**Proposition 2** ([19])**.**

*1. For any λ* ∈ Spec(*E*)*, it holds that* |*λ*| ≤ 1*, i.e.,*

$$\mathcal{H}\_{\mathbb{S}} = \{ \psi \mid \exists \ m \in \mathbb{N}, \exists \ |\lambda| < 1, \ (\mathcal{U} - \lambda)^{m} \psi ) = 0 \}.$$


We call H*c* and H*s* the *centered eingenspace* and the *stable eigenspace* [18], respectively.

**Corollary 3.** *For any ψ* ∈ H*s and φ* ∈ H*c, it holds that ψ*, *φ* = 0*.*

Now, let us see the stationary states from the viewpoint of the *orthogonal* decomposition of H*c* ⊕ H*<sup>s</sup>*.
