Non-relativistic Lee model in two-dimensional Riemannian manifolds

This work is a continuation of our previous work [F. Erman and O. T. Turgut, J. Math. Phys. 48, 122103 ( 2007)], where we constructed the non-relativistic Lee model in three-dimensional Riemannian manifolds. Here we renormalize the two-dimensional version by using the same methods and the results are shortly given since the calculations are basically the same as in the three-dimensional model. We also show that the ground state energy is bounded from below due to the upper bound of the heat kernel for compact and Cartan-Hadamard manifolds. In contrast to the construction of the model and the proof of the lower bound of the ground state energy, the mean field approximation to the two-dimensional model is not similar to the one in three dimensions and it requires a deeper analysis, which is the main result of this paper. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4705355]