Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs

We develop a Logvinenko--Sereda theory for one-dimensional vector-valued self-adjoint operators. We thus deliver upper bounds on $L^2$-norms of eigenfunctions -- and linear combinations thereof -- in terms of their $L^2$- and $W^{1,2}$-norms on small control sets that are merely measurable and suitably distributed along each interval. An essential step consists in proving a Bernstein-type estimate for Laplacians with rather general vertex conditions. Our results carry over to a large class of Schr\"odinger operators with magnetic potentials; corresponding results are unknown in higher dimension. We illustrate our findings by discussing the implications in the theory of quantum graphs.