Floquet-Bloch theory provides "nice" spectral decompositions for operators which commute with group actions (usually abelian groups). The simplest example is that of periodic Schrodinger operators on $\ell^2(\mathbb Z)$, where the group $\mathbb Z$ acts by translations.
I was wondering if there are cases where a similar analysis can be done for operators on spaces with a "nice" semigroup action. A case analogous to the simple example above would be a Schrodinger operator on $\ell^2(\mathbb N)$, which is "periodic" in the $\mathbb N$ sense. Is there a nice way to analyze such operators (even partially) using something similar to Floquet-Bloch theory? Or do you immediately lose everything once you don't have an actual group?
Thanks in advance.