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 \(H_A\).