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| − | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w1201101.png" /> be a classical Hamiltonian (cf. also [[Hamilton operator|Hamilton operator]]) defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w1201102.png" />. The Weyl quantization rule associates to this function the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w1201103.png" /> defined on functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w1201104.png" /> as
| + | <!--This article has been texified automatically. Since there was no Nroff source code for this article, |
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| | + | If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. |
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| − | <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/w/w120/w120110/w1201105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
| + | Out of 284 formulas, 257 were replaced by TEX code.--> |
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| − | <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/w/w120/w120110/ w1201106.png" /> </td> </tr></table> | + | {{TEX|semi-auto}}{{TEX|partial}} |
| | + | Let $a ( x , \xi )$ be a classical Hamiltonian (cf. also [[Hamilton operator|Hamilton operator]]) defined on $\mathbf{R} ^ { n } \times \mathbf{R} ^ { n }$. The Weyl quantization rule associates to this function the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/ w1201103.png"/> defined on functions $u ( x )$ as |
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| − | For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w1201107.png" />, with
| + | \begin{equation} \tag{a1} ( a ^ { w } u ) ( x ) = \end{equation} |
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| − | <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/w/w120/w120110/w1201108.png" /></td> </tr></table>
| + | \begin{equation*} = \int \int e ^ { 2 i \pi ( x - y ) . \xi } a \left( \frac { x + y } { 2 } , \xi \right) u ( y ) d y d \xi. \end{equation*} |
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| − | whereas the classical quantization rule would map the Hamiltonian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w1201109.png" /> to the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011010.png" />. A nice feature of the Weyl quantization rule, introduced in 1928 by H. Weyl [[#References|[a12]]], is the fact that real Hamiltonians get quantized by (formally) self-adjoint operators. Recall that the classical quantization of the Hamiltonian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011011.png" /> is given by the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011012.png" /> acting on functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011013.png" /> by
| + | For instance, $( x . \xi ) ^ { w } = ( x . D _ { x } + D _ { x } .x ) / 2$, with |
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| − | <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/w/w120/w120110/w12011014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
| + | \begin{equation*} D _ { x } = \frac { 1 } { 2 i \pi } \frac { \partial } { \partial x }, \end{equation*} |
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| − | where the [[Fourier transform|Fourier transform]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011015.png" /> is defined by
| + | whereas the classical quantization rule would map the Hamiltonian $x \cdot \xi$ to the operator $x . D _ { x }$. A nice feature of the Weyl quantization rule, introduced in 1928 by H. Weyl [[#References|[a12]]], is the fact that real Hamiltonians get quantized by (formally) self-adjoint operators. Recall that the classical quantization of the Hamiltonian $a ( x , \xi )$ is given by the operator $\operatorname{Op} ( a )$ acting on functions $u ( x )$ by |
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| − | <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/w/w120/w120110/w12011016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
| + | \begin{equation} \tag{a2} ( \operatorname{Op} ( a ) u ) ( x ) = \int e ^ { 2 i \pi x \cdot \xi } a ( x , \xi ) \hat { u } ( \xi ) d \xi, \end{equation} |
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| − | so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011017.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011018.png" />. In fact, introducing the one-parameter group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011019.png" />, given by the integral formula
| + | where the [[Fourier transform|Fourier transform]] $\widehat{u}$ is defined by |
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| − | <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/w/w120/w120110/w12011020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
| + | \begin{equation} \tag{a3} \hat { u } ( \xi ) = \int e ^ { - 2 i \pi x . \xi } u ( x ) d x, \end{equation} |
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| − | <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/w/w120/w120110/w12011021.png" /></td> </tr></table>
| + | so that $\check{\widehat { u }} = u$, with $\check{v} ( x ) = v ( - x )$. In fact, introducing the one-parameter group $J ^ { t } = \operatorname { exp } 2 i \pi t D _ { x } . D _ { \xi }$, given by the integral formula |
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| | + | \begin{equation} \tag{a4} ( J ^ { t } a ) ( x , \xi ) = \end{equation} |
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| | + | \begin{equation*} = | t | ^ { - n } \int \int e ^ { - 2 i \pi t ^ { - 1 } y . \eta } { a ( x + y , \xi + \eta ) d y d \eta }, \end{equation*} |
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| | one sees that | | one sees that |
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| − | <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/w/w120/w120110/w12011022.png" /></td> </tr></table>
| + | \begin{equation*} ( \operatorname{Op} ( J ^ { t } a ) u ) ( x ) = \end{equation*} |
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| − | <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/w/w120/w120110/w12011023.png" /></td> </tr></table>
| + | \begin{equation*} = \int \int e ^ { 2 i \pi ( x - y ) . \xi } a ( ( 1 - t ) x + t y , \xi ) u ( y ) d y d \xi. \end{equation*} |
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| − | In particular, one gets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011024.png" />. Moreover, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011025.png" /> one obtains | + | In particular, one gets $a ^ { w } = \text{Op} ( J ^ { 1 / 2 } a )$. Moreover, since $( \operatorname{Op} ( a ) ) ^ { * } = \operatorname{Op} ( J \overline { a } )$ one obtains |
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| − | <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/w/w120/w120110/w12011026.png" /></td> </tr></table>
| + | \begin{equation*} ( a ^ { w } ) ^ { * } = \operatorname { Op } \left( J ( \overline { ( J ^ { 1 / 2 } a ) } \right) = \operatorname { Op } ( J ^ { 1 / 2 } \overline { a } ) = ( \overline { a } ) ^ { w }, \end{equation*} |
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| − | yielding formal self-adjointness for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011027.png" /> (cf. also [[Self-adjoint operator|Self-adjoint operator]]). | + | yielding formal self-adjointness for real $a$ (cf. also [[Self-adjoint operator|Self-adjoint operator]]). |
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| | ==Wigner functions.== | | ==Wigner functions.== |
| | Formula (a1) can be written as | | Formula (a1) can be written as |
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| − | <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/w/w120/w120110/w12011028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
| + | \begin{equation} \tag{a5} ( a ^ { w } u , v ) = \int \int a ( x , \xi ) {\cal H} ( u , v ) ( x , \xi ) d x d \xi, \end{equation} |
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| − | where the Wigner function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011029.png" /> is defined as | + | where the Wigner function $\mathcal{H}$ is defined as |
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| − | <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/w/w120/w120110/w12011030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
| + | \begin{equation} \tag{a6} \mathcal{H} ( u , v ) ( x , \xi ) = \end{equation} |
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| − | <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/w/w120/w120110/w12011031.png" /></td> </tr></table>
| + | \begin{equation*} = \int u \left( x + \frac { y } { 2 } \right) \overline{v} \left( x - \frac { y } { 2 } \right) e ^ { - 2 i \pi y \cdot \xi } d y. \end{equation*} |
