The Lie Group of Fourier Integral Operators on Open Manifolds
نویسنده
چکیده
The theory of pseudodifferential operators and Fourier integral operators on compact manifolds is well established and their applications in mathematical physics well known , see for example Hörmander , Duistermaat , Treves 12 . For open (non compact) manifolds this is not the case, and that’s what I would like to focus on in this paper. We are interested in the geometry of the spaces of pseudodifferential operators and Fourier integral operators and their associated diffeomorphism groups. Unfortunately, the theory on open manifolds is very technical and much more complicated than in the compact case . I will give here only a overview of the results and refer for technicalities to the published papers Eichhorn and Schmid 6 , . Instead of going through the details of the construction of the Lie group structures of pseudodifferential and Fourier integral operators on open manifolds I will first review the compact case and then explain what goes wrong in the non compact case and how we fixed these problems. REVIEW COMPACT CASE In 1985 we proved the following theorem Adams, Ratiu , Schmid ,,: Theorem: The group FIO∗(M) of invertible Fourier integral operators on a compact manifold M is a graded ∞-dim Lie group with graded ∞-dim Lie algebra ΨDO(M) of pseudodifferential operators on M . FIO∗(M) is and ∞-dim principal fiber bundle over the base manifold Diffθ(Ṫ ∗M) of contact transformations of Ṫ ∗M with gauge group ΨDO∗(M) of invertible pseudodifferential operators.
منابع مشابه
Lie Groups of Fourier Integral Operators on Open Manifolds
We endow the group of invertible Fourier integral operators on an open manifold with the structure of an ILH Lie group. This is done by establishing such structures for the groups of invertible pseudodiierential operators and contact transformations on an open manifold of bounded geometry, and gluing those together via a local section.
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تاریخ انتشار 2002