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 × [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.

De Lellis, C (2008). *The Euler equations from the point of view of differential inclusions.* Bollettino dell'Unione Matematica Italiana, (9) 1(3):873-879.

## 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 × [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.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | Italian |

Date: | 2008 |

Deposited On: | 06 Feb 2009 10:20 |

Last Modified: | 05 Apr 2016 12:57 |

Publisher: | Unione Matematica Italiana |

ISSN: | 1972-6724 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=2455350 http://arxiv.org/abs/math/0702079v3 |

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