奇异扰动算子预解式差的谱性质
Spectral properties of the resolvent difference for singularly perturbed operators
AI总结：
本文研究了奇异扰动椭圆算子预解式差的谱估计问题，在测度满足Ahlfors条件及权函数满足积分条件下获得了阶锐估计，并证明了特征值的Weyl型渐近性。
摘要：
我们在域 $Ω\subseteq \mathbb{R}^\mathbf{N}$ 中获得了奇异扰动椭圆算子 $\mathbf{A}+\mathbf{V}_1$ 和 $\mathbf{A}+\mathbf{V}_2$ 预解式差的阶锐谱估计，其中扰动 $\mathbf{V}_1, \mathbf{V}_2$ 由 $V_1μ,V_2μ$ 生成，$μ$ 是关于 Lebesgue 测度奇异的测度，满足 Ahlfors 类型的双边或单边条件，而 $V_1,V_2$ 是满足某些积分条件的权函数。作为重要的特殊情形，获得了算子 $\mathbf{A}$ 的两个具有不同权函数的 Robin 实现的预解式差的谱估计。对于测度支集是 $Ω$ 中的紧 Lipschitz 超曲面或更一般地 Haußdorff 维数 $d=\mathbf{N}-1$ 的可求长集的情形，证明了特征值的 Weyl 型渐近性。
英文摘要：
We obtain order sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators $\mathbf{A}+\mathbf{V}_1$ and $\mathbf{A}+\mathbf{V}_2$ in a domain $Ω\subseteq \mathbb{R}^\mathbf{N}$ with perturbations $\mathbf{V}_1, \mathbf{V}_2$ generated by $V_1μ,V_2μ,$ where $μ$ is a measure singular with respect to the Lebesgue measure and satisfying two-sided or one-sided conditions of Ahlfors type, while $V_1,V_2$ are weight functions subject to some integral conditions. As an important special case, spectral estimates for the difference of resolvents of two Robin realizations of the operator $\mathbf{A}$ with different weight functions are obtained. For the case when the support of the measure is a compact Lipschitz hypersurface in $Ω$ or, more generally, a rectifiable set of Haußdorff dimension $d=\mathbf{N}-1$, the Weyl type asymptotics for eigenvalues is justified.
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