Abstract
We study here various expansions and approximations of the spectrum of non-commutative harmonic oscillators of the type
Q (x, Dx) = A (−∂x2/2 + x2/2) + B (x∂x + 1/2), x ∈ R,
where A, B are constant 2 × 2 matrices such that A is real symmetric positive (or negative) definite and B ≠ 0 is real skew-symmetric, when the Hermitian matrix A + iB is positive (or negative) definite. Special emphasis is put on the lowest eigenvalue. These expansions are written in terms of det(A )/pf(B )2 in the limit det(A ) → +∞ for constant pf(B ) and constant Tr(A )/√det(A ).