## Abstract

Let H=-δ+V be a nonnegative Schrödinger operator on L^{2}(R^{N}), where N≥3 and V is a radially symmetric nonpositive function in R^{N} decaying quadratically at the space infinity. For any 1≤p≤q≤∞, we denote by {norm of matrix}e^{-tH}{norm of matrix}_{q,p} the operator norm of the Schrödinger heat semigroup e^{-tH} from L^{p}(R^{N}) to L^{q}(R^{N}). In this paper, under suitable conditions on V, we give the exact and optimal decay rates of {norm of matrix}e^{-tH}{norm of matrix}_{q,p} as t→∞ for all 1≤p≤q≤∞. The decay rates of {norm of matrix}e^{-tH}{norm of matrix}_{q,p} depend on whether the operator H is subcritical or critical and on the behavior of the positive harmonic function for the operator H.

Original language | English |
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Pages (from-to) | 2764-2783 |

Number of pages | 20 |

Journal | Journal of Functional Analysis |

Volume | 264 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2013 Jun 15 |

## Keywords

- L-L estimates
- Quadratically decaying potential
- Schrödinger heat semigroups

## ASJC Scopus subject areas

- Analysis