Helffer–Sjöstrand formula explained

In mathematics, more specifically, in functional analysis, the Helffer-Sjöstrand formula is a formula for computing a function of a self-adjoint operator.

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 .^[1]

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 ^[2]

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

References

• Lecture notes on Weyl's law

Further reading

• Spectral Measures: Helffer-Sjöstrand

Notes and References

1. Dimassi, M., & Sjöstrand, J. (1999). Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series (268). Cambridge University Press. Chapter 8. ISBN 9780511662195.
2. Hörmander, L. (1983). The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer Verlag. Theorem 3.1.11. ISBN 9783540123274.