数学物理学报
數學物理學報
수학물이학보
ACTA MATHEMATICA SCIENTIA
2010年
1期
18-30
,共13页
椭圆算子%周期位势%绝对连续谱%一致Sobolev不等式
橢圓算子%週期位勢%絕對連續譜%一緻Sobolev不等式
타원산자%주기위세%절대련속보%일치Sobolev불등식
Elliptic operator%Periodic potential%Absolute continuous spectrum%Uniform Sobolev inequalities
该文研究周期椭圆算子d∑j,l=1 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[22]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~(2pd/(d+2p))_(loc)(R~d)且w ∈A~(p,∞)_(1+α)(T~d)∩L~∞(T~d)(α>0,p≥d),或者V ∈L~(2d/3)_(loc)(R~d),w ∈C~1(T~d),或者V ∈L~(d/2)_(loc)(R~d),w∈L~(d/2)_(2,loc)(T~d).
該文研究週期橢圓算子d∑j,l=1 D_jw(x)a_(jl)D_l+V(x)在R~d(d≥3)中的譜性質,其中A=(a_(jl))是d×d階的實常值正定矩陣, V(x)和w(x)是關于相同格點的週期標量函數,併且w(x)是正的.利用文中第一作者[22]建立的d-環麵上的一緻Sobolev不等式,證明瞭該算子的譜是純絕對連續的,如果V ∈L~(2pd/(d+2p))_(loc)(R~d)且w ∈A~(p,∞)_(1+α)(T~d)∩L~∞(T~d)(α>0,p≥d),或者V ∈L~(2d/3)_(loc)(R~d),w ∈C~1(T~d),或者V ∈L~(d/2)_(loc)(R~d),w∈L~(d/2)_(2,loc)(T~d).
해문연구주기타원산자d∑j,l=1 D_jw(x)a_(jl)D_l+V(x)재R~d(d≥3)중적보성질,기중A=(a_(jl))시d×d계적실상치정정구진, V(x)화w(x)시관우상동격점적주기표량함수,병차w(x)시정적.이용문중제일작자[22]건립적d-배면상적일치Sobolev불등식,증명료해산자적보시순절대련속적,여과V ∈L~(2pd/(d+2p))_(loc)(R~d)차w ∈A~(p,∞)_(1+α)(T~d)∩L~∞(T~d)(α>0,p≥d),혹자V ∈L~(2d/3)_(loc)(R~d),w ∈C~1(T~d),혹자V ∈L~(d/2)_(loc)(R~d),w∈L~(d/2)_(2,loc)(T~d).
In this paper we consider the spectral properties of periodic elliptic operator d∑j,l=1 D_j w(x)a_(jl) D_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 V ∈L~(2pd/(d+2p))_(loc)(R~d) and w ∈A~(p,∞)_(1+α)(T~d)∩ L~∞ (T~d) for some α>0,p≥d,or V ∈L~(2d/3)_(loc)(R~d),w ∈C~1 (T~d),or V ∈~(d/2)_(loc)(R~d),w ∈~(d/2)_(2,loc)(T~d).
