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The lowest eigenvalue of non-commutative harmonic oscillators $Q$ is studied.It is shown that $Q$ can be decomposed into four self-adjoint operators, andall the eigenvalues of each operator are simple. We show that the lowesteigenvalue $E$ of $Q$ is simple. Furthermore a Jacobi matrix representation of$Q$ is given and spectrum of $Q$ is considered numerically.

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