Generalized eigenvectors are key tools in the theory of rigged Hilbert spaces. Let $\mathcal{H}$ be a Hilbert space and let Φ be a dense subspace of $\mathcal{H}$. Let A be a densely defined linear operator in $\mathcal{H}$ such that Φ ⊂ D_A and AΦ ⊂ Φ. The generalized eigenvectors of A are the eigenvectors of the algebraic dual of A |_Φ. In the case where Φ is endowed with a topology τ finer than the norm topology inherited from $\mathcal{H}$, generalized eigenvectors that are τ-continuous may be of great interest. We discuss conditions which ensure the existence of representations associated to generalized eigenvectors of A. As an application, we review and refine Böhm's study of the algebra of the quantum harmonic oscillator.

广义特征向量是装配的希尔伯特空间理论中的关键工具。让 $\mathcal{H}$ 是希尔伯特空间，令Φ是 $\mathcal{H}$。设A是一个严格定义的线性算子 $\mathcal{H}$使得Φ⊂ d_阿和甲Φ⊂Φ。A的广义特征向量是A |的代数对偶的特征向量 _Φ 。在Φ具有比其继承的范数拓扑更精细的拓扑τ的情况下 $\mathcal{H}$，τ连续的广义特征向量可能引起人们的极大兴趣。我们讨论确保与A的广义特征向量相关的表示的存在的条件。作为应用，我们回顾并完善了博姆对量子谐波振荡器的代数的研究。