Relativistic N-boson systems bound by pair potentials V(r_{ij}) = g(r_{ij}^2)

TL;DR

This paper develops bounds for the ground state energy of a relativistic N-boson system with pairwise potentials of the form g(r^2), using envelope theory and variational methods, achieving high accuracy for oscillator potentials.

Contribution

It introduces a method combining envelope theory and variational techniques to accurately bound the energy of relativistic N-boson systems with convex pair potentials.

Findings

Derived formulas for lower energy bounds.

Established upper bounds valid for all N >= 2.

Achieved less than 0.15% error for oscillator potential energies.

Abstract

We study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form V(r_{ij}) = g(r_{ij}^2) in three spatial dimensions. In natural units hbar = c = 1 the system has the semirelativistic `spinless-Salpeter' Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N g(|r_i - r_j|^2), where g is monotone increasing and has convexity g'' >= 0. We use `envelope theory' to derive formulas for general lower energy bounds and we use a variational method to find complementary upper bounds valid for all N >= 2. In particular, we determine the energy of the N-body oscillator g(r^2) = c r^2 with error less than 0.15% for all m >= 0, N >= 2, and c > 0.