On the one-dimensional asymptotic models of thin Neumann lattices

We consider the spectral Neumann problem for the Laplace operator on a thin lattice comprised of nodes and ligaments. We pose the classical Pauling model on a one-dimensional graph which describes the multidimensional problem in the first approximation contains ordinary differential equations on its edges with Kirchhoff transmission conditions at its vertices. We construct two-term asymptotics for the spectral pairs $\{$eigenvalue, eigenfunction$\}$ of the problem on the lattice. Basing on this analysis, we propound some refined asymptotic model on the graph with shortened edges that includes certain integral characteristics of the junction zones and actually accounts in the first approximation not only for the edge lengths but also for their arrangement, as well as for the shape and size of the nodes.