### Abstract

We consider a vector-valued Hermite-type basis for which the eigenvalue problem associated to the operator H_{A,B}:=B(-∂_{x} ^{2}) + Ax^{2} acting on L^{2}(ℝ; ℂ^{2}) becomes a three-terms recurrence. Here A and B are 2 x 2 constant positive definite matrices. Our main result provides an explicit characterization of the eigenvectors of H_{A,B} that lie in the span of the first four elements of this basis when AB ≠ BA.

Original language | English |
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Pages (from-to) | 121-131 |

Number of pages | 11 |

Journal | Letters in Mathematical Physics |

Volume | 70 |

Issue number | 2 |

DOIs | |

Publication status | Published - Nov 2004 |

### Keywords

- Eigenfunction expansion
- Higher dimensional Hermite basis
- Non-commutative harmonic oscillators

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Boulton, L., Marcantognini, S. A. M., & Morán, M. D. (2004). Finite lifetime eigenfunctions of coupled systems of harmonic oscillators.

*Letters in Mathematical Physics*,*70*(2), 121-131. https://doi.org/10.1007/s11005-004-4291-6