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

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