  In dimensions d= 1, 2, 3 the Laplacian can be perturbed by a point potential.
In higher dimensions the Laplacian with a point potential cannot be defined as
a self-adjoint operator. However, for 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. Their construction uses the so-called generalized integral, a
concept going back to Riesz and Hadamard.
  We consider the Laplace(-Beltrami) operator on the Euclidean space, the
hyperbolic space and the sphere in any dimension. We describe the corresponding
Green's functions, also perturbed by a point potential. We describe their limit
as the scaled hyperbolic space and the scaled sphere approach the Euclidean
space. Especially interesting is the behavior of positive eigenvalues of the
spherical Laplacian, which undergo a shift proportional to a negative power of
the radius of the sphere.
  We expect that in any dimension our constructions yield possible behaviors of
the integral kernel of the resolvent of a perturbed Laplacian far from the
support of the perturbation. Besides, they can be viewed as toy models
illustrating various aspects of renormalization in Quantum Field Theory,
especially the point-splitting method and dimensional regularization.
