For a compact Levi non-degenerate CR manifold, the semiclassical spectral projector of a Levi-elliptic Toeplitz operator is, modulo a negligible kernel, the sum of two oscillatory integrals with complex phases, with explicit leading coefficients.
Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudo-differential operators , Comm
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Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds
For a compact Levi non-degenerate CR manifold, the semiclassical spectral projector of a Levi-elliptic Toeplitz operator is, modulo a negligible kernel, the sum of two oscillatory integrals with complex phases, with explicit leading coefficients.