# On the virtual level of two-body interactions and applications to three-body systems in higher dimensions

## TL;DR

This paper investigates three-particle quantum systems in four or more dimensions, demonstrating that unlike lower dimensions, these systems do not exhibit the Efimov effect and have finitely many bound states.

## Contribution

It proves the finiteness of the discrete spectrum for three-body Hamiltonians in higher dimensions, contrasting with the infinite bound states in lower dimensions.

## Key findings

- No Efimov effect in dimensions 4 and higher.
- Finiteness of the number of bound states in higher dimensions.
- Influence of dimension and symmetry on three-body interactions.

## Abstract

We consider a system of three particles in dimension 4 and higher interacting via short-range potentials, where the two-body Hamiltonians have a virtual level at the bottom of the essential spectrum. In dimensions 2 (in case of fermions) and 3 the corresponding three-body Hamiltonian admits an infinite number of bound states, which is known as the Efimov effect. In this work we prove that this is not the case in higher dimensions. We investigate how the dimension and symmetries of the system influence this effect and prove the finiteness of the discrete spectrum of the corresponding three-body Hamiltonian.