Unbounded Hamiltonians generated by Parseval frames

In Bagarello and Ku˙ zel (Acta Appl Math 171:4, 2021) 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 constructed 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 Bagarello and Ku˙ zel (Acta Appl Math 171:4, 2021) 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 analyze the eigenvalues of the Hamiltonians, and we discuss some physical applications of our framework.