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Quantum isotonic nonlinear oscillator as a Hermitian counterpart of Swanson Hamiltonian and pseudo-supersymmetry

Within the ideas of pseudo-supersymmetry, we have studied a non-Hermitian Hamiltonian , where α ≠ β and ξ is a first-order differential operator, to obtain the partner potentials V+(x) and V−(x) which are new isotonic and isotonic nonlinear oscillators, respectively, as the Hermitian equivalents of the non-Hermitian partner Hamiltonians H±. We have provided an algebraic way to obtain the spectrum and wavefunctions of a nonlinear isotonic oscillator. The solutions of V−(x) which are Hermitian counterparts of Swanson Hamiltonian are obtained under some parameter restrictions that are found. Also, we have checked whether the intertwining operator satisfies η1H− = H+η1, where and is the first-order differential operator, which factorizes Hermitian equivalents of H±.