AI 中文总结
本研究证明正复辛矩阵恰好是那些存在非零有界薛定谔交错子的矩阵，由此为振子半群提供了与实亚仿射群经典刻画平行的薛定谔表示层面的算子论刻画。
AI 中文摘要
振子（或称亚仿射）半群在经典上被定义为正复辛矩阵半群的二重覆盖。尽管该定义在几何上是精确的，但它并未给出与正辛矩阵对应的有界算子的内在算子论刻画，这与实亚仿射群形成对比——实亚仿射群与辛群的关系可直接通过海森堡群的薛定谔表示及其交错性质来表述。在复情形下，此类关系在现有文献中主要以无穷小形式或适用于各类高斯函数的形式出现。本研究证明，（复化的）薛定谔交错关系对所有 $L^2(\mathbb{R}^d)$ 中的函数均成立。核心在于其逆命题：若任意复辛矩阵 $S$ 在 $L^2(\mathbb{R}^d)$ 上存在非零有界算子满足该交错关系，则 $S$ 必为正矩阵，且该算子在非零标量因子下与振子半群的对应元素一致。因此，正复辛矩阵恰好是那些存在非零有界薛定谔交错子的矩阵，且相关的交错空间为一维。这为振子半群提供了一种薛定谔表示层面的刻画，与实亚仿射群的经典刻画平行。
英文摘要
The oscillator (or metaplectic) semigroup is classically defined as the two-fold cover of the semigroup of positive complex symplectic matrices. Although geometrically precise, this definition does not provide an intrinsic operator-theoretic characterization of the bounded operators corresponding to positive symplectic matrices. This is in contrast with the real metaplectic group, whose relation with the symplectic group can be expressed directly in terms of the Schrödinger representation of the Heisenberg group through its intertwining property. In the complex setting, such a relation appears in the existing literature mainly in infinitesimal form or on suitable classes of Gaussian functions. In this work we prove that the (complexified) Schrödinger intertwining relation holds for every function in $L^2(\mathbb{R}^d)$. The main point is the converse statement: if an arbitrary complex symplectic matrix $S$ admits a nonzero bounded operator on $L^2(\mathbb{R}^d)$ satisfying this intertwining relation, then $S$ is necessarily positive and the operator coincides, up to a nonzero scalar, with the corresponding element of the oscillator semigroup. Consequently, positive complex symplectic matrices are exactly those admitting a nonzero bounded Schrödinger intertwiner, and the associated intertwining space is one-dimensional. This provides a Schrödinger-representation characterization of the oscillator semigroup, parallel to the classical one for the real metaplectic group.
