  We study planar graphs with large negative curvature outside of a finite set
and the spectral theory of Schr{\"o}dinger operators on these graphs. We obtain
estimates on the first and second order term of the eigenvalue asymptotics.
Moreover, we prove a unique continuation result for eigenfunctions and decay
properties of general eigenfunctions. The proofs rely on a detailed analysis of
the geometry which employs a Copy-and-Paste procedure based on the
Gau{\ss}-Bonnet theorem.
