Klein-Gordon算子的Eigenvalue不等式
Eigenvalue inequalities for Klein-Gordon Operators
摘要：
我们考虑由量子力学的规定关联到Klein-Gordon哈密顿量$\sqrt{|{\bf P}|^2+m^2}$的伪微分算子$H_{m,Ω}$，当被限制到${\mathbb R}^d$中的紧域$Ω$时。当质量$m$为0时，算子$H_{0,Ω}$与在$\partial Ω$上具有杀死条件的柯西随机过程的生成元重合。(The运算符$H_{0,Ω}$有时被称为{ \it分数拉普拉斯算子}，幂为1/2，参见。\cite{Gie} .）证明了$H_{m,Ω}$的特征值$0 < β_1 < β_2 \le >...$与其平均值$\overline{β_k} := \frac{1}{k} \sum_{\ell=1}^k{β_\ell}$的几个普适不等式.其中证明的不等式有：{ \overline{β_k} } \ge { \rm cst.}（\frac{k}{|Ω|}）^{1/d}，对于任意维数$d \ge 2$和任意$k$：β_{k+1} \le \frac{d+1}{d-1} \overline{β_k} .此外，当$d \ge 2$和$k \ge 2j$时，\frac{\overlineβ_{k}}{\overlineβ_{j}} \leq \frac{d}{2^{1/d}(d-1)}（\frac{k}{j}）^{ \frac{1}{d} }.最后，我们提出了一些类似的估计，允许一个外部的势能场，即$H_{m,Ω}+ V(\bf x)$，$V(\bf x)$在某些功能类。
英文摘要：
We consider the pseudodifferential operators $H_{m,Ω}$ associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian $\sqrt{|{\bf P}|^2+m^2}$ when restricted to a compact domain $Ω$ in ${\mathbb R}^d$. When the mass $m$ is 0 the operator $H_{0,Ω}$ coincides with the generator of the Cauchy stochastic process with a killing condition on $\partial Ω$. (The operator $H_{0,Ω}$ is sometimes called the {\it fractional Laplacian} with power 1/2, cf. \cite{Gie}.) We prove several universal inequalities for the eigenvalues $0 < β_1 < β_2 \le >...$ of $H_{m,Ω}$ and their means $\overline{β_k} := \frac{1}{k} \sum_{\ell=1}^k{β_\ell}$. Among the inequalities proved are: {\overline{β_k}} \ge {\rm cst.} (\frac{k}{|Ω|})^{1/d} for an explicit, optimal "semiclassical" constant, and, for any dimension $d \ge 2$ and any $k$: β_{k+1} \le \frac{d+1}{d-1} \overline{β_k}. Furthermore, when $d \ge 2$ and $k \ge 2j$, \frac{\overlineβ_{k}}{\overlineβ_{j}} \leq \frac{d}{2^{1/d}(d-1)}(\frac{k}{j})^{\frac{1}{d}}. Finally, we present some analogous estimates allowing for an external potential energy field, i.e, $H_{m,Ω}+ V(\bf x)$, for $V(\bf x)$ in certain function classes.
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