The hydrogen identity for Laplacians

Abstract

For any finite simple graph G, the hydrogen identity H=L-L^(-1) holds, where H=(d+d^*)^2 is the sign-less Hodge Laplacian defined by sign-less incidence matrix d and where L is the connection Laplacian. Any spectral information about L directly leads to estimates for the Hodge Laplacian H=(d+d^*)^2 and allows to estimate the spectrum of the Kirchhoff Laplacian H_0=d^* d. The hydrogen identity implies that the random walk u(n) = L^n u with integer n solves the one-dimensional Jacobi equation Delta u=H^2 with (Delta u)(n)=u(n+2)-2 u(n)+u(n-2). Every solution is represented by such a reversible path integral. Over a finite field, we get a reversible cellular automaton. By taking products of complexes such processes can be defined over any lattice Z^r. Since L^2 and L^(-2) are isospectral, by a theorem of Kirby, the matrix L^2 is always similar to a symplectic matrix if the graph has an even number of simplices. The hydrogen relation is robust: any Schr\"odinger operator K close to H with the same support can still can be written as $K=L-L^{-1}$ where both L(x,y) and L^-1(x,y) are zero if x and y do not intersect.