arXiv 1209.28522012-09-13math.FA

Fock空间中的有界Weyl伪微分算子

Bounded Weyl pseudodifferential operators in Fock space

Laurent Amour, Lisette Jager, Jean Nourrigat

摘要：

我们的目标是在无限维的环境中构建Weyl演算的模拟，其中通常的配置和相空间最终被无限维的测度空间，即所谓的抽象Wiener空间所取代。算子作用的希尔伯特空间可以看作是一个福克空间，或者等价地，看作是一个在位形空间上的平方可积〔的〕函数空间。该构造并不简单，需要将位形空间分成两个因子，其中第一个因子是有限维的。然后定义一个方便的符号$F$，一个混合演算，作用于有限维因子作为外尔算子，另一个作为反威克算子，定义感谢无限维Segal-Bargmann变换。可以建立混合算子的界。这些界限使我们能够证明收敛的任何序列的混合运营商与有限维因素的增加序列。它们的共同极限是Weyl算子$OP_h^{weyl}(F)$，类似于Calderón-Vaillancourt定理，也是上述上界的结果。

英文摘要：

We aim at constructing an analog of the Weyl calculus in an infinite dimensional setting, in which the usual configuration and phase spaces are ultimately replaced by infinite dimensional measure spaces, the so-called abstract Wiener spaces. The Hilbert space on which the operators act can be seen as a Fock space or, equivalently, as a space of square integrable functions on the configuration space. The construction is not straightforward and needs to split the configuration space into two factors, of which the first one is finite dimensional. Then one defines, for a convenient symbol $F$, a hybrid calculus, acting on the finite dimensional factor as a Weyl operator and on the other one as an anti-Wick operator, defined thanks to an infinite dimensional Segal-Bargmann transformation. One can establish bounds on the hybrid operators. These bounds enable us to prove the convergence of any sequence of hybrid operators associated with an increasing sequence of finite dimensional factors. Their common limit is the Weyl operator $OP_h^{weyl}(F)$, the analog of Calderón-Vaillancourt Theorem being a consequence of the upper mentionned bounds as well.

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摘要：

英文摘要：

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