Helffer–Sjöstrand formula

The Helffer–Sjöstrand formula is a mathematical tool used in spectral theory and functional analysis to represent functions of self-adjoint operators. Named after Bernard Helffer and Johannes Sjöstrand, this formula provides a way to calculate functions of operators without requiring the operator to have a simple or explicitly known spectrum. It is especially useful in quantum mechanics, condensed matter physics, and other areas where understanding the properties of operators related to energy or observables is important.[1]

Background

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]

The formula

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

See also

• Cauchy's integral formula

References

1. Mbarek, Aiman (June 2015). Helffer-Sjöstrand formula for Unitary Operators. HAL (open archive).
2. Dimassi, M.; Sjostrand, J. (1999). Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511662195. ISBN 978-0-521-66544-5.
3. Hörmander, Lars (1983). The Analysis of Linear Partial Differential Operators I. Classics in Mathematics. Springer Nature (published 2003). doi:10.1007/978-3-642-61497-2. ISBN 978-3-540-00662-6.

Further reading

• Lecture notes on Weyl's law
• Spectral Measures: Helffer-Sjöstrand