Semi-classical weights and equivariant spectral theory
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Guillemin_Semi-classical weights.pdf
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410.59 KB
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Author(s)
Dryden, Emily
Guillemin, Victor W
Sena-Dias, Rosa Isabel
Date Issued
May 2016
Journal
Advances in Mathematics
Publisher
Elsevier BV
Citation
Dryden, Emily B. et al. “Semi-Classical Weights and Equivariant Spectral Theory.” Advances in Mathematics 299 (August 2016): 202–246 © 2016 Elsevier Inc
Version
Original manuscript
Abstract
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
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/J.AIM.2016.02.037
