$L^p$ estimates for fractional schrodinger operators with kato class potentials

Abstract

Let $\alpha >0$, $H=(-△{)}^{\alpha }+V(x)$, $V(x)$ belongs to the higher order Kato class $K_{2α}(\mathbbm{R}^n)$. For $1\le p\le \infty$, we prove a polynomial upper bound of $\Vert e^{-itH}(H+M{)}^{-\beta }{\Vert }_{L^p,L^p}$ in terms of time $t$. Both the smoothing exponent $\beta$ and the growth order in $t$ are almost optimal compared to the free case. The main ingredients in our proof are pointwise heat kernel estimates for the semigroup $e^{-tH}$. We obtain a Gaussian upper bound with sharp coefficient for integral $\alpha$ and a polynomial decay for fractal $\alpha$.

37 pages. Final version, to appear in J. Differential Equations

References in corpus (1)

• On Fractional Schrodinger Equations in sobolev spaces