In Bagarello and Kużel (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 Kużel (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.

在 Bagarello 和 Kużel (Acta Appl Math 171:4, 2021) 中，Parseval 框架用于定义有限和无限维度希尔伯特空间中的有界哈密顿量。在这里，我们继续进行这种分析，特别关注哈密顿算子的离散谱，这些算子被构造为由存在于无限维希尔伯特空间中的某些帕塞瓦尔框架定义的一级算子的加权无限和。与 Bagarello 和 Kużel (Acta Appl Math 171:4, 2021) 的主要区别在于，在这里，我们考虑的运算符大多是无界的。相对于我们之前的结果，这是一个有用的升级，因为具有物理意义的哈密顿量确实通常是无界的。然而，由于框架（一般而言）不是基，所以哈密顿量的定义并不那么容易，我们的部分结果也朝着这个方向发展。还，