Wandering range of robust quantum symmetries

This paper introduces the concept of the wandering range of a robust symmetry $S$ of a Hamiltonian $H$. This quantity measures how the perturbed time evolution ${\mathrm{e}}^{\mathrm{i}t(H+\epsilon V)}S{\mathrm{e}}^{-\mathrm{i}t(H+\epsilon V)}$ deviates from its unperturbed counterpart ${\mathrm{e}}^{\mathrm{i}tH}S{\mathrm{e}}^{-\mathrm{i}tH}=S$. Although the wandering range does not necessarily scale linearly with the perturbation strength $\epsilon$, we identify conditions under which this linear behavior is recovered and we obtain explicit nonperturbative bounds.