GLT matrix-sequences and few emblematic applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914144116015104 |
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| author | Khan, Muhammad Faisal |
| author_facet | Khan, Muhammad Faisal |
| contents | This thesis advances the spectral theory of structured matrix-sequences within the framework of Generalized Locally Toeplitz (GLT) $*$-algebras, focusing on the geometric mean of Hermitian positive definite (HPD) GLT sequences and its applications in mathematical physics. For two HPD sequences $\{A_n\}_n \sim_{\mathrm{GLT}} κ$ and $\{B_n\}_n \sim_{\mathrm{GLT}} ξ$ in the same $d$-level, $r$-block GLT $*$-algebra, we prove that when $κ$ and $ξ$ commute, the geometric mean sequence $\{G(A_n,B_n)\}_n$ is GLT with symbol $(κξ)^{1/2}$, without requiring invertibility of either symbol, settling \cite[Conjecture 10.1]{garoni2017} for $r=1$, $d\ge1$. In degenerate cases, we identify conditions ensuring $\{G(A_n,B_n)\}_n \sim_{\mathrm{GLT}} G(κ,ξ)$. For $r>1$ and non-commuting symbols, numerical evidence shows the sequence still admits a spectral symbol, indicating maximality of the commuting result. Numerical experiments in scalar and block settings confirm the theory and illustrate spectral behaviour. We also sketch the extension to $k\ge2$ sequences via the Karcher mean, obtaining $\{G(A_n^{(1)},\ldots,A_n^{(k)})\}_n \sim_{\mathrm{GLT}} G(κ_1,\ldots,κ_k)$. Finally, we apply the GLT framework to mean-field quantum spin systems, showing that matrices from the quantum Curie--Weiss model form GLT sequences with explicitly computable spectral distributions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06312 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | GLT matrix-sequences and few emblematic applications Khan, Muhammad Faisal Numerical Analysis Mathematical Physics This thesis advances the spectral theory of structured matrix-sequences within the framework of Generalized Locally Toeplitz (GLT) $*$-algebras, focusing on the geometric mean of Hermitian positive definite (HPD) GLT sequences and its applications in mathematical physics. For two HPD sequences $\{A_n\}_n \sim_{\mathrm{GLT}} κ$ and $\{B_n\}_n \sim_{\mathrm{GLT}} ξ$ in the same $d$-level, $r$-block GLT $*$-algebra, we prove that when $κ$ and $ξ$ commute, the geometric mean sequence $\{G(A_n,B_n)\}_n$ is GLT with symbol $(κξ)^{1/2}$, without requiring invertibility of either symbol, settling \cite[Conjecture 10.1]{garoni2017} for $r=1$, $d\ge1$. In degenerate cases, we identify conditions ensuring $\{G(A_n,B_n)\}_n \sim_{\mathrm{GLT}} G(κ,ξ)$. For $r>1$ and non-commuting symbols, numerical evidence shows the sequence still admits a spectral symbol, indicating maximality of the commuting result. Numerical experiments in scalar and block settings confirm the theory and illustrate spectral behaviour. We also sketch the extension to $k\ge2$ sequences via the Karcher mean, obtaining $\{G(A_n^{(1)},\ldots,A_n^{(k)})\}_n \sim_{\mathrm{GLT}} G(κ_1,\ldots,κ_k)$. Finally, we apply the GLT framework to mean-field quantum spin systems, showing that matrices from the quantum Curie--Weiss model form GLT sequences with explicitly computable spectral distributions. |
| title | GLT matrix-sequences and few emblematic applications |
| topic | Numerical Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2511.06312 |