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


Colding, T; De Lellis, C; Minicozzi, W (2008). Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications. Communications on Pure and Applied Mathematics, 61(11):1540-1602.

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.

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
Uncontrolled Keywords:Applied Mathematics, General Mathematics
Language:English
Date:2008
Deposited On:14 Jan 2009 08:35
Last Modified:17 Aug 2018 12:00
Publisher:Wiley-Blackwell
ISSN:0010-3640
Additional Information:The attached file is a preprint (accepted version) of an article published in Communications on Pure and Applied Mathematics
OA Status:Green
Publisher DOI:https://doi.org/10.1002/cpa.20232
Related URLs:http://arxiv.org/abs/math/0701302

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