Continuum states in generalized Swanson models

A one-to-one correspondence is known to exist between the spectra of the discrete states of the non Hermitian Swanson-type Hamiltonian $H = {\cal{A}}^{\dagger} {\cal{A}} + \alpha {\cal{A}} ^2 + \beta {\cal{A}}^{\dagger 2}$, ($\alpha \neq \beta$), and an equivalent Hermitian Schr\"{o}dinger Hamiltonian $h$, the two Hamiltonians being related through a similarity transformation. In this work we consider the continuum states of $h$, and examine the nature of the corresponding states of $H$.