具有奇异性的周期椭圆算子的绝对连续性

摘要/Abstract

摘要：

该文研究周期椭圆算子∑^d_j,l₌₁D_jw(x)a_jlD_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_loc^2pd/d+2p(R^d)且w ∈ ∧_1+α^p, ∞(T^d)∩L^∞(T^d)(α>0, p≥ d), 或者V ∈ L _loc^2d/3(R^d), w ∈ C¹(T^d), 或者V ∈ L_loc^d/2(R^d), w ∈ L_2, loc^d/2(T^d).

关键词: 椭圆算子, 周期位势, 绝对连续谱, 一致Sobolev不等式

Abstract:

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).

Key words: Elliptic operator, Periodic potential, Absolute continuous spectrum, Uniform Sobolev inequalities

中图分类号:

35J10

赵培浩, 刘舞龙. 具有奇异性的周期椭圆算子的绝对连续性[J]. 数学物理学报, 2010, 30(1): 18-30.

ZHAO Pei-Hao, LIU Wu-Long. On Absolute Continuity of Periodic Elliptic Operators with Singularity[J]. Acta mathematica scientia,Series A, 2010, 30(1): 18-30.

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