Non-commutative harmonic oscillators-I

Using representation-theoretic methods, we study the spectrum (in the tempered distributions) of the formally self-adjoint 2 × 2 system Q(x, D_x) = A ( - ∂_x²/2 + x²/2) + B (x∂_x + 1/2), x ∈ ℝ, with A, B ∈ Mat₂(ℝ) constant matrices such that A = ^tA > 0 (or < 0) and B = -^tB ≠ 0, in terms of invariants of the matrices A and B. In fact, if the Hermitian matrix A + iB is positive (or negative) definite, we determine the structure of the spectrum of the associated system Q(x,D_x) through suitable vector-valued Hermite functions. In the final sections we indicate how to generalize the results to analogous N × N systems and to particular multivariable cases.