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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