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| − | The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011032.png" /> is sesquilinear continuous from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011033.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011034.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011035.png" /> makes sense for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011036.png" /> (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011038.png" /> stands for the anti-dual): | + | The mapping $( u , v ) \mapsto \mathcal{H} ( u , v )$ is sesquilinear continuous from $\mathcal{S} ( \mathbf{R} ^ { n } ) \times \mathcal{S} ( \mathbf{R} ^ { n } )$ to $\mathcal{S} ( \mathbf{R} ^ { 2 n } )$, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011035.png"/> makes sense for $a \in {\cal S} ^ { \prime } ( {\bf R} ^ { 2 n } )$ (here, $u , v \in \mathcal{S} ( \mathbf{R} ^ { n } )$ and $\mathcal{S} ^ { * }$ stands for the anti-dual): |
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| − | <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/w/w120/w120110/w12011039.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center; " valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011039.png"/></td> </tr></table> |
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| | The Wigner function also satisfies | | The Wigner function also satisfies |
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| − | <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/w/w120/w120110/w12011040.png" /></td> </tr></table>
| + | \begin{equation*} \| \mathcal{H} ( u , v ) \| _ { L ^ { 2 } ( \mathbf{R} ^{n}) } = \| u \|_ { L ^ { 2 } ( \mathbf{R} ^{n}) } \| v \| _ { L ^ { 2 } ( \mathbf{R} ^{n}) } , \end{equation*} |
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| − | <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/w/w120/w120110/w12011041.png" /></td> </tr></table>
| + | \begin{equation*} \mathcal{H} ( u , v ) ( x , \xi ) = 2 ^ { n } \langle \sigma _ { x , \xi }u , v \rangle _ { L^2 ( \mathbf{R} ^ { n } )} , ( \sigma _ { x , \xi} u ) ( y ) = u ( 2 x - y ) \operatorname { exp } ( - 4 i \pi ( x - y ) . \xi). \end{equation*} |
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| − | and the phase symmetries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011042.png" /> are unitary and self-adjoint operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011043.png" />. Also ([[#References|[a10]]], [[#References|[a12]]]), | + | and the phase symmetries $\sigma _{X}$ are unitary and self-adjoint operators on $L ^ { 2 } ( \mathbf{R} ^ { n } )$. Also ([[#References|[a10]]], [[#References|[a12]]]), |
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| − | <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/w/w120/w120110/w12011044.png" /></td> </tr></table>
| + | \begin{equation*} \alpha ^ { w } = \int _ { \mathbf{R} ^ { 2 n } } a ( X ) 2 ^ { n } \sigma _ { X } d X = \end{equation*} |
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| − | <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/w/w120/w120110/w12011045.png" /></td> </tr></table>
| + | \begin{equation*} = \int _ { \mathbf{R} ^ { 2 n } } \hat { a } ( \Xi ) \operatorname { exp } ( 2 i \pi \Xi . M ) d \Xi , $ \end{equation*} |
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011046.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011047.png" />). These formulas give, in particular, | + | where $\Xi \cdot M = \hat{x} \cdot x + \widehat { \xi } \cdot D _{x}$ (here $ \Xi = ( \hat { x } , \hat { \xi } )$). These formulas give, in particular, |
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| − | <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/w/w120/w120110/w12011048.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center; " valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011048.png"/></td> </tr></table> |
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011049.png" /> stands for the space of bounded linear mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011050.png" /> into itself. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011051.png" /> is in the Hilbert–Schmidt class (cf. also [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011052.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011053.png" /> and <img align= "absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011054.png" />. To get this, it suffices to notice the relationship between the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011056.png" /> and its distribution kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011057.png" />: | + | where $\mathcal{L} ( L ^ { 2 } )$ stands for the space of bounded linear mappings from $L ^ { 2 } ( \mathbf{R} ^ { n } )$ into itself. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011051.png"/> is in the Hilbert–Schmidt class (cf. also [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]]) if and only if $a$ belongs to $L ^ { 2 } ( \mathbf{R} ^ { 2 n } )$ and $\| a\| _{\text{HS}} = \| a \| _ { L } 2 _ { ( \mathbf{R} ^ { 2 n }) } $. To get this, it suffices to notice the relationship between the symbol $a$ of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011056.png"/> and its distribution kernel $k$: |
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| − | <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/w/w120/w120110/w12011058.png" /></td> </tr></table>
| + | \begin{equation*} \a ( x , \xi ) = \int k \left( x + \frac { t } { 2 } , x - \frac { t } { 2 } \right) e ^ { - 2 i \pi t \xi } d t. \end{equation*} |
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| | The Fourier transform of the Wigner function is the so-called ambiguity function | | The Fourier transform of the Wigner function is the so-called ambiguity function |
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| − | <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/w/w120/w120110/w12011059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
| + | \begin{equation} \tag{a7} {\cal A} ( u , v ) ( \xi , x ) = \int u \left( z - \frac { x } { 2 } \right) \bar{v} \left( z + \frac { x } { 2 } \right) e ^ { - 2 i \pi z . \xi } d z. \end{equation} |
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| − | For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011060.png" />, the Wigner function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011061.png" /> is the Weyl symbol of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011062.png" /> (cf. also [[Symbol of an operator|Symbol of an operator]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011063.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011064.png" /> (Hermitian) dot-product, so that from (a5) one finds | + | For $\varphi , \psi \in L ^ { 2 } ( \mathbf{R} ^ { n} )$, the Wigner function $\mathcal{H} ( \varphi , \psi )$ is the Weyl symbol of the operator $u \mapsto ( u , \psi ) \varphi$ (cf. also [[Symbol of an operator|Symbol of an operator]]), where $( u , \psi )$ is the $L^{2}$ (Hermitian) dot-product, so that from (a5) one finds |
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| − | <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/w/w120/w120110/w12011065.png" /></td> </tr></table>
| + | \begin{equation*} (u, \psi ) _ { L ^ { 2 } ( {\bf R} ^ { n } ) } ( \varphi , u ) _ { L ^ { 2 } ( {\bf R} ^ { n } ) } = ( {\cal H} ( u , v ) , {\cal H} ( \psi , \varphi ) ) _ { L ^ { 2 } ( {\bf R} ^ { 2 n } ) }. \end{equation*} |
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| | As is shown below, the symplectic invariance of the Weyl quantization is actually its most important property. | | As is shown below, the symplectic invariance of the Weyl quantization is actually its most important property. |
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| | ==Symplectic invariance.== | | ==Symplectic invariance.== |
| − | Consider a finite-dimensional real [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011066.png" /> (the configuration space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011067.png" />) and its dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011068.png" /> (the momentum space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011069.png" />). The phase space is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011070.png" />; its running point will be denoted, in general, by a capital letter (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011071.png" />). The symplectic form (cf. also [[Symplectic connection|Symplectic connection]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011072.png" /> is given by | + | Consider a finite-dimensional real [[Vector space|vector space]] $E$ (the configuration space ${\bf R} _ { x } ^ { n }$) and its dual space $E ^ { * }$ (the momentum space ${\bf R} _ { \xi } ^ { n }$). The phase space is defined as $\Phi = E \oplus E ^ { * }$; its running point will be denoted, in general, by a capital letter ($X = ( x , \xi ) , Y = ( y , \eta )$). The symplectic form (cf. also [[Symplectic connection|Symplectic connection]]) on $\Phi$ is given by |
