Uniform Resolvent Estimates And Higher Order Schrodinger Operators

The thesis is mainly concerned of uniform resolvent estimates of Laplacian in-cluding its fractional power and LP estimates of fractional Schrodinger equation.This thesis is divided into six chapters.Chapter 1 introduces the original background as well as development status of Schrodinger equation and uniform Sobolev estimates, then presents the main results of the thesis.Chapter 2 considers the generalization of uniform Sobolev estimates on Euclidean space to the simply connected manifolds of constant curvature. Our methods relies on the connection between resolvent and the solution of wave equation. In particular, when considering the case of the standard sphere, the new ingredient is that we take advantage of the fact that cos t (?)-?Sn+(n-1/2)2 is periodic with respect to t. In the case of negative curvature, We obtain uniform resolvent estimates by proving a Stien-Tomas type theorem on hyperbolic space.Chapter 3 is devoted to generalize the uniform Sobolev estimates to the fractional case, we use the expression for the resolvent of the fractional power of Laplacian, hence essentially relying on the results of the second order case. Besides, when adding a potential, we establish the limiting absorption principle by exploiting Fredholm theory.Chapter 4 is dealing with quantitative uniqueness for some higher order Schrodinger operators. The main idea is to prove Carleman inequality by choosing suitable weight function. The results improve and generalize some known conclusions.In Chapter 5, we take into account of the point-wise heat kernel estimates for fractional Schrodinger operators, during the proof, we note the facet that the Kato type potentials are closely related to the resolvent of the fractional Laplacian. Then using point-wise heat kernel estimates, we obtain the LP estimates for its solutionsIn the final Chapter 6, a summary is presented and further questions are also discussed.