Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem

In a previous paper [1] we introduced a very simple PT-symmetric non-Hermitian Hamiltonian with a real spectrum and derived a closed formula for the metric operator relating the problem to a Hermitian one. In this paper we propose an alternative formula for the metric operator, which we believe is more elegant and whose construction-based on a backward use of the spectral theorem for self-adjoint operators-provides new insights into the nature of the model.