  We deduce the eigenvalues and the eigenvectors of a parameter-dependent
Hamiltonian $H_\theta$ which is closely related to the Swanson Hamiltonian, and
we construct bi-coherent states for it. After that, we show how and in which
sense the eigensystem of the Hamiltonian $H$ of the inverted quantum harmonic
oscillator can be deduced from that of $H_\theta$. We show that there is no
need to introduce a different scalar product using some ad hoc metric operator,
as suggested by other authors. Indeed we prove that a distributional approach
is sufficient to deal with the Hamiltonian $H$ of the inverted oscillator.
