Helffer–Sjöstrand formula

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

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

Such a function ${\displaystyle {\tilde {f}}}$ is called an almost analytic extension of ${\displaystyle f}$.^[2]

If ${\displaystyle f\in C_{0}^{\infty }(\mathbb {R} )}$ and ${\displaystyle A}$ is a self-adjoint operator on a Hilbert space, then

${\displaystyle f(A)={\frac {1}{\pi }}\int _{\mathbb {C} }{\bar {\partial }}{\tilde {f}}(z)(z-A)^{-1}\,dx\,dy}$^[3]

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

• Cauchy's integral formula