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
, then we can find a function
such that
, and for each
, there exists a
such that
|\bar{\partial}\tilde{f}|\leqCN|\operatorname{Im}z|N.
Such a function
is called an almost analytic extension of
.
[1] The Formula
If
and
is a self-adjoint operator on a Hilbert space, then
f(A)=
\intC\bar{\partial}\tilde{f}(z)(z-A)-1dxdy
where
is an almost analytic extension of
, and
\bar{\partial}z:=
(\partialRe(z)+i\partialIm(z))
.
See also
References
Further reading
Notes and References
- 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.
- Hörmander, L. (1983). The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer Verlag. Theorem 3.1.11. ISBN 9783540123274.