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| − | <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/w/w120/w120110/w12011073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
| + | \begin{equation} \tag{a8} [ ( x , \xi ) , ( y , \eta ) ] = \langle \xi , y \rangle _ { E ^{ * } , E } - \langle \eta , x \rangle _ { E ^{ * } , E }, \end{equation} |
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011074.png" /> stands for the bracket of duality. The [[Symplectic group|symplectic group]] is the subgroup of the linear group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011075.png" /> preserving (a8). With | + | where $\langle \cdot , \cdot \rangle _ { E ^ { * } , E}$ stands for the bracket of duality. The [[Symplectic group|symplectic group]] is the subgroup of the linear group of $\Phi$ preserving (a8). With |
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| − | <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/w/w120/w120110/w12011076.png" /></td> </tr></table>
| + | \begin{equation*} \sigma = \left( \begin{array} { c c } { 0 } & { \operatorname{Id} ( E ^ { * } ) } \\ { - \operatorname{Id} ( E ) } & { 0 } \end{array} \right), \end{equation*} |
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| − | for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011077.png" /> one has | + | for $X , Y \in \Phi$ one has |
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| − | <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/w/w120/w120110/w12011078.png" /></td> </tr></table>
| + | \begin{equation*} [ X , Y ] = \langle \sigma X , Y \rangle _ { \Phi ^ { * } , \Phi }, \end{equation*} |
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| | so that the equation of the symplectic group is | | so that the equation of the symplectic group is |
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| − | <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/w/w120/w120110/w12011079.png" /></td> </tr></table>
| + | \begin{equation*} A ^ { * } \sigma A = \sigma. \end{equation*} |
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| − | One can describe a set of generators for the symplectic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011080.png" />, identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011081.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011082.png" />: the mappings | + | One can describe a set of generators for the symplectic group $\operatorname{Sp} ( n )$, identifying $\Phi$ with $\mathbf{R} _ { x } ^ { n } \times \mathbf{R} _ { \xi } ^ { n }$: the mappings |
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| − | i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011083.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011084.png" /> is an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011085.png" />; | + | i) $( x , \xi ) \mapsto ( T x , \square ^ { t } T ^ { - 1 } \xi )$, where $T$ is an automorphism of $E$; |
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| − | ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011086.png" /> and the other coordinates fixed; | + | ii) $( x _ { k } , \xi _ { k } ) \mapsto ( \xi _ { k } , - x _ { k } )$ and the other coordinates fixed; |
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| − | iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011087.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011088.png" /> is symmetric from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011089.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011090.png" />. One then describes the metaplectic group, introduced by A. Weil [[#References|[a11]]]. The metaplectic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011091.png" /> is the subgroup of the group of unitary transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011092.png" /> generated by | + | iii) $( x , \xi ) \mapsto ( x , \xi + S x )$, where $S$ is symmetric from $E$ to $E ^ { * }$. One then describes the metaplectic group, introduced by A. Weil [[#References|[a11]]]. The metaplectic group $\operatorname{ Mp } ( n )$ is the subgroup of the group of unitary transformations of $L ^ { 2 } ( \mathbf{R} ^ { n } )$ generated by |
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| − | j) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011094.png" />; | + | j) $( M _ { T } u ) ( x ) = | \operatorname { det } T \rceil ^ { - 1 / 2 } u ( T ^ { - 1 } x )$, where $T \in \operatorname{GL} ( n , \mathbf{R} )$; |
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| | jj) partial Fourier transformations; | | jj) partial Fourier transformations; |
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| − | jjj) multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011096.png" /> is a symmetric matrix. There exists a two-fold covering (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011097.png" /> of both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w12011099.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110100.png" />) | + | jjj) multiplication by $\operatorname { exp } ( i \pi \langle S x , x \rangle )$, where $S$ is a symmetric matrix. There exists a two-fold covering (the $\pi_{1}$ of both $\operatorname{ Mp } ( n )$ and $\operatorname{Sp} ( n )$ is $\bf Z$) |
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| − | <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/w/w120/w120110/w120110101.png" /></td> </tr></table>
| + | \begin{equation*} \pi : \operatorname { Mp} ( n ) \rightarrow \operatorname { Sp} ( n ) \end{equation*} |
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| − | such that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110104.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110105.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110106.png" /> is their Wigner function, then | + | such that, if $\chi = \pi ( M )$ and $u$, $v$ are in $L ^ { 2 } ( \mathbf{R} ^ { n } )$, while $\mathcal{H} ( u , v )$ is their Wigner function, then |
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| − | <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/w/w120/w120110/w120110107.png" /></td> </tr></table>
| + | \begin{equation*} \mathcal{H} ( M u , M v ) = \mathcal{H} ( u , v ) \circ \chi ^ { - 1 }. \end{equation*} |
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| − | This is Segal's formula [[#References|[a9]]], which can be rephrased as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110109.png" />. There exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110110.png" /> in the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110111.png" /> such that | + | This is Segal's formula [[#References|[a9]]], which can be rephrased as follows. Let $a \in {\cal S} ^ { \prime } ( {\bf R} ^ { 2 n } )$ and $\chi \in \operatorname { Sp } ( n )$. There exists an $M$ in the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110111.png"/> such that |
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| − | <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/w/w120/w120110/w120110112.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
| + | \begin{equation} \tag{a9} ( a \circ \chi ) ^ { w } = M ^ { * } a ^ { w } M. \end{equation} |
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| − | In particular, the images by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110113.png" /> of the transformations j), jj), jjj) are, respectively, i), ii), iii). Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110114.png" /> is the phase translation, <img align= "absmiddle" border="0 " src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110115.png" />, (a9) is fulfilled with <img align= "absmiddle" border="0 " src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110116.png" /> and phase translation given by | + | In particular, the images by $\pi$ of the transformations j), jj), jjj) are, respectively, i), ii), iii). Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110114.png"/> is the phase translation, $\chi ( x , \xi ) = ( x + x _ { 0 } , \xi + \xi _ { 0 } )$, (a9) is fulfilled with $M = \tau _ { x _ { 0 } , \xi _ { 0 }}$ and phase translation given by |
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| − | <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/w/w120/w120110/w120110117.png" /></td> </tr></table>
