Title | Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications. |

Publication Type | Journal Article |

Year of Publication | 2008 |

Authors | Colding T.H, De Lellis C., W.P. Minicozzi II |

Journal | Communications on Pure and Applied Mathematics |

Volume | 61 |

Pagination | 1540–1602 |

Publisher | Wiley-Blackwell |

Type of Article | minimal |

ISSN | 0010-3640 |

Keywords | Applied Mathematics, General Mathematics |

Abstract | We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends, the space of solutions that grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently, for a surface for a fixed potential and a dense set of metrics), the constant function 0 is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity. One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution u to a Schrödinger equation on a product N $\times$ [0, T], where N is a closed manifold with a certain spectral gap. Examples of such N's are all (round) spheres n for n 1 and all Zoll surfaces. Finally, we discuss some examples arising in geometry of such manifolds and Schrödinger operators. |

Notes | Comm. Pure Appl. Math. 61 (2008), no. 11, 1540–1602. |

URL | http://onlinelibrary.wiley.com/doi/10.1002/cpa.20232/abstract |

DOI | 10.1002/cpa.20232 |

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