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Local asymptotics for slowly shrinking spectral bands of a Berezin-Toeplitz operator. (English) Zbl 1230.47055
Author’s abstract: Motivated by geometric quantization, we study how the asymptotics of certain spectral projectors of an invariant self-adjoint Toeplitz operator relate to the full level-\(k\) Szegő kernel associated to a positive line bundle on a compact symplectic manifold. These operators quantize classical observables on the “phase space” \(M\), and the semiclassical regime corresponds to \(k\to+\infty\). We focus on the asymptotics of the spectral projectors for \(\xi<\frac{1}{2}\), spectral bands of width \(O(k^{-\xi})\) centered at \(E\) asymptotically capture all of the Szegő kernel at points of classical energy \(E\). On the same classical locus, these bands also realize a version of Tian’s almost isometry [G. Tian, J. Differ. Geom. 32, No. 1, 99–130 (1990; Zbl 0706.53036)].

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
53D50 Geometric quantization
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