Quantitative uniqueness for Schrödinger operator with regular potentials

Abstract

We give a sharp upper bound on the vanishing order of solutions to Schrödinger equation with C 1 electric and magnetic potentials on a compact smooth manifold. Our method is based on quantitative Carleman type inequalities developed by Donnelly and Fefferman. It also extends the first author's previous work to the magnetic potential case.

Semiclassical Cauchy estimates and applications

“…This type of results have been proved in more general setting, e.g. [1] for C 1 -potentials and are closely related to the unique continuation problems. There are two different ways to achieve such kind of results: the usual approach is through the Carleman-type estimates which establish a priori estimates with a weight; another approach was developed by Garofalo and Lin [7] based on a combination of geometric and variational ideas.…”

“…From the argument in [14], we can also see that this estimate only depends on the seminorms of p and ψ in S 2n (1). In other words, if p and ψ varies in a way such that every sup…”

“…In [1], it is proved that this is true even when (M, g) and V are only smooth and the constant C only depends on (M, g) and the C 1 -norm of V .…”

“…We do not assume the analyticity of either the manifold or the potential. See [1] for a general setting where the potential is only assumed to be C 1 . From now on, for simplicity, we shall use u U to represent the L 2 -norm of the function u in the set U .…”

“…We can remove the condition r > c 1 h in part (ii) by carefully constructing a weight involving logarithmic terms near the origin in Carleman estimates. For the details, see [1]. For our purpose, the weak version above will be sufficient.…”