辛扩张映射与一类新的Shubin伪微分算子
A symplectic extension map and a new Shubin class of pseudo-differential operators
摘要：
对于任意的伪微分算子$A:\mathcal{S}(\mathbb{R}% ^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$，其Weyl符号为$a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$，我们考虑与Weyl符号$\widetilde{a}=(a\otimes1_{2k})\circ{s}$相关的伪微分算子$\widetilde{A}:\mathcal{S}(\mathbb{R}% ^{n+k})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n+k})$，其中$1_{2k}(x)=1$对所有的$x\in\mathbb{R}^{2k}$和${s}$是$\mathbb{R}^{2(n+k)}$的线性辛同构.我们称这些算子为$A$的$\widetilde{A}$辛维扩张。本文详细研究了$A$和$\widetilde{A}$之间的关系，特别是它们的正则性、可逆性和谱性质。我们得到一个明确的公式，允许表达的特征函数的$\widetilde{A}$的$A$。我们使用这种形式主义，以构建新的类的伪微分算子，这是扩展的舒宾类$HG_ρ^{m_{1},m_{0}}$的全球亚椭圆算子。我们证明了新类中的算子具有$HG_{ρ}^{m_{1},m_{0}}$中算子的可逆性和谱性质，但不具有全局亚椭圆性质。最后，我们研究了一些属于新类的算子的例子，这些算子在数学物理中很重要。
英文摘要：
For an arbitrary pseudo-differential operator $A:\mathcal{S}(\mathbb{R}% ^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$ with Weyl symbol $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$, we consider the pseudo-differential operators $\widetilde{A}:\mathcal{S}(\mathbb{R}% ^{n+k})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n+k})$ associated with the Weyl symbols $\widetilde{a}=(a\otimes1_{2k})\circ{s}$, where $1_{2k}(x)=1$ for all $x\in\mathbb{R}^{2k}$ and ${s}$ is a linear symplectomorphism of $\mathbb{R}^{2(n+k)}$. We call the operators $\widetilde{A}$ symplectic dimensional extensions of $A$. In this paper we study the relation between $A$ and $\widetilde{A}$ in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of $\widetilde{A}$ in terms of those of $A$. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes $HG_ρ^{m_{1},m_{0}}$ of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in $HG_{ρ}^{m_{1},m_{0}}$ but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.
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