Unique Continuation Properties for One Dimensional Higher Order Schrodinger Equations

We study two types of unique continuation properties for the higher order Schrodinger equation with potential i partial derivative(t)u = (-Delta(x))(m)u + V(t, x)u, (t, x) is an element of R1+n, 2 <= m is an element of N+. The first one says if u has certain exponential decay at two times, then u 0, and this result is sharp by constructing critical non-trivial solutions. The second one says if u 0 in an arbitrary half-space of R1+n, then u 0 identically. The uniqueness theorems are given when n = 1, but we also prove partial results when n is an element of N+ for their own interests. Possibility or obstacles to proving these unique continuation properties in higher spatial dimensions are also discussed.