TY - JOUR

T1 - Non-commutative harmonic oscillators-I

AU - Parmeggiani, Alberto

AU - Wakayama, Masato

N1 - Funding Information:
Work in part supported by N.A.T.O.-C.N.R., by C.N.R.-G.N.A.F.A., by Grant-in Aid for Scientific Research (B) No. 09440022, the Ministry of Education, Science and Culture of Japan.

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

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U2 - 10.1515/form.2002.025

DO - 10.1515/form.2002.025

M3 - Article

AN - SCOPUS:0036266266

VL - 14

SP - 539

EP - 604

JO - Forum Mathematicum

JF - Forum Mathematicum

SN - 0933-7741

IS - 4

ER -