See here. I present Theorem 6 and Corollary 7 as follows.
Theorem 6. For large $s$, $\Delta_s$ and $H$ have the same number of eigenvalues in the interval $[0, s^{-2/5}]$.
Corollary 7. $\dim \ker (\Delta_s) \le \# \text{ (eigenvalues of }H\text{ in }[0, s^{-2/5}]\text{).}$
It seems to me these two results are pulled out of nowhere in the text. Can someone explain to me the intuition behind them?
Best Answer
Why not go to Witten's actual paper? He explains it well, and here is a sketch:
The operator $\Delta_s$ depends on a Morse function $f:M\to\mathbb{R}$, and it takes on the form $$\Delta_s = \Delta+ s^2|df|^2 + s\cdot F$$ where $F$ is something depending on derivatives of $f$ (basically the Hessian). The point is that for $s>>0$ this is dominated by $|df|^2$, except at the critical points (where $df$ vanishes). So the collection of 'small' eigenvectors of $\Delta_s$ will be localized around those critical points (to compensate). We can thus approximate $\Delta_s$ near those critical points to analyze the eigenvalues (and that approximation is your "model operator" $H$). Your stated theorem is quantifying this approximation. Your stated corollary follows because those eigenvalues actually vanish for large $s$.
I forgot where I learned the rigorous approach, but to isolate around these critical points a partition of unity is used, where the size of the supports scales as $s^{2/5}$ I think.
There is "physics intuition" behind this approach (also explained in Witten's paper), but mathematically it is a restatement of the above. Roughly, the "Hamiltonian" $\Delta_s$ looks like the "potential energy" $s^2|df|^2$ for large $s$, and so this system is modeled by a bunch of "harmonic oscillators" for which we know the energy states.