Eigenvectors of Toeplitz matrices from Fisher-Hartwig symbols with greater than, or equal to, one singularity
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914902965223424 |
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| author | Rigas, Pete |
| author_facet | Rigas, Pete |
| contents | Asymptotically, we analytically derive the form of eigenvectors for two Fisher-Hartwig symbols besides those which were previously investigated in a $2016$ work due to Movassagh and Kadanoff, in which the authors characterized the eigenpairs of Toeplitz matrices generated by Fisher-Hartwig symbols with one singularity. To perform such computations, we extend their methods which consists of formulating an eigenvalue problem, obtained by a Wiener-Hopf method, from which a suitable winding number is defined for passing to Fourier space, and introducing a factorization dependent upon the winding number, for other Fisher-Hartwig symbols which have previously been defined in the literature. Following the computations required for the proof after obtaining the asymptotic approximation of the eigenvectors, we provide a table from which eigenvalues of one Fisher-Hartwig symbol for different complex valued functions $b$ can be inferred. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_01842 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Eigenvectors of Toeplitz matrices from Fisher-Hartwig symbols with greater than, or equal to, one singularity Rigas, Pete Mathematical Physics Statistical Mechanics Spectral Theory 82D02, 81V02 Asymptotically, we analytically derive the form of eigenvectors for two Fisher-Hartwig symbols besides those which were previously investigated in a $2016$ work due to Movassagh and Kadanoff, in which the authors characterized the eigenpairs of Toeplitz matrices generated by Fisher-Hartwig symbols with one singularity. To perform such computations, we extend their methods which consists of formulating an eigenvalue problem, obtained by a Wiener-Hopf method, from which a suitable winding number is defined for passing to Fourier space, and introducing a factorization dependent upon the winding number, for other Fisher-Hartwig symbols which have previously been defined in the literature. Following the computations required for the proof after obtaining the asymptotic approximation of the eigenvectors, we provide a table from which eigenvalues of one Fisher-Hartwig symbol for different complex valued functions $b$ can be inferred. |
| title | Eigenvectors of Toeplitz matrices from Fisher-Hartwig symbols with greater than, or equal to, one singularity |
| topic | Mathematical Physics Statistical Mechanics Spectral Theory 82D02, 81V02 |
| url | https://arxiv.org/abs/2308.01842 |