The following argument is contained in Benzi et al. (2013).
Proposition 1 (Combes–Thomas estimate for banded matrices) Suppose \(H\) is banded: \(H_{\ell k} = 0\) for all \(r_{\ell k} \geq m\). Then, for \(z \notin \sigma(H)\), \[ \left| (z - H)^{-1}_{\ell k} \right| \lesssim e^{-\frac{\eta}{m} \, r_{\ell k}}, \qquad \text{where } \eta \sim \mathrm{dist}\big( z, \sigma(H) \big). \]
Proof. Since \(H_{\ell k} = 0\) for all \(r_{\ell k} \geq m\), we have
\[ \begin{align} [H^n]_{\ell k} = \sum_{\ell_1,\dots,\ell_{n-1}} H_{\ell \ell_1} H_{\ell_1\ell_2} \cdots H_{\ell_{n-1} k} = 0 \end{align} \]
for all \(r_{\ell k} \geq nm\).
Therefore, for all \(n \leq \frac1m r_{\ell k}\), we have
\[ \begin{align} \left| (z - H)^{-1}_{\ell k} \right| % &= \inf_{P_n \in \mathcal P_n} \left| \left[ (z - H)^{-1} - P_n(H) \right]_{\ell k} \right| \nonumber\\ % &\leq \inf_{P_n \in \mathcal P_n} \left\| (z - \cdot)^{-1} - P_n \right\|_{L^\infty(\sigma(H))} \nonumber\\ % &\lesssim e^{-\eta \, n} \approx e^{-\frac{\eta}m \, r_{\ell k} } \end{align} \]
where \(\eta \sim \mathrm{dist}\big( z, \sigma(H) \big)\).
Example 1 Here, we let \(p_n\) be the polynomial interpolation of \(x \mapsto (z - x)^{-1}\) on Chebyshev nodes \(\{ \cos \frac{j\pi}{n} \}_{j=0}^n\) and plot the \(L^\infty\) error on \([-1,1]\) for different choices of \(z\) (this gives an upper bound on the locality of the resolvent as above):
This is \(O(e^{-\eta \, n})\) decay with
| \(\mathrm{dist}(z, \sigma(H))\) | \(\eta\) |
|---|---|
| 1.0 | 0.880853 |
| 0.5 | 0.479982 |
| 0.25 | 0.24404 |
| 0.125 | 0.116845 |
| 0.0625 | 0.0505203 |
i.e. it roughly halves when the distance to the “spectrum” is halved.