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Prequantum transfer operator for symplectic Anosov diffeomorphism by Faure, Frédéric; Tsujii, Masato is a scholarly article available to read on EtoBox.

What is Prequantum transfer operator for symplectic Anosov diffeomorphism about?

We define the prequantization of a symplectic Anosov diffeomorphism f:M-> M, which is a U(1) extension of the diffeomorphism f preserving an associated specific connection, and study the spectral properties of the associated transfer operator, called prequantum transfer operator. This is a model for the transfer operators associated to geodesic flows on negatively curved manifolds (or contact Anosov flows). We restrict the prequantum transfer operator to the N-th Fourier mode with respect to the U(1) action and investigate the spectral property in the limit N->infinity, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, we show a " band structure " of the spectrum, that is, the spectrum is contained in a few separated annuli and a disk concentric at the origin. We show that, with the special (H\"older continuous) potential V0=1/2 log |det Df_x|_{E_u}|, the outermost annulus is the unit circle and separated from the other parts. For this, we use an extension of the transfer operator to the Grassmanian bundle. Using Atiyah-Bott trace formula, we establish the Gutzwiller trace formula with exponentially small reminder

Author

Faure, Frédéric; Tsujii, Masato

Published

2012

Language

EN