Helffer–Sjöstrand formula explained

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

f \in C ^{inf ty} _{0} ( R ) , then we can find a function \tilde f \in C ^{inf ty} _{0} ( C ) such that \tilde{f}| _{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 .^[2]

The formula

If

f \in C ^{inf ty( R )} _{0} and A is a self-adjoint operator on a Hilbert space, thenf(A) = (1)/(\pi) \int _{C} \bar{\partial} \tilde{f}(z) (z - A) ^{-1} dx dy ^[3]

where

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

See also

• Cauchy's integral formula

Further reading

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

Notes and References

1. Book: Mbarek, Aiman . Helffer-Sjöstrand formula for Unitary Operators . June 2015 . . 2015 . hal-01163568.
2. Book: Dimassi, M. . Spectral Asymptotics in the Semi-Classical Limit . Sjostrand . J. . 1999 . Cambridge University Press . 978-0-521-66544-5 . London Mathematical Society Lecture Note Series . Cambridge . 10.1017/CBO9780511662195.
3. Book: Hörmander, Lars . Lars Hörmander . The Analysis of Linear Partial Differential Operators I . 1983 . . Classics in Mathematics . 2003 . en . 10.1007/978-3-642-61497-2.