Helffer–Sjöstrand formula

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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.{{Cite book |last=Mbarek |first=Aiman |url=https://hal.science/hal-01163568/ |title=Helffer-Sjöstrand formula for Unitary Operators |date=June 2015 |publisher=HAL (open archive) |ref=hal-01163568}}

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 $f$ .{{Cite book |last1=Dimassi |first1=M. |url=https://www.cambridge.org/core/books/spectral-asymptotics-in-the-semiclassical-limit/1E49D44B72B94C4ED55304A1C1CD7E9F |title=Spectral Asymptotics in the Semi-Classical Limit |last2=Sjostrand |first2=J. |date=1999 |publisher=Cambridge University Press |isbn=978-0-521-66544-5 |series=London Mathematical Society Lecture Note Series |location=Cambridge |doi=10.1017/CBO9780511662195}}

The formula

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

$f(A) = \frac{1}{\pi} \int_{\mathbb{C}} \bar{\partial} \tilde{f}(z) (z - A)^{-1} \, dx \, dy$ {{Cite book |last=Hörmander |first=Lars |author-link=Lars Hörmander |url=https://link.springer.com/book/10.1007/978-3-642-61497-2 |title=The Analysis of Linear Partial Differential Operators I |date=1983 |publisher=Springer Nature |series=Classics in Mathematics |publication-date=2003 |language=en |doi=10.1007/978-3-642-61497-2|isbn=978-3-540-00662-6 }}

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

Helffer–Sjöstrand formula

Background

The formula

See also

References

Further reading