Point potentials on Euclidean space, hyperbolic space and sphere in any dimension
Jan Dereziński, Christian Gaß, Błażej Ruba
TL;DR
This work provides a unified framework for Green's functions of the Laplacian perturbed by point potentials on ${f R}^d$, ${f H}^d$, and ${f S}^d$ in all dimensions. It develops explicit Krein-type resolvent formulas and renormalized self-energies via two parallel schemes—point splitting and generalized integrals (dimensional regularization)—which are compatible in odd dimensions and necessitate subtle renormalization in even dimensions. The authors establish flat (Euclidean) limits of the curved-space Green's functions, analyze the spectral consequences of the perturbations (including shifts of spherical eigenvalues and potential new poles), and interpret the results as tractable toy models for long-distance behavior and renormalization in quantum field theory. The work highlights new closed-form expressions for hyperbolic and spherical Green's functions with point interactions and clarifies how renormalization constants govern the asymptotics and pole structure across geometries, contributing to the mathematical understanding of singular perturbations in curved spaces and their QFT analogies.
Abstract
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.