Exact results for the Spectra of Bosons and Fermions with Contact Interaction

TL;DR

This paper provides exact solutions for the energy spectra of bosons and fermions with contact interactions in Landau levels, including novel eigenstates and their mathematical properties, advancing understanding of quantum many-body systems.

Contribution

It introduces a method to exactly determine energy levels for N-body bosonic and fermionic systems with contact interactions, including new eigenstates and their algebraic properties.

Findings

Exact energy levels obtained via finite matrix diagonalization.

Novel analytic eigenstates related to Catalan numbers.

Complete solution for the three-body problem.

Abstract

An N-body bosonic model with delta-contact interactions projected on the lowest Landau level is considered. For a given number of particles in a given angular momentum sector, any energy level can be obtained exactly by means of diagonalizing a finite matrix: they are roots of algebraic equations. A complete solution of the three-body problem is presented, some general properties of the N-body spectrum are pointed out, and a number of novel exact analytic eigenstates are obtained. The FQHE N-fermion model with Laplacian-delta interactions is also considered along the same lines of analysis. New exact eigenstates are proposed, along with the Slater determinant, whose eigenvalues are shown to be related to Catalan numbers.