Localization and landscape functions on quantum graphs

Abstract:

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 \begin{equation*} |\psi (x)| \le \Upsilon (x). \end{equation*} 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 $E$.