具有奇异性的周期椭圆算子的绝对连续性

On Absolute Continuity of Periodic Elliptic Operators with Singularity

该文研究周期椭圆算子sun from(j,l=1) to d D_(jw)(x)a_(jl)D_l+V(x)在R^d(d≥3)中的谱性质,其中A=(a_(jl))是d×d阶的实常值正定矩阵,V(x)和w(x)是关于相同格点的周期标量函数,并且w(x)是正的.利用文中第一作者建立的d-环面上的一致Sobolev不等式,证明了该算子的谱是纯绝对连续的,如果V∈L_(loc)^(2pd/(d+2p))(R^d)且w∈A_(1+α)^(p,∞)(T^d)∩L~∞(T^d)(α>0,p≥d),或者V∈L_(loc)^(2d/3)/(R^d),ω∈C^1(T^d),或者V∈L_(loc)^(d/2)(R^d),w∈L_(2,loc)^(d/2)(T^d).

In this paper we consider the spectral properties of periodic elliptic operator d∑j,l=1Djw（x）αjlDl＋V（x）in R^d, d ≥ 3, where A = （ajz） is a d x d positive definite matrix with real constant entries, V（x） and w（x） are periodic scalar function with respect to the same lattice, and w（x） is positive. Using a new uniform Sobolev inequalities on the d-torus established in [22], we prove that the spectrum of the operator is purely absolutely continuous if V∈L2pd/d＋2p loc（R^d）and w∈A^p,∞1＋α（T^d）∩L^∞（T^d）for some α 〉 O, p ≥ d, or V∈L^2d/3loc（R^d）,w∈C^1（T^d）or V∈L^d/2loc（R^d）,w∈L^d/2 2,loc（T^d）.