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

TL;DR

This paper analyzes the spectral properties of operators derived from the metaplectic representation of the unitary group and its Lie algebra, providing explicit formulas and asymptotic laws for their eigenvalues.

Contribution

It offers a detailed spectral analysis of operators from the metaplectic representation, including conditions for discrete spectrum and eigenvalue multiplicities, linking them to combinatorial and geometric polynomials.

Findings

Eigenvalues expressed in terms of matrix eigenvalues

Eigenvalue multiplicities follow a quasi polynomial pattern

Weyl's law established for these operators

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 $H_A$.