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Helffer-Sjöstrand Formula
editThe Helffer-Sjöstrand formula is a useful formula for computing a function of a self-adjoint operator.
Background
editIf , then we can find a function such that , and for each , there exists a such that
Such a function is called an almost analytic extension of [1].
The Formula
editIf and is a self-adjoint operator on a Hilbert space, then
where is an almost analytic extension of , and .
References
edit- ^ 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.