Spectral analysis of non-commutative harmonic oscillators: The lowest eigenvalue and no crossing

The lowest eigenvalue of non-commutative harmonic oscillators Q(α,β)Q(α,β) (α>0α>0, β>0β>0, αβ>1αβ>1) is studied. It is shown that Q(α,β)Q(α,β) can be decomposed into four self-adjoint operators,Q(α,β)=⨁σ=±,p=1,2Qσp, and all the eigenvalues of each operator QσpQσp are simple. We show that the lowest eigenvalue of Q(α,β)Q(α,β) is simple whenever α≠βα≠β. Furthermore a Jacobi matrix representation of QσpQσp is given and spectrum of QσpQσp is considered numerically.