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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.18728 |
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| _version_ | 1866912506178437120 |
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| author | Belmonte, Fabián de Nittis, Giuseppe |
| author_facet | Belmonte, Fabián de Nittis, Giuseppe |
| contents | In this article we compute and analyze the spectrum of operators defined by the metaplectic representation $μ$ on the unitary group $\mathbb{U}(d)$ or operators defined by the corresponding induced representation $dμ$ 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 $H_A=-i dμ(A)$ 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 $H_A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18728 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 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 Belmonte, Fabián de Nittis, Giuseppe Spectral Theory In this article we compute and analyze the spectrum of operators defined by the metaplectic representation $μ$ on the unitary group $\mathbb{U}(d)$ or operators defined by the corresponding induced representation $dμ$ 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 $H_A=-i dμ(A)$ 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 $H_A$. |
| title | 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 |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2412.18728 |