We discuss the Calderón problems for the fractional Schrödinger equations. We show that the fractional Laplacian of any positive Sobolev scale has the Poincaré inequality on all domains that are bounded in one direction when integrability scale is at least two. This uses results for the Gagliardo seminorms, complex interpolation inequalities and standard embedding theorems. The Poincaré inequalities allow us to study the fractional Calderón problems on these unbounded domains for certain limited classes of Sobolev multipliers and linear lower order perturbations, following the earlier works of Rüland-Salo and Covi-Mönkkönen-R.-Uhlmann. We establish uniqueness results for these inverse problems based on the ideas originating from the work of Ghosh-Salo-Uhlmann.
The talk is based on work with Philipp Zimmermann (ETHZ) and earlier works with Giovanni Covi (Heidelberg), Keijo Mönkkönen (Jyväskylä) and Gunther Uhlmann (UW).