Some special solutions of the Schr\"odinger equation

Abstract

In this paper we consider the inverse problem of determining on a compact Riemannian mani-fold the electric potential or the magnetic field in a Schrödinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the magnetic Schrödinger equation. We prove in dimension n ě 2 that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the magnetic field and the electric potential and we establish Hölder-type stability.

A Note on Weighted Estimates for the Schrödinger Operator

“…Theorem 1.5 contains a result which is similar to the one given in Proposition 3.4 of [1], but for the unit sphere in R n instead of the section of the paraboloid…”

“…As we will see in the proof of Theorem 1.1, any estimate of type (4) implies an estimate (3) for the same values of α and p. This allow us to get necessary conditions for the restriction of the Fourier transform (4) to hold by constructing counterexamples for the evolution Schrödinger equation (3), although the result that we get is not so good as the result obtained directly in [1] for the section of the paraboloid given in (5). …”

“…This result for the estimate (4) was proved in [1,Lemma 3.3] by using standard counterexamples, that is, the characteristic funcion of a δ-cell on S n−1 . But these standard counterexamples do not work for the estimates (3) because of the homogeneity.…”

“…In [1] we pointed out that the proof of the result for the mentioned section of the paraboloid do not appear to extend, in a routine manner, to other curved submanifolds of the type we considered there, as for example the unit sphere. As we will see in the proof of Theorem 1.1, any estimate of type (4) implies an estimate (3) for the same values of α and p. This allow us to get necessary conditions for the restriction of the Fourier transform (4) to hold by constructing counterexamples for the evolution Schrödinger equation (3), although the result that we get is not so good as the result obtained directly in [1] for the section of the paraboloid given in (5).…”

“…This problem is closely related to a certain weighted restriction theorem for the Fourier transform of the measure on the unit sphere. This allow us to give partial answers to some open questions in [1,5,7].…”