Non-commutative harmonic oscillators-I

Alberto Parmeggiani, Masato Wakayama

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)


Using representation-theoretic methods, we study the spectrum (in the tempered distributions) of the formally self-adjoint 2 × 2 system Q(x, Dx) = 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,Dx) 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.

Original languageEnglish
Pages (from-to)539-604
Number of pages66
JournalForum Mathematicum
Issue number4
Publication statusPublished - 2002
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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