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.
Parmeggiani, A., Wakayama, M. (2002). Non-commutative harmonic oscillators-I. FORUM MATHEMATICUM, 14(4), 539-604 [10.1515/form.2002.025].
Non-commutative harmonic oscillators-I
Parmeggiani A.;
2002
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 ( - ∂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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
