The IMS formula of localization cited in Ch. 3 of Schroedinger Operators with Application to Quantum Mechanics and Global Geometry by H. Cycon, R. Froese, W. Kirsch, and B. Simon depends on the partition of unity of $\mathcal{R}^{n}$ with functions $J_{a}\in C_{0}^{\infty}(\mathcal{R}^{n})$ indexed by elements $a$ of a set $A$ and having these properties
- $0 \leq J_{a} \leq 1$
- $\sum_{a}J_{a}^{2} = 1$
- $J_{a}$ is locally finite
- $\sup_{x\in \mathcal{R}^{n}}\sum_{a}|\nabla J_{a}(x)|^{2} < \infty$
I am focused on the item 4 - this condition is hard to verify, and I am seeking references regarding the questions below for a manifold $M$. I have a partition of unity $J_{a}$ whose cover contains balls of a fixed radius $r_0$.
- Can I make the set $A$ countable?
- Suppose I have a uniform partition with $|\nabla J_{a}(x)| < C$ with some constant $C$. Are there any conditions on $M$ ensuring that we have a limited number of not more than $N$ open covers of any point $x$? I need this for the condition 4 to hold?
Thank you!