We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in ℝn, n ≥ 3, for the magnetic Schrödinger operator with L ∞ magnetic and electric potentials, determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives. © 2014 Springer-Verlag Berlin Heidelberg.

We prove uniform Lpestimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding result of [3] in the case of Laplace-Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1] very closely. We also show that spectral regions in our Lpresolvent estimates are optimal.

We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $R^n$ for the perturbed polyharmonic operator $(-\Delta)^m +q$ with $q\in L^{n/2m}$, $n>2m$, determines the potential $q$ in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted $L^2$ and $L^p$ spaces. The $L^p$ estimates for the special Green function are derived from $L^p$ Carleman estimates with linear weights for the polyharmonic operator.

We show that the knowledge of the set of the Cauchy data on the boundary of a $C^1$ bounded open set in $\R^n$, $n\ge 3$, for the Schr\"odinger operator with continuous magnetic and bounded electric potentials determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schr\"odinger operator with a gain of two derivatives.

We show that the spectrum of a Schr\"odinger operator on $\mathbb{R}^n$, $n\ge 3$, with a periodic smooth Riemannian metric, whose conformal multiple has a product structure with one Euclidean direction, and with a periodic electric potential in $L^{n/2}_{\text{loc}}(\mathbb{R}^n)$, is purely absolutely continuous. Previously known results in the case of a general metric are obtained in [12], see also [8], under the assumption that the metric, as well as the potential, are reflection symmetric.

We show that a first order perturbation A(x) · D + q(x) of the polyharmonic operator (-Δ)m, m ≥ 2, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in ℝn, n ≥ 3. Notice that the corresponding result does not hold in general when m = 1. © 2013 American Mathematical Society.