Eigenvalue asymptotics and unique continuation of eigenfunctions on planar graphs

2021/04/08 by Bonnefont, Michel, Golenia, Sylvain, Keller, Matthias
#Combinatorics (math.CO) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2104.03582

Abstract

We study planar graphs with large negative curvature outside of a finite set and the spectral theory of Schrödinger 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 Gauß-Bonnet theorem.

Related

• Elliptic operators on planar graphs: Unique continuation for eigenfunctions and nonpositive curvature
• Curvature, geometry and spectral properties of planar graphs
• Eigenvalue asymptotics for Schrödinger operators on sparse graphs
• Growth of Eigenfunctions and R-limits on Graphs
• Spectral asymptotics of all the eigenvalues of Schrödinger operators on flat tori