Difference between revisions of "Fedosov trace formula"
From Encyclopedia of Mathematics
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| + | 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, | ||
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| + | \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*} | ||
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| + | Here, the meaning of $x _ { k }$, $d _ { k }$ and $b _ { k }$ is the following. First, $A$ is a [[Pseudo-differential operator|pseudo-differential operator]] with compactly supported Weyl symbol (cf. also [[Symbol of an operator|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|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 [[#References|[a1]]]. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> B. Fedosov, "Trace formula for Schrödinger operator" ''Russian J. Math. Phys.'' , '''1''' (1993) pp. 447–463</td></tr></table> |
Latest revision as of 17:03, 1 July 2020
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 |
How to Cite This Entry:
Fedosov trace formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fedosov_trace_formula&oldid=50490
Fedosov trace formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fedosov_trace_formula&oldid=50490
This article was adapted from an original article by Victor Nistor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article