Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha )$ of the form $U_{\chi }=\Pi A\chi \Pi$, where $\Pi :H^2\left(X\right)\to L^2\left(X\right)$ 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 $\mathrm{ind}\left(U_{\chi }\right)$ when the principal symbol is unitary, or equivalently to determine...

We study the invariant symbolic calculi associated with the unitary irreducible representations of a compact Lie group.

Let $G$ be the semidirect product $V⋊K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$. Let $\pi$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$. We construct an invariant symbolic calculus for $\pi$, under some technical hypothesis. We give some examples including the Poincaré group.

We give a full description of the semiclassical spectral theory of quantum toric integrable systems using microlocal analysis for Toeplitz operators. This allows us to settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of the system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and commuting Hamiltonians. This type of...