Exact Results in Quantum Theory & Gravity

In dimensions d≥4, the Laplacian perturbed by a point potential cannot be defined as a self-adjoint operator. However, in any dimension, there exists a natural family of functions that can be interpreted as Green’s functions of the Laplacian with a spherically symmetric point potential. In dimensions 1, 2, 3 they are the integral kernels of the resolvent of well-defined self-adjoint operators. In higher dimensions they are not even integral kernels of bounded operators.
I will describe two methods to construct these Green functions. The first resembles dimensional regularization in the minimal subtraction scheme, the second resembles the point-splitting method. Both methods can be viewed as toy models illustrating various aspects of renormalization in Quantum Field Theory. It is expected that the obtained Green functions approximate the Green functions of true Schrödinger operators with a potential of shrinking support.