| + | \begin{equation*} ( \tau _ { x _ { 0 } , \xi _ { 0 } } u ) ( y ) = u ( y - x _ { 0 } ) e ^ { 2 i \pi \langle y - x _ { 0 } / 2 , \xi _ { 0 } \rangle }. \end{equation*} |
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110118.png" /> is the symmetry with respect to <img align="absmiddle" border="0 " src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110119.png" />, <img align="absmiddle" border="0 " src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110120.png" /> in (a9) is, up to a unit factor, the phase symmetry <img align="absmiddle" border="0 " src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110121.png" /> defined above. This yields the following composition formula: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110122.png" /> with | + | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110118.png"/> is the symmetry with respect to $( x _ { 0 } , \xi _ { 0 } )$, $M$ in (a9) is, up to a unit factor, the phase symmetry $\sigma _ { x _ { 0 } , \xi _ { 0 } }$ defined above. This yields the following composition formula: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110122.png"/> with |
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| − | <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/w/w120/w120110/w120110123.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
| + | \begin{equation} \tag{a10} ( a \sharp b ) ( X ) = \end{equation} |
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| − | <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/w/w120/w120110/w120110124.png" /></td> </tr></table>
| + | \begin{equation*} = 2 ^ { 2 n } \int \int e ^ { - 4 i \pi [ X - Y , X - Z ] } { a } ( Y ) b ( Z ) d Y d Z , \end{equation*} |
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| − | with an integral on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110125.png" />. One can compare this with the classical composition formula, | + | with an integral on $\mathbf{R} ^ { 2 n } \times \mathbf{R} ^ { 2 n }$. One can compare this with the classical composition formula, |
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| − | <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/w/w120/w120110/w120110126.png" /></td> </tr></table>
| + | \begin{equation*} \operatorname {Op} ( a ) \operatorname {Op} ( b ) = \operatorname {Op} ( a \circ b ) \end{equation*} |
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| | (cf. (a2)) with | | (cf. (a2)) with |
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| − | <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/w/w120/w120110/w120110127.png" /></td> </tr></table>
| + | \begin{equation*} ( a \circ b ) ( x , \xi ) = \int \int e ^ { - 2 i \pi y . \eta } a ( x , \xi + \eta ) b ( y + x , \xi ) d y d \eta, \end{equation*} |
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| − | with an integral on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110128.png" />. It is convenient to give an asymptotic version of these compositions formulas, e.g. in the semi-classical case. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110129.png" /> be a real number. A smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110130.png" /> defined on <img align="absmiddle" border="0 " src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110131.png" /> is in the symbol class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110133.png" /> if | + | with an integral on $\mathbf{R} ^ { n } \times \mathbf{R} ^ { n }$. It is convenient to give an asymptotic version of these compositions formulas, e.g. in the semi-classical case. Let $m$ be a real number. A smooth function $a ( x , \xi , h )$ defined on $\mathbf{R} _ { x } ^ { n } \times \mathbf{R} _ { \xi } ^ { n } \times ( 0 ,1 ]$ is in the symbol class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110133.png"/> if |
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| − | <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/w/w120/w120110/w120110134.png" /></td> </tr></table>
| + | \begin{equation*} \operatorname { sup } _ {\substack{ ( x , \xi ) \in {\bf R} ^ { 2 n }}, \\{0<h\leq 1} } \left| D _ { x } ^ { \alpha } D _ { \xi } ^ { \beta } a ( x , \xi , h ) \right| h ^ { m - | \beta | } < \infty . \end{equation*} |
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| − | Then one has for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110135.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110136.png" /> the expansion | + | Then one has for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110135.png"/> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110136.png"/> the expansion |
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| − | <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/w/w120/w120110/w120110137.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
| + | \begin{equation} \tag{a11} ( a \sharp b ) ( x , \xi ) = r _ { N } ( a , b ) + \end{equation} |
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| − | <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/w/w120/w120110/w120110138.png" /></td> </tr></table>
| + | \begin{equation*} + \sum _ { 0 \leq k < N } 2 ^ { - k } \sum _ { | \alpha | + | \beta | = k } \frac { ( - 1 ) ^ { | \beta | } } { \alpha ! \beta ! } D _ { \xi } ^ { \alpha } \partial _ { x } ^ { \beta } a D _ { \xi } ^ { \beta } \partial _ { x } ^ { \alpha } b, \end{equation*} |
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| − | with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110139.png" />. The beginning of this expansion is thus | + | with $r _ { N } ( a , b ) \in S _ { \text{scl} } ^ { m _ { 1 } + m _ { 2 } - N}$. The beginning of this expansion is thus |
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| − | <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/w/w120/w120110/w120110140.png" /></td> </tr></table>
| + | \begin{equation*} a b + \frac { 1 } { 2 \iota} \{ a , b \}, \end{equation*} |
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110141.png" /> denotes the [[Poisson brackets|Poisson brackets]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110142.png" />. The sums inside (a11) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110143.png" /> even are symmetric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110144.png" /> and skew-symmetric for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110145.png" /> odd. This can be compared to the classical expansion formula | + | where $\{ a , b \}$ denotes the [[Poisson brackets|Poisson brackets]] and $\iota = 2 \pi {i} $. The sums inside (a11) with $k$ even are symmetric in $a , b$ and skew-symmetric for $k$ odd. This can be compared to the classical expansion formula |
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| − | <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/w/w120/w120110/w120110146.png" /></td> </tr></table>
| + | \begin{equation*} ( a \circ b ) ( x , \xi ) = \sum _ { | \alpha | < N } \frac { 1 } { \alpha ! } D _ { \xi } ^ { \alpha } a \partial _ { x } ^ { \alpha } b + t _ { N } ( a , b ), \end{equation*} |
| | | | |
| − | with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110147.png" />. Moreover, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110148.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110149.png" />, the multiple composition formula gives | + | with $t _ { N } ( a , b ) \in S _ { \text{scl} } ^ { m _ { 1 } + m _ { 2 } - N }$. Moreover, for $a _ { 1 } , \dots , a _ { 2k } $ in $L ^ { 1 } ( \Phi = \mathbf{R} ^ { 2 n } )$, the multiple composition formula gives |
| | | | |
| − | <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/w/w120/w120110/w120110150.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center; " valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110150.png"/></td> </tr></table> |
| | | | |
| − | <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/w/w120/w120110/w120110151.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center; " valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110151.png"/></td> </tr></table> |
| | | | |
| − | <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/w/w120/w120110/w120110152.png" /></td> </tr></table>
| + | \begin{equation*} .d Y _ { 1 } \ldots d Y _ { 2 k }, \end{equation*} |
| | | | |
| − | and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110153.png" />, | + | and if $a _ { 2 k + 1 } \in L ^ { 1 } ( \Phi )$, |
| | | | |
| − | <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/w/w120/w120110/w120110154.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center; " valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110154.png"/></td> </tr></table> |
| | | | |
