2 research outputs found
Localization and landscape functions on quantum graphs
6 figures6 figures6 figuresWe discuss explicit landscape functions for quantum graphs. By a "landscape function" we mean a function that controls the localization properties of normalized eigenfunctions through a pointwise inequality of the form The ideal is a function that a) responds to the potential energy 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 r\'egime, as we show with simple examples. We therefore apply different methods in different r\'egimes determined by the values of the potential energy and the eigenvalue parameter