Oscillator representations and systems of ordinary differential equations

Using representation-theoretic methods, we determine the spectrum of the 2 × 2 system [Formula: see text] with A, B ∈ Mat(2)(ℝ) constant matrices such that A = (t)A > 0 (or <0), B = −(t)B ≠ 0, and the Hermitian matrix A + iB positive (or negative) definite. We also give results that generalize (in a possible direction) the main construction.