关于正则度量树上薛定谔算子负特征值计数的注记
Remarks on counting negative eigenvalues of Schrödinger operator on regular metric trees
摘要：
讨论了正则度量树上薛定谔算子$-Δ-αV$的负特征值个数$N_-(α)$的估计，它依赖于势$V\ge 0$的性质和大参数$α$的值.我们得到的条件$V$保证行为$N_-(α)=O(α^p)$为任何给定的$p\ge 1/2$。对于一类特殊的树，我们证明了这些条件不仅是充分的，而且是必要的。对于$p>1/2$的阶尖估计涉及$V$在某个“弱”$L_p$ -或$\ell_p(L_1)$ -空间中的（拟）范数。我们表明，这些结果可以很容易地从Naimark和作者的早期论文中得到，Proc. London Math. Soc.（3）80（2000），690-724。结果大大改善了Ekholm，Frank和Kovaík最近论文中的估计，arXive：0710.5500。
英文摘要：
We discuss estimates on the number $N_-(α)$ of negative eigenvalues of the Schrödinger operator $-Δ-αV$ on regular metric trees, as depending on the properties of the potential $V\ge 0$ and on the value of the large parameter $α$. We obtain conditions on $V$ guaranteeing the behavior $N_-(α)=O(α^p)$ for any given $p\ge 1/2$. For a special class of trees we show that these conditions are not only sufficient but also necessary. For $p>1/2$ the order-sharp estimates involve a (quasi-)norm of $V$ in some `weak' $L_p$- or $\ell_p(L_1)$-space. We show that the results can be easily derived from the ones of an earlier paper by Naimark and the author, Proc. London Math. Soc. (3) 80 (2000), 690-724. The results considerably improve the estimates found in the recent paper by Ekholm, Frank, and Kovařík, arXive:0710.5500.
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