The Study Of Compactness And Spectrum For Some Classes Of Symmetric Nonlocal Operators

In this thesis,we will study the compactness and spectral properties for some classes of symmetric nonlocal operators.The thesis mainly contains the following four parts.In Chapter 1,we focus on the compactness of semigroups corresponding to a time-change of truncated symmetric α-stable processes.We establish sufficient conditions for Orlicz-Sobolev inequalities for general Dirichlet forms via super-Poincare inequalities,which yield the criterion with the form of integral type for the compactness of semigroups corresponding to time-changed truncated symmetric α-stable processes.In Chapter 2,we establish several functional inequalities corresponding to timechanged symmetric α-stable processes on and give sufficient and necessary conditions for the compactness of the associated semigroups and upper bounds for the corresponding heat kernel.In particular,for the special weighted function W(x)=(1+|x|)β with β>α,we obtain an optimal Nash-type inequality,which in turn indicates that upper bounds for heat kernel we obtained are optimal.The Chapter 3,analogous to Chapter 2,is concerned on the compactness of Markov semigroups corresponding to a time-change of nonlocal Dirichlet forms with finite second moment jumping kernels.The necessary conditions for compactness of the semigroup are given by adopting the comparison technique,while sufficient conditions are presented by establishing essential super-Poincaré inequality as well as introducing a class of auxiliary Dirichlet forms.In Chapter 4,asymptotic properties of the integrated density of states N(λ)with respect to the spectrum for the random Schrodinger operator Hω=(-Δ)α/2+Vω are studied,where Vω(x)=∑i∈Zd ξi(ω)W(x-i)is a random potential term generated by a sequence of independent and identically distributed random variables {ξi}i∈Zd and a non-negative measurable function W(x).According to different properties of W(x)and spectrum properties of(-Δ)α/2,we present asymptotic properties of N(λ)with exact order.