Draft:Helffer–Sjöstrand formula

Helffer-Sjöstrand Formula
The Helffer-Sjöstrand formula is a useful formula for computing a function of a self-adjoint operator.

Background
If $$ f \in C_0^\infty (\mathbb{R}) $$, then we can find a function $$ \tilde f \in C_0^\infty (\mathbb{C}) $$ such that $$ \tilde{f}|_{\mathbb{R}} = f $$, and for each $$ N \ge 0$$, there exists a $$ C_N > 0$$ such that

$$ |\bar{\partial} \tilde{f}| \leq C_N |\operatorname{Im} z|^N. $$

Such a function $$\tilde{f} $$ is called an almost analytic extension of $$ f$$.

The Formula
If $$f \in C_0^\infty(\mathbb{R})$$ and $$A$$ is a self-adjoint operator on a Hilbert space, then

$$ f(A) = \frac{1}{\pi} \int_{\mathbb{C}} \bar{\partial} \tilde{f}(z) (z - A)^{-1} \, dx \, dy $$

where $$ \tilde{f} $$ is an almost analytic extension of $$ f $$, and $$ \bar{\partial}_z := \frac{1}{2}(\partial_{Re(z)} + i\partial_{Im(z)}) $$.