Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913680891838464 |
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| author | Finski, Siarhei |
| author_facet | Finski, Siarhei |
| contents | We study small eigenvalues of Toeplitz operators on polarized complex projective manifolds. For Toeplitz operators whose symbols are supported on proper subsets, we prove the existence of eigenvalues that decay exponentially with respect to the semiclassical parameter. We moreover, establish a connection between the logarithmic distribution of these eigenvalues and the Mabuchi geodesic between the fixed polarization and the Lebesgue envelope associated with the polarization and the non-zero set of the symbol. As an application of our approach, we also obtain analogous results for Toeplitz matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry Finski, Siarhei Complex Variables Differential Geometry Functional Analysis Spectral Theory 32U15, 32A25, 15B05, 53D50, 58J50 We study small eigenvalues of Toeplitz operators on polarized complex projective manifolds. For Toeplitz operators whose symbols are supported on proper subsets, we prove the existence of eigenvalues that decay exponentially with respect to the semiclassical parameter. We moreover, establish a connection between the logarithmic distribution of these eigenvalues and the Mabuchi geodesic between the fixed polarization and the Lebesgue envelope associated with the polarization and the non-zero set of the symbol. As an application of our approach, we also obtain analogous results for Toeplitz matrices. |
| title | Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry |
| topic | Complex Variables Differential Geometry Functional Analysis Spectral Theory 32U15, 32A25, 15B05, 53D50, 58J50 |
| url | https://arxiv.org/abs/2502.01554 |