Non-commutative harmonic oscillators-I

Using representation-theoretic methods, we study the spectrum (in the tempered distributions) of the formally self-adjoint 2 x 2 system Q(x, D-x) = A(-partial derivative(x)(2)/2 + x(2)/2) + B(xpartial derivative(x) + 1/2), x is an element of R, with A, B is an element of Mat(2)(R) constant matrices such that A = (t)A > 0 (or < 0) and B = -B-t ¬equal; 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 x N systems and to particular multivariable cases.