具有奇异性的周期椭圆算子的绝对连续性

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)(α0,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).

On Absolute Continuity of Periodic Elliptic Operators with Singularity

In this paper we consider the spectral properties of periodic elliptic operator ∑^d_j,l=1D_jw(x)a_jlD_l+V(x) in R^d, d ≥  3, where A=(a_jl) is a d×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 L_loc^2pd/d+2p(R^d) and w ∈ ∧_1+α^p, ∞(T^d)∩L^∞(T^d) for some α>0, p≥ d, or V ∈ L _loc^2d/3(R^d), w ∈ C¹(T^d), or V ∈ L_loc^d/2(R^d), w ∈ L_2, loc^d/2(T^d).