Spectral analysis of non commutative harmonic oscillators: The lowest eigenvalue and no crossing
Abstract The lowest eigenvalue of non commutative harmonic oscillators Q a b a 0 b 0 a b 1 is studied. It is shown that Q a b can be decomposed into four self adjoint operators, Q a b s p 1 2 Q s p and all the eigenvalues of each operator Q s p are simple. We show that the lowest eigenvalue of Q a b is simple whenever a b Furthermore a Jacobi matrix representation of Q s p is given and spectrum of Q s p is considered numerically.