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Index and dynamics of quantized contact transformations. (English) Zbl 0865.47018

Summary: Quantized contact transformations are Toeplitz operators over a contact manifold \((X,\alpha)\) of the form \(U_{\chi} = \Pi A \chi \Pi\), where \(\Pi : H^2(X) \rightarrow L^2(X)\) is a Szegö projector, where \(\chi\) is a contact transformation and where \(A\) is a pseudodifferential operator over \(X\). They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine \(\text{ind} (U_{\chi})\) when the principal symbol is unitary, or equivalently to determine whether \(A\) can be chosen so that \(U_{\chi}\) is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms \(g\) by showing that \(U_g\) duplicates the classical transformation laws on theta functions. Using the Cauchy-Szegö kernel on the Heisenberg group, we calculate the traces on theta functions of each degree \(N\). We also study the quantum dynamics generated by a general q.c.t. \(U_{\chi}\), i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on \(\chi.\) Our principal results are proofs of equidistribution of eigenfunctions \(\phi_{Nj}\) and weak mixing properties of matrix elements \((B\phi_{Ni}, \phi_{Nj})\) for quantizations of mixing symplectic maps.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
58C40 Spectral theory; eigenvalue problems on manifolds
81S10 Geometry and quantization, symplectic methods
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
47G30 Pseudodifferential operators
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