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 ∂ t + Δ ) u = V ( x ) u (i\partial _t+\Delta )u=V(x)u . In general the problem is to find a class of potentials V 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 L^2 resolvent estimates ‖ ( − Δ − z ) − 1 f ‖ L 2 ( | V | ) ≤ C ‖ f ‖ L 2 ( | V | − 1 ) \|(-\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 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 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.

On absolute continuity of the spectrum of periodic Schrödinger operators

“…The estimate (3.5) is a uniform Sobolev inequality on the torus T 3 for the second-order elliptic operator H 0 (δ 0 + iρ). Similar inequalities were obtained in the setting of R n by many authors ( [15,5,6,23]) to study unique continuation properties of differential operators. Also, (3.5) was shown in [27] for w in the Morrey class.…”

“…The class F and the characterization of (1.6) has been recently used in various problems concerning Schrödinger operators and equations ( [1,22,23,24]). As is well known from [8], (1.6) holds for w ∈ M p (R n ) with…”

“…[6,23]), and will be needed here for our resolvent estimates in Proposition 3.2 which is a key ingredient in the proof of the following theorem which gives a new condition concerning the absolute continuity:…”

A note on the Schrödinger smoothing effect

“…The quantity was already appeared in and concerning the unique continuation for the Schrödinger equation and eigenvalue bounds for the Schrödinger operator, respectively. We also refer the reader to for some weighted L 2 estimates in which a time‐dependent weight is involved.…”

“…Estimate makes this extension more sharp as in Remark . See also for the case where and in , which just implies …”