From resolvent estimates to unique continuation for the Schrödinger equation

Abstract:

In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation $(i\partial _t+\Delta )u=V(x)u$. In general the problem is to find a class of potentials $V$ which allows the unique continuation. The key point of our work is to make a direct link between the problem and the weighted $L^2$ resolvent estimates $\|(-\Delta -z)^{-1}f\|_{L^2(|V|)}\leq C\|f\|_{L^2(|V|^{-1})}$. We carry it out in an abstract way, and thereby we do not need to deal with each of the potential classes. To do so, we will make use of the limiting absorption principle and Kato $H$-smoothing theorem in spectral theory, and employ some tools from harmonic analysis. Once the resolvent estimate is set up for a potential class, from our abstract theory the unique continuation would follow from the same potential class. Also, it turns out that there can be no dented surface on the boundary of the maximal open zero set of the solution $u$. In this regard, another main issue for us is to know which class of potentials allows the resolvent estimate. We establish such a new class which contains previously known ones, and we will also apply it to the problem of well-posedness for the equation.