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

Abstract

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 $H_A$.