Semi-classical weights and equivariant spectral theory
We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigensp...
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Elsevier BV
2018
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| Online Access: | http://hdl.handle.net/1721.1/118447 https://orcid.org/0000-0003-2641-1097 |
| _version_ | 1826201076660961280 |
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| author | Dryden, Emily Guillemin, Victor W Sena-Dias, Rosa Isabel |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Dryden, Emily Guillemin, Victor W Sena-Dias, Rosa Isabel |
| author_sort | Dryden, Emily |
| collection | MIT |
| description | We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show that the asymptotic equivariant spectrum of the Laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold; we also show that the asymptotic equivariant spectrum of a Tn-invariant Schrödinger operator on Rn determines its potential in some suitably convex cases. In addition, we prove that the asymptotic equivariant spectrum of an S1-invariant metric on S2determines the metric itself in many cases. Finally, we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces. As a crucial ingredient in these inverse results, we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi-classical differential operators invariant under a torus action. Keywords: Laplacian; Asymptotic equivariant spectrum; Semi-classical weights; Toric manifold; Symplectic orbifold |
| first_indexed | 2024-09-23T11:45:56Z |
| format | Article |
| id | mit-1721.1/118447 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T11:45:56Z |
| publishDate | 2018 |
| publisher | Elsevier BV |
| record_format | dspace |
| spelling | mit-1721.1/1184472022-10-01T05:53:04Z Semi-classical weights and equivariant spectral theory Dryden, Emily Guillemin, Victor W Sena-Dias, Rosa Isabel Massachusetts Institute of Technology. Department of Mathematics Dryden, Emily Guillemin, Victor W Sena-Dias, Rosa Isabel We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show that the asymptotic equivariant spectrum of the Laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold; we also show that the asymptotic equivariant spectrum of a Tn-invariant Schrödinger operator on Rn determines its potential in some suitably convex cases. In addition, we prove that the asymptotic equivariant spectrum of an S1-invariant metric on S2determines the metric itself in many cases. Finally, we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces. As a crucial ingredient in these inverse results, we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi-classical differential operators invariant under a torus action. Keywords: Laplacian; Asymptotic equivariant spectrum; Semi-classical weights; Toric manifold; Symplectic orbifold 2018-10-11T19:26:16Z 2018-10-11T19:26:16Z 2016-05 2016-01 2018-09-25T17:53:38Z Article http://purl.org/eprint/type/JournalArticle 0001-8708 1090-2082 http://hdl.handle.net/1721.1/118447 Dryden, Emily B. et al. “Semi-Classical Weights and Equivariant Spectral Theory.” Advances in Mathematics 299 (August 2016): 202–246 © 2016 Elsevier Inc https://orcid.org/0000-0003-2641-1097 http://dx.doi.org/10.1016/J.AIM.2016.02.037 Advances in Mathematics Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier BV arXiv |
| spellingShingle | Dryden, Emily Guillemin, Victor W Sena-Dias, Rosa Isabel Semi-classical weights and equivariant spectral theory |
| title | Semi-classical weights and equivariant spectral theory |
| title_full | Semi-classical weights and equivariant spectral theory |
| title_fullStr | Semi-classical weights and equivariant spectral theory |
| title_full_unstemmed | Semi-classical weights and equivariant spectral theory |
| title_short | Semi-classical weights and equivariant spectral theory |
| title_sort | semi classical weights and equivariant spectral theory |
| url | http://hdl.handle.net/1721.1/118447 https://orcid.org/0000-0003-2641-1097 |
| work_keys_str_mv | AT drydenemily semiclassicalweightsandequivariantspectraltheory AT guilleminvictorw semiclassicalweightsandequivariantspectraltheory AT senadiasrosaisabel semiclassicalweightsandequivariantspectraltheory |