2018/03/05 by Oliver Knill, Knill, Oliver · 2 citations
Computer Science · Mathematics · #05Exx #15A36 #58J50 #Cellular Automata and Applications #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph theory and applications #Spectral Theory (math.SP) #Topological and Geometric Data Analysis #cs.DM #math.SP #msc:05Exx #msc:15A36 #msc:58J50
paper · pdf · doi:10.48550/arxiv.1803.01464
29 pages, 8 figures
arxiv created 2018/03/05 · openalex publication_date 2018/03/05 · arxiv updated 2018/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 H0=d^* d. The hydrogen identity implies that the random walk u(n) = Ln u with integer n solves the one-dimensional Jacobi equation Delta u=H2 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 Zr. Since L2 and L^(-2) are isospectral, by a theorem of Kirby, the matrix L2 is always similar to a symplectic matrix if the graph has an even number of simplices. The hydrogen relation is robust: any Schrödinger 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.