Bound states in the compactified gravity

TL;DR

This paper derives explicit gravitational potentials in compactified multidimensional spaces and investigates bound states of particles on a 3D brane, combining analytical solutions with approximate methods.

Contribution

It provides explicit solutions for gravitational potentials in 2D and 4D compactified spaces and analyzes bound states in these geometries.

Findings

Explicit potential solutions in 2D and 4D compactified spaces.

Bound states of particles on a 3D brane are identified.

Approximate methods effectively analyze bound states in compactified geometries.

Abstract

A potential of pointlike mass in the partially compactified multidimensional space is considered. The problem is reduced to the multidimensional Poisson equation with the Dirac comb source in r.h.s. Explicit solutions are built in the cases of 2D and 4D spaces with one compact dimension. The last example of the potential is used in the Schr\"odinger equation. Bound states of a gravitating test particle on 3D brane of 4D compactified space are studied by means of various approximate methods.