Namespaces
Variants
Actions

Difference between revisions of "Fedosov trace formula"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 18 formulas out of 18 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
 
Line 1: Line 1:
An asymptotic formula as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300401.png" /> for the "localized"  trace of the exponential of a Hamiltonian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300402.png" />. The leading terms of this expansion can be calculated in terms of the fixed points of the classical Hamiltonian flow associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300403.png" /> (provided that it has only isolated fixed points, see below). Explicitly,
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300404.png" /></td> </tr></table>
+
Out of 18 formulas, 18 were replaced by TEX code.-->
  
Here, the meaning of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300407.png" /> is the following. First, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300408.png" /> is a [[Pseudo-differential operator|pseudo-differential operator]] with compactly supported Weyl symbol (cf. also [[Symbol of an operator|Symbol of an operator]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f1300409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004010.png" /> be the homogeneous components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004011.png" />, and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004012.png" /> the Hamiltonian flow associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004013.png" /> (cf. also [[Hamiltonian system|Hamiltonian system]]). The formula above is proved under the assumption that, on the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004014.png" />, the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004015.png" /> has only isolated fixed points, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004016.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130040/f13004018.png" />. See [[#References|[a1]]].
+
{{TEX|semi-auto}}{{TEX|done}}
 +
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|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><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>
+
<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
This article was adapted from an original article by Victor Nistor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article