On the invertibility of Born-Jordan quantization

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• 作者：Elena Cordero^a ; ^{elena.cordero@unito.it" class="auth_mail" title="E-mail the corresponding author} ; Maurice de Gosson^b ; ^{maurice.de.gosson@univie.ac.at" class="auth_mail" title="E-mail the corresponding author} ; Fabio Nicola^c ; ^{fabio.nicola@polito.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：47G30 ; 81S05 ; 46F10
• 刊名：Journal de mathematiques pures et appliquees
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：105
• 期：4
• 页码：537-557
• 全文大小：491 K

文摘

As a consequence of the Schwartz kernel Theorem, any linear continuous operator 1604db2063e432cca68498563f69">$\hat{A}$: S(Rⁿ)⟶S^′(Rⁿ)$\mathcal{S}({\mathbb{R}}^n)⟶{\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ can be written in Weyl form in a unique way, namely it is the Weyl quantization of a unique symbol a∈S^′(R²ⁿ)$a\in {\mathcal{S}}^{\prime }({\mathbb{R}}^{2n})$. Hence, dequantization can always be performed, and in a unique way. Despite the importance of this topic in Quantum Mechanics and Time-frequency Analysis, the same issue for the Born–Jordan quantization seems simply unexplored, except for the case of polynomial symbols, which we also review in detail. In this paper we show that any operator 1604db2063e432cca68498563f69">$\hat{A}$ as above can be written in Born–Jordan form, although the representation is never unique if one allows general temperate distributions as symbols. Then we consider the same problem when the space of temperate distributions is replaced by the space of smooth slowly increasing functions which extend to entire function in C²ⁿ${\mathbb{C}}^{2n}$, with a growth at most exponential in the imaginary directions. We prove again the validity of such a representation, and we determine a sharp threshold for the exponential growth under which the representation is unique. We employ techniques from the theory of division of distributions.