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 (determined by a Lagrangian plane ) with potential on a manifold. The isomorphism is given by , where is an (explicitly calculated) Lagrangian plane. A similar isomorphism also holds for the Laplace operator on a decorated graph. The inequality is established for the Laplace operator on a decorated graph (obtained by decorating a connected finite graph with edges and vertices) with `continuity' conditions. It is also shown that the quantity does not decrease when new edges or manifolds are added. Bibliography: 12 titles.