Exact Diagonalization of the Fractional Quantum Hall Many-Body Hamiltonian in the Lowest Landau Level

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For a gaussian interaction V(x,y)=λe⁻⁽ˣ²⁺ʸ²⁾/ʳ² with long range r>>lB, lB the magnetic length, we rigorously prove that the eigenvalues of the finite volume Hamiltonian HN,LL=PLL Hₙ PLL, Hₙ=∑ᵢ₌₁ⁿ [-iℏ ∇ₓᵢ-eA(xᵢ)]²+∑ᵢ,ⱼ; ᵢ≠ ⱼ V(xᵢ-xⱼ), =(0,0,B), and PLL the projection onto the lowest Landau level, are given by the following set: Let M be the number of flux quanta flowing through the sample such that ν=N/M is the filling factor. Then each eigenvalue is given by E=E(n₁,...,nₙ)=∑ᵢ,ⱼ₌₁;ᵢ≠ ⱼⁿ W(nᵢ-nⱼ). Here nᵢ∈ 1,2,...,M, n₁<...<nₙ and the function W is given by W(n)=λ∑ⱼ∈ Z e-1/r²(Ln/M-jL)² if the system is kept in a volume [0,L]². The eigenstates are also explicitely given.