Eigenmodes of the Laplacian on Hyperbolic Lattices

We examine a specific category of eigenfunctions of the lattice Laplacian on $\{p,q\}$-tessellations of the Poincar\'e disk that bear resemblance to plane waves in the continuum case. Our investigation reveals that the lattice eigenmodes deviate from the continuum solutions by a factor that depends solely on the local inclination of the vertex in relation to the wave's propagation direction. This allows us to compute certain eigenfunctions by numerical and analytical methods. For various special cases we find explicit exact eigenfunctions and their eigenvalues on the infinite lattice.