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We re®ne our study of the spectrum of non-commutative harmonic oscillatorswhere AY B e Mat 2 R are constant 2 Â 2 matrices such that A t A b 0 (or`0) and B À t B H 0, and the Hermitian matrix A iB b 0 (or`0). We introduce a new family of L 2 -bases and study the relation between the coe½cients of the eigenfunction obtained by means of these bases, and the ones obtained by means of the bases introduced in [4]. We hence completely determine the spectrum and its multiplicity.2000 Mathematics Subject Classi®cation: 22E46, 34A30, 35P.

Using representation-theoretic methods, we determine the spectrum of the 2 ؋ 2 systemwith A, B ʦ Mat2(‫)ޒ‬ 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.