Fedosov trace formula

An asymptotic formula as $h \rightarrow 0$ for the "localized" trace of the exponential of a Hamiltonian $H ( t )$. The leading terms of this expansion can be calculated in terms of the fixed points of the classical Hamiltonian flow associated to $H$ (provided that it has only isolated fixed points, see below). Explicitly,

\begin{equation*} \operatorname { Tr } [ A \operatorname { exp } ( - i h ^ { - 1 } H ( t ) ) ] = \sum _ { k = 1 } ^ { n } a _ { 0 } ( x _ { k } ) d _ { k } e ^ { b _ { k } } + O ( h ). \end{equation*}

Here, the meaning of $x _ { k }$, $d _ { k }$ and $b _ { k }$ is the following. First, $A$ is a pseudo-differential operator with compactly supported Weyl symbol (cf. also Symbol of an operator). Let $H _ { 0 }$ and $H _ { 1 }$ be the homogeneous components of $H$, and denote by $f _ { t }$ the Hamiltonian flow associated to $H _ { 0 }$ (cf. also Hamiltonian system). The formula above is proved under the assumption that, on the support of $A$, the flow $f _ { t }$ has only isolated fixed points, denoted by $x _ { 1 } , \ldots , x _ { n }$. Then $d _ { k } = \operatorname { det } ( 1 - f _ { t } ^ { \prime } ( x _ { k } ) ) ^ { 1 / 2 }$ and $b _ { k } = - i h ^ { - 1 } H _ { 0 } ( x _ { k } ) t - i H _ { 1 } ( x _ { k } ) t$. See [a1].

References

[a1]	B. Fedosov, "Trace formula for Schrödinger operator" Russian J. Math. Phys. , 1 (1993) pp. 447–463