Localization and landscape functions on quantum graphs

We discuss explicit landscape functions for quantum graphs. By a "landscape function" Υ (x) we mean a function that controls the localization properties of normalized eigenfunctions ψ (x) through a pointwise inequality of the form {equation*} |ψ (x)| ≤ Υ (x). {equation*} The ideal Υ 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.