It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a \({\mathcal{D}}\)-quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of \(E\)-connection for observables. Several examples are discussed.

众所周知，具有纯离散特征值的自伴哈密顿量可以写成互为正交的投影仪的（无限）线性组合，其特征值作为展开系数。投影仪由哈密顿量的特征向量定义。在最近的一些论文中，这种扩展已经扩展到以下情况：这些特征向量形成Riesz基，或者最近形成\（{\ mathcal {D}} \）-拟基（Bagarello and Bellomonte in J. Phys。 A 50：145203，2017，Bagarello et al。in J.Math.Phys。59：033506，2018），而不是正交基础。在这里，我们讨论将这些集合替换为Parseval帧时可以做什么。这种兴趣是出于物理原因，尤其是由于数学上的事实最初定义物理系统的希尔伯特空间有时还包含状态，这些状态实际上不能由物理系统本身占用。特别是，我们展示了从正交基准到Parseval框架时，可观测范围的变化。从这个角度来看，我们提出了可观察物\（E \） -connection的概念。讨论了几个例子。