| − | <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/w/w120/w120110/w120110155.png" /></td> </tr></table>
| + | \begin{equation*} = 2 ^ { 2 n k } \int _ { \Phi ^ { 2 k } } a _ { 1 } ( Y _ { 1 } ) \ldots a _ { 2 k } ( Y _ { 2 k } ) \text{..} a _ { 2 k + 1 } \left( X + \sum _ { 1 \leq j < l \leq 2 k } ( - 1 ) ^ { j + l } ( Y _ { j } - Y _ { l } ) \right). \end{equation*} |
| | | | |
| − | <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/w/w120/w120110/w120110156.png" /></td> </tr></table>
| + | \begin{equation*} .\operatorname { exp } 4 i \pi \sum _ { 1 \leq j < l \leq 2 k } ( - 1 ) ^ { j + l } [ X - Y _ { j } , X - Y _ { l } ] .. d Y _ { 1 } \ldots d Y _ { 2 k }. \end{equation*} |
| | | | |
| − | Consider the standard sum of homogeneous symbols defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110157.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110158.png" /> is an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110159.png" />, | + | Consider the standard sum of homogeneous symbols defined on $\Omega \times \mathbf{R} ^ { n }$, where $\Omega$ is an open subset of ${\bf R} ^ { n }$, |
| | | | |
| − | <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/w/w120/w120110/w120110160.png" /></td> </tr></table>
| + | \begin{equation*} a = a _ { m } + a _ { m - 1 } + r _ { m - 2 }, \end{equation*} |
| | | | |
| − | with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110161.png" /> smooth on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110162.png" /> and homogeneous in the following sense: | + | with $a _ { j }$ smooth on $\Omega \times \mathbf{R} ^ { n }$ and homogeneous in the following sense: |
| | | | |
| − | <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/w/w120/w120110/w120110163.png" /></td> </tr></table>
| + | \begin{equation*} a _ { j } ( x , \lambda \xi ) = \lambda ^ { j } a _ { j } ( x , \xi ) , \text { for } | \xi | \geq 1 , \lambda \geq 1, \end{equation*} |
| | | | |
| − | and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110164.png" />, i.e. for all compact subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110165.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110166.png" />, | + | and $r _ { m - 2} \in S _ { \text{loc} } ^ { m - 2 } ( \Omega )$, i.e. for all compact subsets $K$ of $\Omega$, |
| | | | |
| − | <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/w/w120/w120110/w120110167.png" /></td> </tr></table>
| + | \begin{equation*} \operatorname { sup } _ { x \in K, \atop \xi \in {\bf R} ^ { n } } \left| ( D _ { x } ^ { \alpha } D _ { \xi } ^ { \beta } r _ { m - 2 } ) ( x , \xi ) \right| ( 1 + | \xi | ) ^ { 2 - m + | \beta | } < \infty. \end{equation*} |
| | | | |
| − | This class of pseudo-differential operators (cf. also [[Pseudo-differential operator|Pseudo-differential operator]]) is invariant under diffeomorphisms, and using the Weyl quantization one gets that the principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110168.png" /> is invariantly defined on the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110169.png" /> whereas the subprincipal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110170.png" /> is invariantly defined on the double characteristic set | + | This class of pseudo-differential operators (cf. also [[Pseudo-differential operator|Pseudo-differential operator]]) is invariant under diffeomorphisms, and using the Weyl quantization one gets that the principal symbol $a _ { m }$ is invariantly defined on the cotangent bundle $T ^ { * } ( \Omega )$ whereas the subprincipal symbol $a_{m - 1}$ is invariantly defined on the double characteristic set |
| | | | |
| − | <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/w/w120/w120110/w120110171.png" /></td> </tr></table>
| + | \begin{equation*} \{ a _ { m } = 0 , d a _ { m } = 0 \} \end{equation*} |
| | | | |
| | of the principal symbol. If one writes | | of the principal symbol. If one writes |
| | | | |
| − | <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/w/w120/w120110/w120110172.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center; " valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110172.png"/></td> </tr></table> |
| | | | |
| − | one gets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110173.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110174.png" />. Moreover, | + | one gets $a = J ^ { - 1 / 2 } b$ and $a _ { m } = b _ { m }$. Moreover, |
| | | | |
| − | <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/w/w120/w120110/w120110175.png" /></td> </tr></table>
| + | \begin{equation*} a _ { m - 1 } = b _ { m - 1 } - \frac { 1 } { 2 \iota } \sum _ { 1 \leq j \leq n } \frac { \partial ^ { 2 } b _ { m } } { \partial x _ { j } \partial \xi _ { j } } = b _ { m - 1 } ^ { s }. \end{equation*} |
| | | | |
| − | Thus, if one defines the subprincipal symbol as the above analytic expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110176.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110177.png" /> is the classical symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110178.png" />, one finds that this invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110179.png" /> is simply the second term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110180.png" /> in the expansion of the Weyl symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110181.png" />. In the same vein, it is also useful to note that when considering pseudo-differential operators acting on half-densities one gets a refined principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110182.png" /> invariant by diffeomorphism. | + | Thus, if one defines the subprincipal symbol as the above analytic expression $b ^ { s } _{m - 1}$ where $b = b _ { m } + b _ { m - 1} + \ldots$ is the classical symbol of $a ^ { w } = \operatorname{Op} ( b )$, one finds that this invariant $b ^ { s } _{m - 1}$ is simply the second term $a_{m - 1}$ in the expansion of the Weyl symbol $a$. In the same vein, it is also useful to note that when considering pseudo-differential operators acting on half-densities one gets a refined principal symbol $a _ { m } + a _ { m - 1 }$ invariant by diffeomorphism. |
| | | | |
| | ==Weyl–Hörmander calculus and admissible metrics.== | | ==Weyl–Hörmander calculus and admissible metrics.== |
| − | The developments of the analysis of partial differential operators in the 1970s required refined localizations in the phase space. E.g., the Beals–Fefferman local solvability theorem [[#References|[a2]]] yields the geometric condition (P) as an if-and-only-if solvability condition for differential operators of principal type (with possibly complex symbols). These authors removed the analyticity assumption used by L. Nirenberg and F. Treves, and a key point in their method is a Calderón–Zygmund decomposition of the symbol, that is, a micro-localization procedure depending on a particular function, yielding a pseudo-differential calculus tailored to the symbol under investigation. Another example is provided by the Fefferman–Phong inequality [[#References|[a6]]], establishing that second-order operators with non-negative symbols are bounded from below on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110183.png" />; a Calderón–Zygmund decomposition is needed in the proof, as well as an induction on the number of variables. These micro-localizations go much beyond the standard homogeneous calculus and also beyond the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110184.png" />, previously called exotic. In 1979, L.V. Hörmander published [[#References|[a7]]], providing simple and general rules for a pseudo-differential calculus to be admissible. Consider a positive-definite quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110185.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110186.png" />. The dual quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110187.png" /> with respect to the symplectic structure is | + | The developments of the analysis of partial differential operators in the 1970s required refined localizations in the phase space. E.g., the Beals–Fefferman local solvability theorem [[#References|[a2]]] yields the geometric condition (P) as an if-and-only-if solvability condition for differential operators of principal type (with possibly complex symbols). These authors removed the analyticity assumption used by L. Nirenberg and F. Treves, and a key point in their method is a Calderón–Zygmund decomposition of the symbol, that is, a micro-localization procedure depending on a particular function, yielding a pseudo-differential calculus tailored to the symbol under investigation. Another example is provided by the Fefferman–Phong inequality [[#References|[a6]]], establishing that second-order operators with non-negative symbols are bounded from below on $L^{2}$; a Calderón–Zygmund decomposition is needed in the proof, as well as an induction on the number of variables. These micro-localizations go much beyond the standard homogeneous calculus and also beyond the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110184.png"/>, previously called exotic. In 1979, L.V. Hörmander published [[#References|[a7]]], providing simple and general rules for a pseudo-differential calculus to be admissible. Consider a positive-definite quadratic form $G$ defined on $\Phi$. The dual quadratic form $G ^ { \sigma }$ with respect to the symplectic structure is |
