### Abstract

We study one-dimensional quantum mechanical systems in the semiclassical limit. We construct a lowest order quasimode ψ(ℏ) for the Hamiltonian H(ℏ) when the energy E and Planck's constant ℏ satisfy the appropriate Bohr - Sommerfeld conditions. This means that ψ(ℏ) is an approximate solution of the Schrödinger equation in the sense that \\[H(ℏ) - E]ψ(ℏ)|| ≤ Cℏ^{3/2}||ψ(ℏ)||. It follows that H(ℏ) has some spectrum within a distance Cℏ^{3/2} of E. Although the result has a long history, our time-dependent construction technique is novel and elementary.

Original language | English (US) |
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Pages (from-to) | 10113-10129 |

Number of pages | 17 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 31 |

Issue number | 50 |

DOIs | |

State | Published - Dec 18 1998 |

Externally published | Yes |

### ASJC Scopus subject areas

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

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## Cite this

Hagedorn, G. A., & Robinson, S. L. (1998). Bohr - Sommerfeld quantization rules in the semiclassical limit.

*Journal of Physics A: Mathematical and General*,*31*(50), 10113-10129. https://doi.org/10.1088/0305-4470/31/50/009