On the Dual Geometry of Laplacian Eigenfunctions
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We discuss the geometry of Laplacian eigenfunctions -Δϕ = λϕ on compact manifolds (M,g) and combinatorial graphs G=(V,E). The 'dual' geometry of Laplacian eigenfunctions is well understood on T^d (identified with Z^d) and R^n (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' α(ϕ_λ, ϕ_μ) between eigenfunctions ϕ_λ and ϕ_μ is given by a global average of local correlations α(ϕ_λ, ϕ_μ)^2 = ϕ_λϕ_μ_L^2^-2∫_M( ∫_M p(t,x,y)( ϕ_λ(y) - ϕ_λ(x))( ϕ_μ(y) - ϕ_μ(x)) dy)^2 dx, where p(t,x,y) is the classical heat kernel and e^-t λ + e^-t μ = 1. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.
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