TitleThree circles theorems for Schrödinger operators on cylindrical ends and geometric applications.
Publication TypeJournal Article
Year of Publication2008
AuthorsColding T.H, De Lellis C., W.P. Minicozzi II
JournalCommunications on Pure and Applied Mathematics
Type of Articleminimal
KeywordsApplied Mathematics, General Mathematics

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.


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