Theorem 2 (Helffer–Sjöstrand)

Let $u$ be a self-adjoint partial operator on E. Let $f\in \mathscr{D}(\mathbf{R})$ and $\widetilde{f}\in \mathscr{D}(\mathbf{C})$ be an almost-analytic extension of $f$. Let $h$ be the mapping from $\mathbf{C}$ into $\mathscr{L}(E)$ defined by

if $\lambda \in \mathbf{C}-\mathbf{R}$ and $h(\lambda ) = 0$ if $\lambda \in \mathbf{R}$. Then $h$ is $\mu$-integrable on $\mathbf{C}$ and

The mapping $h$ is measurable and its support is compact. Since $u$ is self-adjoint, one has $\|R(u, \lambda )\|\leqslant |\mathscr{I}(\lambda )|^{-1}$ for every $\lambda \in \mathbf{C}-\mathbf{R}$ (Prop. 17 of IV, p. 248). The mapping $h$ is therefore bounded by formula (13), and consequently it is integrable over $\mathbf{C}$.

Let $\varepsilon \in \mathbf{R}^*_+$. Denote by $F^+_{\varepsilon}$ (resp. $F^-_{\varepsilon}$) the closed set in $\mathbf{C}$ of the $\lambda \in \mathbf{C}$ such that $\mathscr{I}(\lambda )\geqslant \varepsilon$ (resp. $\mathscr{I}(\lambda )\leqslant -\varepsilon$ ). One has

$\int_{\mathbf{C}}h(\lambda )d\mu(\lambda ) =$ lim$_{\varepsilon\rightarrow 0}(\int_{F^+_{\varepsilon}}h(\lambda )d\mu(\lambda ) +\int_{F^-_{\varepsilon}}h(\lambda )d\mu(\lambda ))$.

Let $r >0$ be such that the support of $h$ is contained in $C = [-r, r]^2$. Let us denote by $R_{\varepsilon}^+$ the box $[-2r,2r]\times [\varepsilon ,2r]$ in $\mathbf{C}$. It is a locally polyhedral subset of $\mathbf{C}$ (VAR, R2, 11.3, p. 48). It satisfies the following conditions:

(i) One has $R^+_{\varepsilon}\subset 2C\cap F^+_{\varepsilon}$;

(ii) The set $R^+_{\varepsilon}$ contains the intersection of $F^+_{\varepsilon}$ and the support of $h$;

(iii) The regular boundary $\partial R^+_{\varepsilon}$ (VAR, R2, 11.3.2, p. 49) contains the segment $S_{\varepsilon}= [-r, r] +i\varepsilon \subset \mathbf{C}$;

(iv) One has $h(\lambda ) = 0$ if $\lambda \in \partial R_{\varepsilon}^+-S_{\varepsilon}$.

Let $d\lambda$ (resp. $d\lambda$ ) denote the differential 1-form on $\mathbf{C}$ which is the differential of the identity mapping of $\mathbf{C}$ (resp. of complex conjugation). Let $g$ denote the function on $\mathbf{C}-\mathbf{R}$ with values in $\mathscr{L}(E)$ such that $g(\lambda ) =\widetilde{f}(\lambda )R(u, \lambda )$ for $\lambda \in \mathbf{C}-\mathbf{R}$. Put $\omega =g d\lambda$; it is a differential 1-form on $\mathbf{C}-\mathbf{R}$, with compact support and with values in $\mathscr{L}(E)$. Since the resolvent of $u$ is holomorphic (Prop. 14 of IV, p. 246), one has

The vector measure associated with $d\omega$ (VAR, R2, 10.4.3, p. 43) is the measure with density $-2ih$ with respect to Lebesgue measure. Applying Stokes's formula to the locally polyhedral set $R_{\varepsilon}^+$ and to the differential form $\omega$ (VAR, R2, 11.3.4, p. 49), one therefore obtains

whence

Reasoning analogously for $F^-_{\varepsilon}$, one obtains

and one concludes that the integral of $h$ over $\mathbf{C}$ is the limit as $\varepsilon \rightarrow 0$ of

By Prop. 7, one has $v_{\varepsilon}=\pi f_{\varepsilon}(u)$, where $f_{\varepsilon}$ is the function defined on $\mathbf{R}$ by

As $f_{\varepsilon}\rightarrow f$ when $\varepsilon \rightarrow 0$ in $\mathscr{C}_b(\mathbf{R})$ (Lemma 7), the endomorphism $v_{\varepsilon}=\pi f_{\varepsilon}(u)$ converges to $\pi f(u)$ in $\mathscr{L}(E)$ (Prop. 5 of IV, p. 275). The theorem is proved.