### 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 |
---|---|

Pages (from-to) | 669-690 |

Number of pages | 22 |

Journal | Forum Mathematicum |

Volume | 14 |

Issue number | 5 |

DOIs | |

Publication status | Published - Jan 1 2002 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Non-commutative harmonic oscillators-II'. Together they form a unique fingerprint.

## Cite this

Parmeggiani, A., & Wakayama, M. (2002). Non-commutative harmonic oscillators-II.

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