Localization and landscape functions on quantum graphs

We discuss explicit landscape functions for quantum graphs. By a ``landscape function'' \Upsilon (x) we mean a function that controls the localization properties of normalized eigenfunctions \psi (x) through a pointwise inequality of the form \displaystyle \vert\psi (x)\vert \le \Upsilon (x). The ideal \Upsilon is a function that (a) responds to the potential energy V(x) and to the structure of the graph in some formulaic way; (b) is small in examples where eigenfunctions are suppressed by the tunneling effect; and (c) relatively large in regions where eigenfunctions may, or may not, be concentrated, as observed in specific examples. It turns out that the connectedness of a graph can present a barrier to the existence of universal landscape functions in the high-energy regime, as we show with simple examples. We therefore apply different methods in different regimes determined by the values of the potential energy V(x) and the eigenvalue parameter E.