| | | | |
| − | <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/w/w120/w120110/w120110188.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>
| + | \begin{equation} \tag{a12} G ^ { \sigma } ( T ) = \operatorname { sup } _ { G ( U ) = 1 } [ T , U ] ^ { 2 } . \end{equation} |
| | | | |
| − | Define an admissible metric on the phase space as a mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110189.png" /> to the set of positive-definite quadratic forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110190.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110191.png" />, such that the following three properties are fulfilled: | + | Define an admissible metric on the phase space as a mapping from $\Phi$ to the set of positive-definite quadratic forms on $\Phi$, $X \mapsto G _ { X }$, such that the following three properties are fulfilled: |
| | | | |
| − | (uncertainty principle) For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110192.png" />, | + | (uncertainty principle) For all $X \in \Phi$, |
| | | | |
| − | <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/w/w120/w120110/w120110193.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>
| + | \begin{equation} \tag{a13} G _ { X } \leq G _ { X } ^ { g }; \end{equation} |
| | | | |
| − | there exist some positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110194.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110195.png" />, such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110196.png" />, | + | there exist some positive constants $\rho$, $C$, such that, for all $X , Y \in \Phi$, |
| | | | |
| − | <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/w/w120/w120110/w120110197.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>
| + | \begin{equation} \tag{a14} G _ { X } ( X - Y) \leq \rho ^ { 2 } \Rightarrow G _ { Y } \leq C G _ { X }; \end{equation} |
| | | | |
| − | there exist some positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110198.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110199.png" />, such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110200.png" />, | + | there exist some positive constants $N$, $C$, such that, for all $X , Y \in \Phi$, |
| | | | |
| − | <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/w/w120/w120110/w120110201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>
| + | \begin{equation} \tag{a15} G _ { X } \leq C ( 1 + G _ { X } ^ { \sigma } ( X - Y ) ) ^ { N } G _ { Y }. \end{equation} |
| | | | |
| − | Property (a13) is clearly related to the uncertainty principle, since for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110202.png" /> one can diagonalize the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110203.png" /> in a symplectic basis so that | + | Property (a13) is clearly related to the uncertainty principle, since for each $X$ one can diagonalize the quadratic form $G _ { X }$ in a symplectic basis so that |
| | | | |
| − | <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/w/w120/w120110/w120110204.png" /></td> </tr></table>
| + | \begin{equation*} G _ { X } = \sum _ { 1 \leq j \leq n } h _ { j } ( | d q _ { j } | ^ { 2 } + | d p _ { j } | ^ { 2 } ), \end{equation*} |
| | | | |
| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110205.png" /> is a set of symplectic coordinates. One then gets | + | where $( q_j , p _ { j } )$ is a set of symplectic coordinates. One then gets |
| | | | |
| − | <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/w/w120/w120110/w120110206.png" /></td> </tr></table>
| + | \begin{equation*} G _ { X } ^ { g } = \sum _ { 1 \leq j \leq n } h _ { j } ^ { - 1 } ( | d q _ { j } | ^ { 2 } + | d p _ { j } | ^ { 2 } ). \end{equation*} |
| | | | |
| − | Condition (a13) thus means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110207.png" />, which can be rephrased in the familiarly vague version as | + | Condition (a13) thus means that $\operatorname{max} h_{j} \leq 1$, which can be rephrased in the familiarly vague version as |
| | | | |
| − | <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/w/w120/w120110/w120110208.png" /></td> </tr></table>
| + | \begin{equation*} \Delta p _ { j } \Delta q_j \sim h _ { j } ^ { - 1 } \geq 1 \end{equation*} |
| | | | |
| − | in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110209.png" />-balls. This condition is relevant to any micro-localization procedure. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110210.png" />, one says that the quadratic form is symplectic. Property (a14) is called slowness of the metric and is usually easy to verify. Property (a15) is the temperance of the metric and is more of a technical character, although very important in handling non-local terms in the composition formula. In particular, this property is useful to verify the assumptions of Cotlar's lemma. Moreover, one defines a weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110211.png" /> as a positive function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110212.png" /> such that there exist positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110213.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110214.png" /> so that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110215.png" />, | + | in the $G$-balls. This condition is relevant to any micro-localization procedure. When $G = G ^ { \sigma }$, one says that the quadratic form is symplectic. Property (a14) is called slowness of the metric and is usually easy to verify. Property (a15) is the temperance of the metric and is more of a technical character, although very important in handling non-local terms in the composition formula. In particular, this property is useful to verify the assumptions of Cotlar's lemma. Moreover, one defines a weight $m$ as a positive function on $\Phi$ such that there exist positive constants $C$, $N$ so that for all $X , Y \in \Phi$, |
| | | | |
| − | <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/w/w120/w120110/w120110216.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a16)</td></tr></table>
| + | \begin{equation} \tag{a16} m ( X ) \leq C ( 1 + G _ { X } ^ { \sigma } ( X - Y ) ) ^ { N } m ( Y ), \end{equation} |
| | | | |
| | and | | and |
| | | | |
| − | <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/w/w120/w120110/w120110217.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a17)</td></tr></table>
| + | \begin{equation} \tag{a17} G _ { X } ( X - Y ) \leq C ^ { - 1 } \Rightarrow C ^ { - 1 } \leq \frac { m ( X ) } { m ( Y ) } \leq C. \end{equation} |
| | | | |
| − | Eventually, one defines the class of symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110218.png" /> as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110219.png" />-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110220.png" /> on the phase space such that | + | Eventually, one defines the class of symbols $S ( m , G )$ as the $C ^ { \infty }$-functions $a$ on the phase space such that |
| | | | |
| − | <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/w/w120/w120110/w120110221.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a18)</td></tr></table>
| + | \begin{equation} \tag{a18} \operatorname { sup } _ { X \in \Phi } \| a ^ { ( k ) } ( X ) \| _ { G _ { X } } m ( X ) ^ { - 1 } < \infty. \end{equation} |
| | | | |
| | It is, for instance, easily checked that | | It is, for instance, easily checked that |
| | | | |
| − | <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/w/w120/w120110/w120110222.png" /></td> </tr></table>
| + | \begin{equation*} S _ { \rho , \delta } ^ { \mu } = S \left( \langle \xi \rangle ^ { \mu } , \langle \xi \rangle ^ { 2 \delta } | d x | ^ { 2 } + \langle \xi \rangle ^ { - 2 \rho } | d \xi | ^ { 2 } \right), \end{equation*} |
| | | | |
