|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|
|Type of Article||minimal|
|Keywords||Applied 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.