Categorical resolutions of a class of derived categories

By using the relative derived categories, we prove that if an Artin algebra $A$ has a module $T$ with ${\rm inj.dim}T<\infty$ such that $^\perp T$ is finite, then the bounded derived category D^b(A\mbox{-}{\rm mod}) admits a categorical resolution in the sense of [Kuz], and a categorical desingularization in the sense of [BO]. For CM-finite Gorenstein algebra, such a categorical resolution is weakly crepant. The similar results hold also for D^b(A\mbox{-}{\rm Mod}).Comment: 23 page

On Landis Conjecture for the Fractional Schr\"{o}dinger Equation

In this paper, we study a Landis-type conjecture for the general fractional Schr\"{o}dinger equation $((-P)^{s}+q)u=0$. As a byproduct, we also proved the additivity and boundedness of the linear operator $(-P)^{s}$ for non-smooth coefficents. For differentiable potentials $q$, if a solution decays at a rate $\exp(-|x|^{1+})$, then the solution vanishes identically. For non-differentiable potentials $q$, if a solution decays at a rate $\exp(-|x|^{\frac{4s}{4s-1}+})$, then the solution must again be trivial. The proof relies on delicate Carleman estimates. This study is an extension of the work by R\"{u}land-Wang (2019).Comment: 44 page