| − | with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110223.png" /> and that this metric is an admissible metric when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110224.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110225.png" />. The metric defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110226.png" /> satisfies (a13)–(a14) but fails to satisfy (a15). Indeed, there are counterexamples showing that for the classical and the Weyl quantization [[#References|[a4]]] there are symbols in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110227.png" /> whose quantization is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110229.png" />-bounded. In fact, one of the building block for the calculus of pseudo-differential operators is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110230.png" />-boundedness of the Weyl quantization of symbols in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110231.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110232.png" /> is an admissible metric. One defines the Planck function of the calculus as | + | with $\langle \xi \rangle = 1 + | \xi |$ and that this metric is an admissible metric when $0 \leq \delta \leq \rho \leq 1$, $\delta < 1$. The metric defining $S _ { 1,1 } ^ { 0 }$ satisfies (a13)–(a14) but fails to satisfy (a15). Indeed, there are counterexamples showing that for the classical and the Weyl quantization [[#References|[a4]]] there are symbols in $S _ { 1,1 } ^ { 0 }$ whose quantization is not $L^{2}$-bounded. In fact, one of the building block for the calculus of pseudo-differential operators is the $L^{2}$-boundedness of the Weyl quantization of symbols in $S ( 1 , G )$, where $G$ is an admissible metric. One defines the Planck function of the calculus as |
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| − | <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/w/w120/w120110/w120110233.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a19)</td></tr></table>
| + | \begin{equation} \tag{a19} H ( X ) = \operatorname { sup } _ { T \neq 0 } \sqrt { \frac { G _{X} ( T ) } { G _ { X } ^ { \sigma } ( T ) } } \end{equation} |
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| − | and notes that from (a13), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110234.png" />. One obtains the composition formula (a11) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110235.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110236.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110237.png" />. In particular, one obtains, with obvious notations, | + | and notes that from (a13), $H ( X ) \leq 1$. One obtains the composition formula (a11) with $\alpha \in S ( m _ { 1 } , G )$, $b \in S ( m _ { 2 } , G )$ and $r _ { N } ( a , b ) \in S ( m _ { 1 } m _ { 2 } H ^ { N } , G )$. In particular, one obtains, with obvious notations, |
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| − | <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/w/w120/w120110/w120110238.png" /></td> </tr></table>
| + | \begin{equation*} a \sharp b \in S ( m _ { 1 } m _ { 2 } , G ), \end{equation*} |
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| − | <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/w/w120/w120110/w120110239.png" /></td> </tr></table>
| + | \begin{equation*} a \sharp b = a b + S ( m _ { 1 } m _ { 2 } H , G ) ,\; a \sharp b = a b + \frac { 1 } { 2 \iota } \{ a , b \} + S ( m _ { 1 } m _ { 2 } H ^ { 2 } , G ), \end{equation*} |
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| − | <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/w/w120/w120110/w120110240.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center; " valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110240.png"/></td> </tr></table> |
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| − | The Fefferman–Phong inequality has also a simple expression in this framework: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110241.png" /> be a non-negative symbol in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110242.png" />, then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110243.png" /> is semi-bounded from below in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110244.png" />. The proof uses a Calderón–Zygmund decomposition and in fact one shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110245.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110246.png" /> is the Planck function related to the admissible metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110247.png" /> defined by | + | The Fefferman–Phong inequality has also a simple expression in this framework: Let $a$ be a non-negative symbol in $S ( H ^ { - 2 } , G )$, then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110243.png"/> is semi-bounded from below in $L ^ { 2 } ( \mathbf{R} ^ { n } )$. The proof uses a Calderón–Zygmund decomposition and in fact one shows that $a \in S ( h ^ { - 2 } , g )$, where $h$ is the Planck function related to the admissible metric $g$ defined by |
| | | | |
| − | <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/w/w120/w120110/w120110248.png" /></td> </tr></table>
| + | \begin{equation*} g_{X} ( T ) = \frac { G _ { X } ( T ) } { H ( X ) [ 1 + a ( X ) + H ( X ) ^ { 2 } \| a ^ { \prime \prime } ( X ) \| ^ { 2 _{ G _ { X }}} ] ^ { 1 / 2 } }. \end{equation*} |
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| − | On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110249.png" /> is an admissible metric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110250.png" /> uniformly with respect to a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110251.png" />, the following metric also satisfies (a13)–(a15): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110252.png" />, with | + | On the other hand, if $G$ is an admissible metric and $q _ { \alpha } \in S ( H ^ { - 1 } , G )$ uniformly with respect to a parameter $\alpha$, the following metric also satisfies (a13)–(a15): $\tilde { g } _ { X } = H ( X ) ^ { - 1 } \tilde { h } ( X ) G _ { X } ( T )$, with |
| | | | |
| − | <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/w/w120/w120110/w120110253.png" /></td> </tr></table>
| + | \begin{equation*} \widetilde { h } ( X ) ^ { - 1 } = 1 + \operatorname { sup } _ { \alpha } | q _ { \alpha } ( X ) | + H ( X ) \operatorname { sup } _ { \alpha } \| q _ { \alpha } ^ { \prime } ( X ) \| _ { G _ { X } } ^ { 2 }. \end{equation*} |
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| − | One gets in this case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110254.png" /> uniformly. A key point in the Beals–Fefferman proof of local solvability under condition (P) can be reformulated through the construction of the previous metric. Sobolev spaces related to this type of calculus were studied in [[#References|[a1]]] (cf. also [[Sobolev space|Sobolev space]]). For an admissible metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110255.png" /> and a weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110256.png" />, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110257.png" /> is defined as | + | One gets in this case that $q _ { \alpha } \in S ( \tilde { h } ^ { - 1 } , \tilde{g} )$ uniformly. A key point in the Beals–Fefferman proof of local solvability under condition (P) can be reformulated through the construction of the previous metric. Sobolev spaces related to this type of calculus were studied in [[#References|[a1]]] (cf. also [[Sobolev space|Sobolev space]]). For an admissible metric $G$ and a weight $m$, the space $H ( m , G )$ is defined as |
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| − | <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/w/w120/w120110/w120110258.png" /></td> </tr></table>
| + | \begin{equation*} \{ u \in \mathcal{S} ^ { \prime } ( \mathbf{R} ^ { n } ) : \forall a \in S ( m , G ) , a ^ { w } u \in L ^ { 2 } ( \mathbf{R} ^ { n } ) \}. \end{equation*} |
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| − | It can be proven that a Hilbertian structure can be set on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110259.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110260.png" /> and that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110261.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110262.png" /> another weight, the mapping | + | It can be proven that a Hilbertian structure can be set on $H ( m , G )$, that $H ( 1 , G ) = L ^ { 2 } ( \mathbf{R} ^ { n } )$ and that for $a \in S ( m , G )$ and $m_1$ another weight, the mapping |
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| − | <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/w/w120/w120110/w120110263.png" /></td> </tr></table>
| + | \begin{equation*} a ^ { w } : H ( m m _ { 1 } , G ) \rightarrow H ( m _ { 1 } , G ) \end{equation*} |
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| | is continuous. | | is continuous. |
| Line 278: |
Line 286: |
