### Abstract

Using representation-theoretic methods, we study the spectrum (in the tempered distributions) of the formally self-adjoint 2 × 2 system Q(x, D_{x}) = A ( - ∂_{x} ^{2}/2 + x^{2}/2) + B (x∂_{x} + 1/2), x ∈ ℝ, with A, B ∈ Mat_{2}(ℝ) constant matrices such that A = ^{t}A > 0 (or < 0) and B = -^{t}B ≠ 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,D_{x}) 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.

Original language | English |
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Pages (from-to) | 539-604 |

Number of pages | 66 |

Journal | Forum Mathematicum |

Volume | 14 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 1 2002 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Forum Mathematicum*,

*14*(4), 539-604. https://doi.org/10.1515/form.2002.025