图上具有稀疏位势的Schr dinger算子的谱估计
Spectral estimates for the Schrödinger operators with sparse potentials on graphs
摘要：
在\cite{RS09}中提出的格$\Z^d,\ d>2$的“稀疏势”的构造被推广到一类全局维数为数$D>2$的组合图和度量图。对于这类图上的薛定谔算子$-\D-\a V$，在具有稀疏势$V$的情况下，我们研究了$-\D-\a V$的负特征值个数$N_-(-\D-\a V)$的行为（作为$\a\to\infty$）.我们表明，通过稀疏的潜力，可以实现任何规定的渐近行为的$N_-(-\D-\a V)$下非常温和的正则性假设。类似的构造也适用于格$\Z^2$，其中D=2。
英文摘要：
The construction of "sparse potentials", suggested in \cite{RS09} for the lattice $\Z^d,\ d>2$, is extended to a wide class of combinatorial and metric graphs whose global dimension is a number $D>2$. For the Schrödinger operator $-\D-\a V$ on such graphs, with a sparse potential $V$, we study the behavior (as $\a\to\infty$) of the number $N_-(-\D-\a V)$ of negative eigenvalues of $-\D-\a V$. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of $N_-(-\D-\a V)$ under very mild regularity assumptions. A similar construction works also for the lattice $\Z^2$, where D=2.
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