| | Further developments of the Weyl calculus were explored in [[#References|[a3]]], with higher-order micro-localizations. Several metrics | | Further developments of the Weyl calculus were explored in [[#References|[a3]]], with higher-order micro-localizations. Several metrics |
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| − | <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/w/w120/w120110/w120110264.png" /></td> </tr></table>
| + | \begin{equation*} g _ { 1 } \leq \ldots \leq g _ { k } \end{equation*} |
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| − | are given on the phase space. All these metrics satisfy (a13)–(a14), but, except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110265.png" />, fail to satisfy globally the temperance condition (a15). Instead, the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110266.png" /> is assumed to be (uniformly) temperate on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110267.png" />-balls. It is then possible to produce a satisfactory quantization formula for symbols belonging to a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110268.png" />. A typical example is given in [[#References|[a5]]], with applications to propagation of singularities for non-linear hyperbolic equations: | + | are given on the phase space. All these metrics satisfy (a13)–(a14), but, except for $g_1$, fail to satisfy globally the temperance condition (a15). Instead, the metric $g_{l+ 1}$ is assumed to be (uniformly) temperate on the $g_{l}$-balls. It is then possible to produce a satisfactory quantization formula for symbols belonging to a class $S ( m , g _ { k } )$. A typical example is given in [[#References|[a5]]], with applications to propagation of singularities for non-linear hyperbolic equations: |
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| − | <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/w/w120/w120110/w120110269.png" /></td> </tr></table>
| + | \begin{equation*} g _ { 1 } = | d x | ^ { 2 } + \frac { | d \xi | ^ { 2 } } { | \xi | ^ { 2 } } \leq g_2 = \frac { | d x | ^ { 2 } } { | x | ^ { 2 } } + \frac { | d \xi | ^ { 2 } } { | \xi | ^ { 2 } }, \end{equation*} |
| | | | |
| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110270.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110271.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110272.png" /> on | + | where $g_1$ is defined on $| \xi | > 1$, and $g_{2}$ on |
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| − | <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/w/w120/w120110/w120110273.png" /></td> </tr></table>
| + | \begin{equation*} \{ | x | < 1 , | x | | \xi | > 1 \} \end{equation*} |
| | | | |
| − | It is then possible to quantize functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110274.png" /> homogeneous of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110275.png" /> in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110276.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110277.png" /> in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110278.png" />, so as to get composition formulas, Sobolev spaces, and the standard pseudo-differential apparatus allowing a commutator argument to work for propagation results. | + | It is then possible to quantize functions $a ( x , \xi )$ homogeneous of degree $\mu$ in the variable $x$, and $\nu$ in the variable $\xi $, so as to get composition formulas, Sobolev spaces, and the standard pseudo-differential apparatus allowing a commutator argument to work for propagation results. |
| | | | |
| | ====References==== | | ====References==== |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.-M. Bony, J.-Y. Chemin, "Espaces fonctionnels associés au calcul de Weyl–Hörmander" ''Bull. Soc. Math. France'' , '''122''' (1994) pp. 77–118 {{MR|}} {{ZBL|0798.35172}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Beals, C. Fefferman, "On local solvability of linear partial differential equations" ''Ann. of Math.'' , '''97''' (1973) pp. 482–498 {{MR|0352746}} {{ZBL|0256.35002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.-M. Bony, N. Lerner, "Quantification asymtotique et microlocalisations d'ordre supérieur" ''Ann. Sci. Ecole Norm. Sup.'' , '''22''' (1989) pp. 377–483</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Boulkhemair, "Remarque sur la quantification de Weyl pour la classe de symboles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120110/w120110279.png" />" ''C.R. Acad. Sci. Paris'' , '''321''' : 8 (1995) pp. 1017–1022 {{MR|1360564}} {{ZBL|0842.35144}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.-M. Bony, "Second microlocalization and propagation of singularities for semi-linear hyperbolic equations" K. Mizohata (ed.) , ''Hyperbolic Equations and Related Topics'' , Kinokuniya (1986) pp. 11–49 {{MR|925240}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> C. Fefferman, D.H. Phong, "On positivity of pseudo-differential operators" ''Proc. Nat. Acad. Sci. USA'' , '''75''' (1978) pp. 4673–4674 {{MR|0507931}} {{ZBL|0391.35062}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> L. Hörmander, "The Weyl calculus of pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''32''' (1979) pp. 359–443 {{MR|517939}} {{ZBL|0388.47032}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> L. Hörmander, "The analysis of linear partial differential operators III-IV" , Springer (1985)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> I. Segal, "Transforms for operators and asymptotic automorphisms over a locally compact abelian group" ''Math. Scand.'' , '''13''' (1963) pp. 31–43</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> A. Unterberger, "Oscillateur harmonique et opérateurs pseudo-différentiels" ''Ann. Inst. Fourier'' , '''29''' : 3 (1979) pp. 201–221 {{MR|0552965}} {{ZBL|0396.47027}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Weil, "Sur certains groupes d'opérateurs unitaires" ''Acta Math.'' , '''111''' (1964) pp. 143–211 {{MR|0165033}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> H. Weyl, "Gruppentheorie und Quantenmechanik" , S. Hirzel (1928) {{MR|0450450}} {{ZBL|54.0954.03}} </TD></TR></table> | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> J.-M. Bony, J.-Y. Chemin, "Espaces fonctionnels associés au calcul de Weyl–Hörmander" ''Bull. Soc. Math. France'' , '''122''' (1994) pp. 77–118 {{MR|}} {{ZBL|0798.35172}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> R. Beals, C. Fefferman, "On local solvability of linear partial differential equations" ''Ann. of Math.'' , '''97''' (1973) pp. 482–498 {{MR|0352746}} {{ZBL|0256.35002}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> J.-M. Bony, N. Lerner, "Quantification asymtotique et microlocalisations d'ordre supérieur" ''Ann. Sci. Ecole Norm. Sup.'' , '''22''' (1989) pp. 377–483</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> A. Boulkhemair, "Remarque sur la quantification de Weyl pour la classe de symboles $S _ { 1,1 } ^ { 0 }$" ''C.R. Acad. Sci. Paris'' , '''321''' : 8 (1995) pp. 1017–1022 {{MR|1360564}} {{ZBL|0842.35144}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> J.-M. Bony, "Second microlocalization and propagation of singularities for semi-linear hyperbolic equations" K. Mizohata (ed.) , ''Hyperbolic Equations and Related Topics'' , Kinokuniya (1986) pp. 11–49 {{MR|925240}} {{ZBL|}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> C. Fefferman, D.H. Phong, "On positivity of pseudo-differential operators" ''Proc. Nat. Acad. Sci. USA'' , '''75''' (1978) pp. 4673–4674 {{MR|0507931}} {{ZBL|0391.35062}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> L. Hörmander, "The Weyl calculus of pseudo-differential operators" ''Commun. Pure Appl. Math.'' , '''32''' (1979) pp. 359–443 {{MR|517939}} {{ZBL|0388.47032}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> L. Hörmander, "The analysis of linear partial differential operators III-IV" , Springer (1985)</td></tr><tr><td valign="top">[a9]</td> <td valign="top"> I. Segal, "Transforms for operators and asymptotic automorphisms over a locally compact abelian group" ''Math. Scand.'' , '''13''' (1963) pp. 31–43</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> A. Unterberger, "Oscillateur harmonique et opérateurs pseudo-différentiels" ''Ann. Inst. Fourier'' , '''29''' : 3 (1979) pp. 201–221 {{MR|0552965}} {{ZBL|0396.47027}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> A. Weil, "Sur certains groupes d'opérateurs unitaires" ''Acta Math.'' , '''111''' (1964) pp. 143–211 {{MR|0165033}} {{ZBL|}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> H. Weyl, "Gruppentheorie und Quantenmechanik" , S. Hirzel (1928) {{MR|0450450}} {{ZBL|54.0954.03}} </td></tr></table> |