Pseudo-Bosons and Their Coherent States

These notes are based on a simple remark, which appeared more and more evident during my recent attempts to use coherent states in the computation of some transition probabilities for some specific quantum mechanical systems driven by non self-adjoint Hamiltonians. My idea was to extend what originally done in \cite{bagfein}, in connection with the Swanson model to other physical interesting situations. Since the Swanson Hamiltonian is manifestly non self-adjoint, and since it can be written in terms of pseudo-bosonic ladder operators, it appeared natural to use coherent states of a special type, the bi-coherent states, in order to set up the path integral for the model. However, looking for these other applications, I realized that many reasonable claims on bi-coherent states were stated in recent years, but only few of them were proved in detail. Many formulas look natural, also because they are the exact counterparts of the analogous formulas for coherent states. But it is clear that if something is {\em reasonable} it does not mean necessarily that it is also {\em true}. For instance, the eigenstates of a non self-adjoint Hamiltonian $H$ with purely discrete spectrum need not being a basis in $\Hil$, the Hilbert space where $H$ is defined. However, in the physical literature this is often taken for granted, exactly because it is reasonable. Etc etc