## Abstract

Superoscillations occur when a globally band-limited function locally oscillates faster than its highest Fourier component. We generalize this effect to arbitrary quantum-mechanical operators as a weak value, where the preselected state is a superposition of eigenstates of the operator with eigenvalues bounded to a range, and the postselection state is a local position. Superbehavior of this operator occurs whenever the operator's weak value exceeds its eigenvalue bound. We give illustrative examples of this effect for total angular momentum and energy. In the latter case, we demonstrate a sequence of harmonic oscillator potentials where a finite-energy state converges everywhere on the real line, using only bounded superpositions of states whose asymptotic energy vanishes - "energy out of nothing."This limit requires postselecting the particle in a region whose size diverges in the considered limit. We further show in this example that the superenergy is associated with superoscillations in time with a rate given by the local superenergy divided by the reduced Planck's constant. This example demonstrates the possibility of mimicking a high-energy state with coherent superpositions of nearly zero-energy states for as wide a spatial region as desired. We provide numerical evidence of these features to further bolster and elucidate our claims.

Original language | English |
---|---|

Article number | 012206 |

Journal | Physical Review A |

Volume | 110 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2024 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2024 American Physical Society.