Non-commutative harmonic oscillators-I

Alberto Parmeggiani, Masato Wakayama

Research output: Contribution to journalArticle

28 Citations (Scopus)

Abstract

Using representation-theoretic methods, we study the spectrum (in the tempered distributions) of the formally self-adjoint 2 × 2 system Q(x, Dx) = A ( - ∂x 2/2 + 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
Volume14
Issue number4
DOIs
Publication statusPublished - Jan 1 2002

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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