We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator \(\frac{1}{2}\, \log (-\Delta )\) in an open set \(\Omega \in \mathbb R^d\), \(d\ge 2\), of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue \(\lambda _1(\Omega )\) and compare them with previously known inequalities.

中文翻译：

---

对数拉普拉斯算子的光谱特性

我们得到了算子\(\frac{1}{2}\, \log (-\Delta )\)在开集\(\Omega \in \mathbb R^d \) , \(d\ge 2\)，具有 Dirichlet 边界条件的有限测度。我们还推导出了一些关于特征值\(\lambda _1(\Omega )\)下界的结果，并将它们与之前已知的不等式进行比较。