### Abstract

We refine our study of the spectrum of non-commutative harmonic oscillators Q(x, D_{x}) = 1/2A(-∂_{x} ^{2} + x^{2}) + B(x∂_{x} + 1/2), x ∈ ℝ, where A, B ∈ Mat_{2}(ℝ) are constant 2 × 2 matrices such that A = ^{t}A > 0 (or <0) and B = -^{t}B ≠ 0, and the Hermitian matrix A + iB > 0 (or <0). We introduce a new family of L^{2}-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 |
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Pages (from-to) | 669-690 |

Number of pages | 22 |

Journal | Forum Mathematicum |

Volume | 14 |

Issue number | 5 |

DOIs | |

Publication status | Published - Jan 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Forum Mathematicum*,

*14*(5), 669-690. https://doi.org/10.1515/form.2002.029

**Non-commutative harmonic oscillators-II.** / Parmeggiani, Alberto; Wakayama, Masato.

Research output: Contribution to journal › Article

*Forum Mathematicum*, vol. 14, no. 5, pp. 669-690. https://doi.org/10.1515/form.2002.029

}

TY - JOUR

T1 - Non-commutative harmonic oscillators-II

AU - Parmeggiani, Alberto

AU - Wakayama, Masato

PY - 2002/1/1

Y1 - 2002/1/1

N2 - We refine our study of the spectrum of non-commutative harmonic oscillators Q(x, Dx) = 1/2A(-∂x 2 + 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(-∂x 2 + 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.

UR - http://www.scopus.com/inward/record.url?scp=0036042664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036042664&partnerID=8YFLogxK

U2 - 10.1515/form.2002.029

DO - 10.1515/form.2002.029

M3 - Article

AN - SCOPUS:0036042664

VL - 14

SP - 669

EP - 690

JO - Forum Mathematicum

JF - Forum Mathematicum

SN - 0933-7741

IS - 5

ER -