Algebras of semiclassical pseudodifferential operators associated with Zoll-type domains in cotangent bundles

We consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a Bohr–Sommerfeld condition (for example, the unit disk bundle of a Zoll metric). Given such a domain, we construct an algebra of associated semiclassical pseudodi...

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Bibliographic Details
Published in:Journal of functional analysis 2015-04, Vol.268 (7), p.1755-1807
Main Authors: Hernandez-Duenas, Gerardo, Uribe, Alejandro
Format: Article
Language:English
Subjects:
Fourier integral operators
Pseudodifferential operators
Quantization of symplectic manifolds
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Summary:We consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a Bohr–Sommerfeld condition (for example, the unit disk bundle of a Zoll metric). Given such a domain, we construct an algebra of associated semiclassical pseudodifferential operators with singular symbols. The Schwartz kernels of the operators have frequency set contained in the union of the diagonal and the flow-out of the null foliation of the boundary of the domain. We develop a symbolic calculus, prove the existence of projectors (under a mild additional assumption) whose range can be thought of as quantizing the domain, give a symbolic proof of a Szegö limit theorem, and study associated propagators.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2014.12.004