(Georgia Institute of Technology, 2016-10-09)
Harrell, Evans M.
Pointwise bounds on eigenfunctions are useful for establishing
localization of quantum states, and they have implications for the
distribution of eigenvalues and for physical properties such as
conductivity. In the low-energy regime, localization is associated with
exponential decrease through potential barriers. We adapt the Agmon
method to control this tunneling effect for quantum graphs with Sobolev
and pointwise estimates. It turns out that as a generic matter, the rate of
decay is controlled by an Agmon metric related to the classical Liouville-
Geen approximation for the line, but more rapid decay is typical, arising
from the geometry of the graph. In the high-energy regime one expects
states to oscillate but to be dominated by a 'landscape function' in terms
of the potential and features of the graph. We discuss the construction of
useful landscape functions for quantum graphs.