The kernel of Laplace-Beltrami operators with zero-radius potential or on decorated graphs

An isomorphism is described for the kernel of the Laplace operator Delta(Lambda) (determined by a Lagrangian plane Lambda subset of C-k circle plus C-k) with potential Sigma(k)(j=1) c(j)delta(qj) (x) on a manifold. The isomorphism is given by Gamma: ker Delta(Lambda) -> Lambda boolean AND L, where L is an (explicitly calculated) Lagrangian plane. A similar isomorphism also holds for the Laplace operator oil a decorated graph. The inequality 1 <= dint ker Delta(Lambda 0) <= n - v + 2 is established for the Laplace. operator Delta(Lambda 0) on a decorated graph (obtained by decorating a connected finite graph with n. edges and v vertices) with 'continuity' conditions. 11, is also shown that the quantity n - n + 1 - dim ker Delta(Lambda 0) does not, decrease when new edges or manifold, are added.