AI 中文总结
研究亚临界情形下多调和薛定谔算子的谱估计，通过关联奥捷巴耶夫函数，得到负特征值分布函数双边估计及负谱离散性条件，还建立李布 - 瑟林型双边估计以改进经典估计。
AI 中文摘要
我们研究亚临界情形\(2l<\mathbf{N}\)下多调和薛定谔算子\(-\Delta^l - \mu\)的谱估计。假设测度势\(\mu\)满足一个容量小性条件，以保证相应算子是半有界自伴的。对于这样的测度\(\mu\)，我们关联一个奥捷巴耶夫函数，它反映了势的局部集中和空间分布。借助这个函数，我们得到了负特征值分布函数的双边估计，并导出了负谱离散性的充分条件和必要条件。作为应用，我们建立了李布 - 瑟林型的双边估计，改进了经典估计。
英文摘要
We study spectral estimates for polyharmonic Schrödinger operators $-Δ^l-μ$ in the subcritical regime $2l<\mathbf{N}$. The measure potential $μ$ is assumed to satisfy a capacitary smallness condition which guarantees that the corresponding operator is semibounded and self-adjoint. With such a measure $μ$ we associate an Otelbaev function, which reflects both the local concentration and the spatial distribution of the potential. In terms of this function, we obtain two-sided estimates for the distribution function of the negative eigenvalues, and derive a sufficient condition and a necessary condition for the discreteness of the negative spectrum. As an application, we establish two-sided estimates of Lieb-Thirring-type, improving the classical ones.
