Exact Ground State of Several N-body Problems With an N-body Potential

TL;DR

This paper demonstrates that certain N-body quantum problems with harmonic oscillator interactions also have exact solutions when replaced with specific N-body potentials, and conjectures a broader solvability pattern.

Contribution

It introduces a class of N-body potentials for which exact ground and excited states can be obtained, extending known solvable models and proposing general conjectures on solvability.

Findings

Exact ground states are obtainable with N-body Coulomb-like potentials.

Solvability extends from harmonic oscillator to specific N-body potentials.

Adding inverse-square potentials reduces degeneracy but preserves solvability.

Abstract

I consider several N-body problems for which exact (bosonic) ground state and a class of excited states are known in case the N-bodies are also interacting via harmonic oscillator potential. I show that for all these problems the exact (bosonic) ground state and a class of excited states can also be obtained in case they interact via an N-body potential of the form $-e^2/\sqrt{\sumr^2_i}$ (or $-e^2/\sqrt{\sum_{i<j} (r_i - r_j)^2}$). Based on these and previously known examples, I conjecture that whenever an N-body problem is solvable in case the N-bodies are interacting via an oscillator potential, the same problem is also solvable in case they are interacting via the N-body potential. Based on several examples, I also conjecture that in either case one can always add an N-body potential of the form $\beta^2/{\sum_{i} r_i^2}$ and the problem is still solvable except that the degeneracy in the bound state spectrum is now much reduced.