We refine our study of the spectrum of non-commutative harmonic oscillators Q(x, Dx) = 1/2A(-∂x2 + x2) + B(x∂x + 1/2), x ∈ ℝ, where A, B ∈ Mat2(ℝ) are constant 2 × 2 matrices such that A = tA > 0 (or <0) and B = -tB ≠ 0, and the Hermitian matrix A + iB > 0 (or <0). We introduce a new family of L2-bases and study the relation between the coefficients of the eigenfunction obtained by means of these bases, and the ones obtained by means of the bases introduced in . We hence completely determine the spectrum and its multiplicity.
All Science Journal Classification (ASJC) codes
- 数学 (全般)