Equivalent Hermitian operator from supersymmetric quantum mechanics

Abstract

Diagonalizable pseudo-Hermitian Hamiltonians with real and discrete spectra, which are superpartners of Hermitian Hamiltonians, must be η-pseudo-Hermitian with Hermitian, positive-definite and invertible η operators. We show that despite the fact that an η operator produced by a supersymmetric transformation, corresponding to the exact supersymmetry, is not invertible, it can be used to find the eigenfunctions of a Hermitian operator equivalent to the given pseudo-Hermitian Hamiltonian. Once the eigenfunctions of the Hermitian operator are found the operator may be reconstructed with the help of the spectral decomposition.