Non-commutative harmonic oscillators-II

Alberto Parmeggiani; Masato Wakayama

doi:10.1515/form.2002.029

Non-commutative harmonic oscillators-II

Alberto Parmeggiani
, Masato Wakayama

Research output: Contribution to journal › Article › peer-review

26 Citations (Scopus)

Abstract

We refine our study of the spectrum of non-commutative harmonic oscillators Q(x, D_x) = 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 [4]. We hence completely determine the spectrum and its multiplicity.

Original language	English
Pages (from-to)	669-690
Number of pages	22
Journal	Forum Mathematicum
Volume	14
Issue number	5
DOIs	https://doi.org/10.1515/form.2002.029
Publication status	Published - 2002
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1515/form.2002.029

Cite this

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title = "Non-commutative harmonic oscillators-II",

abstract = "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 [4]. We hence completely determine the spectrum and its multiplicity.",

author = "Alberto Parmeggiani and Masato Wakayama",

note = "Funding Information: Work in part supported by Istituto Nazionale di Alta Matematica (Gruppo G.N.A.F.A.), by Grant-in Aid for Scientific Research (B) No. 11440010, the Ministry of Education, Science and Culture of Japan.",

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doi = "10.1515/form.2002.029",

language = "English",

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TY - JOUR

T1 - Non-commutative harmonic oscillators-II

AU - Parmeggiani, Alberto

AU - Wakayama, Masato

N1 - Funding Information: Work in part supported by Istituto Nazionale di Alta Matematica (Gruppo G.N.A.F.A.), by Grant-in Aid for Scientific Research (B) No. 11440010, the Ministry of Education, Science and Culture of Japan.

PY - 2002

Y1 - 2002

N2 - 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 [4]. We hence completely determine the spectrum and its multiplicity.

AB - 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 [4]. We hence completely determine the spectrum and its multiplicity.

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U2 - 10.1515/form.2002.029

DO - 10.1515/form.2002.029

M3 - Article

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SN - 0933-7741

VL - 14

SP - 669

EP - 690

JO - Forum Mathematicum

JF - Forum Mathematicum

IS - 5

ER -

Non-commutative harmonic oscillators-II

Abstract

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