  In \cite{BK} Parseval frames were used to define bounded Hamiltonians, both
in finite and in infinite dimesional Hilbert spaces. Here we continue this
analysis, with a particular focus on the discrete spectrum of Hamiltonian
operators defined as a weighted infinite sum of rank one operators defined by
some Parseval frame living in an infinite dimensional Hilbert space. The main
difference with \cite{BK} is that, here, the operators we consider are mostly
unbounded. This is an useful upgrade with respect to our previous results,
since physically meaningful Hamiltonians are indeed often unbounded. However,
due to the fact that frames (in general) are not bases, the definition of an
Hamiltonian is not so easy, and part of our results goes in this direction.
Also, we discuss the eigenvalues of the Hamiltonians, and we discuss some
physical applications of our framework.
