My motivation is physical, but my question in mathematical. In quantum mechanics, time evolution of states is the action of a one-parameter subgroup of projective-unitary transformations of a projective Hilbert space, $\mathcal P(\mathcal H)$. Stationary states are the fixed-points of the action of this one-parameter subgroup.
By Wigner's theorem this one-parameter group can be lifted to a strongly continuous one-parameter subgroup of the unitary group. The action of this one-parameter subgroup on $\mathcal H$ is described by the Schrödinger equation. Stationary state vectors are such eleents of $\mathcal H$ that remain in a one-dimensional subspace of $\mathcal H$ under this action. These one-dimensional subspaces are the stationary states in $\mathcal P(\mathcal H)$. In many important cases, stationary state vectors form a basis of $\mathcal H$, so the trajectory of any element of $\mathcal H$ is the linear combination of the trajectories of stationary states. Specifically, the fixed points of the acion of this strongly continuous one-parameter subgroup of unitary operators are also stationary state vectors, but they clearly cannot form a basis except when this one-parameter subgroup is trivial. In the non-trivial cases, what can we know about these fixed points?