We extend the Bargmann transform to the magnetic pseudodifferential calculus, using gauge-covariant families of coherent states. We also introduce modulation mappings, a first step towards adapting modulation spaces to the magnetic case.
Coherent states are special types of wavefunctions that minimize a generalized uncertainty principle for a suitable pair of operators. Equivalently, they are eigenstates of an appropriate annihilation operator. Their applications are extensive throughout physics including in quantum optics, nuclear physics, quantum field theory, path integral formulations, and quantum information through the study of entanglement and quantum measurement. This thesis explores two main topics. First, we consider the Schrödinger evolution of a Gaussian coherent state under a non-Hermitian Hamiltonian. We develop a symbol calculus and use it to construct an approximate solution to the time-dependent Schrödinger equation. We find the evolution equations of the center and the Gaussian matrix of the coherent state, which form a system. This result generalizes the previously-known case where the classical Hamiltonian is quadratic. In the second part of the thesis, we apply a quantum version of dimensional reduction to construct Gaussian coherent states in the Bargmann space of complex projective space. The semiclassical properties of these reduced states are controlled by a suitable notion of symbol. Making use of these properties, we provide norm estimates and a propagation result for Hermitian Hamiltonians. As a special case of these reduced states, we define and examine spin-squeezed states that live naturally in the Bargmann space of the Riemann sphere.
We construct an abstract pseudodifferential calculus with operator-valued symbol, adapted to the treatment of Coulomb-type interactions, and we apply it to study the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case where the electronic Hamiltonian admits a local gap in its spectrum. In particular, we show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, we study the propagation of certain wave packets up to long time values of Ehrenfest order.
Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.