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

Eigenvalue asymptotics and unique continuation of eigenfunctions on planar graphs