitian. We show the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator, similar to the direct approach to invariant vector spaces used in Refs [3,4].