Abstract: As an example of exact renormalization in field theory we will study a simplified version of the Lee model following a method proposed by Rajeev. We can show that the ground state energy is bounded from below on certain classes of manifolds using some bounds on heat kernels. To improve these bounds we use a kind of mean field theory analysis, first in three dimensions and then in two dimensions. We also briefly talk about a relativistic version of this model, essentially on a two dimensional manifold.