标题： 酉群单位算子$\mathbb{U}(d)$及其李代数$\mathfrak{u}(d)$通过余 metaplectic 表示和 Weyl 量子化表示的显式谱分析 显示英文标题

标题： Explicit Spectral Analysis for Operators Representing the unitary group $\mathbb{U}(d)$ and its Lie algebra $\mathfrak{u}(d)$ through the Metaplectic Representation and Weyl Quantization

摘要： 本文中我们计算并分析了由度量表示 $\mu$ 在酉群 $\mathbb{U}(d)$ 上定义的算子，或者由相应的李代数 $\mathfrak{u}(d)$ 的诱导表示 $d\mu$ 定义的算子的谱。 结果表明这两种类型的算子的点谱都可以用相应矩阵的特征值来表达。 对于每个 $A\in\mathfrak{u}(d)$，已知自伴算子 $H_A=-i d\mu(A)$ 具有二次型的魏尔符号，并且我们将给出保证它具有离散谱的条件。 在这些条件下，利用组合学中的一个已知结果，我们证明了$H_A$的特征值重数（经过一些明确的平移和标量乘法后）是一个次数为$d-1$的拟多项式。 此外，我们证明了计数特征值的函数表现得像一个Ehrhart多项式。 利用这一结果，我们证明了对于算子$H_A$的Weyl定律。 显示英文摘要

摘要： In this article we compute and analyze the spectrum of operators defined by the metaplectic representation $\mu$ on the unitary group $\mathbb{U}(d)$ or operators defined by the corresponding induced representation $d\mu$ of the Lie algebra $\mathfrak{u}(d)$. It turns out that the point spectrum of both types of operators can be expressed in terms of the eigenvalues of the corresponding matrices. For each $A\in\mathfrak{u}(d)$, it is known that the selfadjoint operator $H_A=-i d\mu(A)$ has a quadratic Weyl symbol and we will give conditions on to guarantee that it has discrete spectrum. Under those conditions, using a known result in combinatorics, we show that the multiplicity of the eigenvalues of $H_A$ is (up to some explicit translation and scalar multiplication) a quasi polynomial of degree $d-1$. Moreover, we show that counting eigenvalues function behaves as an Ehrhart polynomial. Using the latter result, we prove a Weyl's law for the operators $H_A$.