Analytic solutions of topologically disjoint systems

TL;DR

This paper presents an analytical method to solve complex topologically disjoint multi-particle systems with arbitrary intra-group interactions and harmonic inter-group potentials, providing explicit spectra and wave functions.

Contribution

It introduces a reduction technique to solve multi-group particle systems analytically, including spectra and wave functions for specific interactions.

Findings

String separation energy increases with number of groups and interaction strength.

Analytical solutions for spectra and wave functions are derived.

The method handles arbitrary intra-group interactions within a harmonic inter-group framework.

Abstract

We describe a procedure to solve an up to $2N$ problem where the particles are separated topologically in $N$ groups with at most two particles in each. Arbitrary interactions are allowed between the (two) particles within one group. All other interactions are approximated by harmonic oscillator potentials. The problem is first reduced to an analytically solvable $N$-body problem and $N$ independent two-body problems. We calculate analytically spectra, wave functions, and normal modes for both the inverse square and delta-function two-body interactions. In particular, we calculate separation energies between two strings of particles. We find that the string separation energy increases with $N$ and interaction strength.