Point Potentials on Euclidean Space, Hyperbolic Space and Sphere in Any Dimension

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.