Identification of Berezin-Toeplitz deformation quantization

We give a complete identification of the deformation quantization which was obtained from the Berezin-Toeplitz quantization on an arbitrary compact Kaehler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose classifying form is explicitly calculated. Its characteristic class (which classifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szegoe kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjoestrand.Comment: 26 page

Deformation Quantization of Geometric Quantum Mechanics

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always ${1 \over \hbar}$.Comment: 27+1 pages, harvmac file, no figure

Felix Alexandrovich Berezin and his work

This is a survey of Berezin's work focused on three topics: representation theory, general concept of quantization, and supermathematics.Comment: LaTeX, 27 page

Toeplitz operators on symplectic manifolds

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.Comment: 40 page

Distribution of resonances for open quantum maps

We analyze simple models of classical chaotic open systems and of their quantizations (open quantum maps on the torus). Our models are similar to models recently studied in atomic and mesoscopic physics. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller (or trapped set) of the system. In a simplified model, a rigorous argument gives the full resonance spectrum, which satisfies the fractal Weyl law. For this model, we can also compute a quantity characterizing the fluctuations of conductance through the system, namely the shot noise power: the value we obtain is close to the prediction of random matrix theory.Comment: 60 pages, no figures (numerical results are shown in other references

Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization

For arbitrary compact quantizable Kähler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential manner. It is shown that the star product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given