  In a recent paper we have introduced several possible inequivalent
descriptions of the dynamics and of the transition probabilities of a quantum
system when its Hamiltonian is not self-adjoint. Our analysis was carried out
in finite dimensional Hilbert spaces. This is useful, but quite restrictive
since many physically relevant quantum systems live in infinite dimensional
Hilbert spaces. In this paper we consider this situation, and we discuss some
applications to well known models, introduced in the literature in recent
years: the extended harmonic oscillator, the Swanson model and a generalized
version of the Landau levels Hamiltonian. Not surprisingly we will find new
interesting features not previously found in finite dimensional Hilbert spaces,
useful for a deeper comprehension of this kind of physical systems.
