## Abstract

The well-known Hadamard method of descent consists of deriving solutions of wave equations in n variables from (known) solutions in more than n variables. Here we show how to build the propagator cos(A^{2}+B^{2}) out of cos(At) and cos(Bt) . The formulas obtained for the case of commuting A and B may be illustrated by deriving solutions of the Cauchy problem for the n dimensional wave equation (and for the Klein-Gordon equation) from the very simple one-dimensional case. In the non-commutative case the formula involves differentiating a very high-dimensional integral to a high order. The result yields expressions for the harmonic oscillator and for certain simple hypoelliptic sum of squares operators, such as the Heisenberg Laplacian.

Original language | English |
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Pages (from-to) | 573-597 |

Number of pages | 25 |

Journal | Bulletin des Sciences Mathematiques |

Volume | 124 |

Issue number | 7 |

DOIs | |

State | Published - Oct 2000 |

Externally published | Yes |

### Bibliographical note

Funding Information:* This research was partially supported by a Minerva foundation grant.

### Funding

* This research was partially supported by a Minerva foundation grant.

Funders | Funder number |
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Minerva Foundation |

## Keywords

- Non-commuting self-adjoint operators
- Propagator
- Wave equation

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