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 $N\geq 0$, 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 ${\bar {\partial }}_{z}:={\frac {1}{2}}(\partial _{Re(z)}+i\partial _{Im(z)})$.