In this short course, we shall review some techniques and recent results regarding the equivariant asymptotics of the Szegö kernels associated to a compact quantizable manifold with a Hamiltonian action; while our focus will actually be on algebro-geometric Szegö kernels, much of what will be discussed admits a generalization to the symplectic almost complex setting.
The thread of the discussion will be offered by the following basic question: if we are given a Hamiltonian action on a quantizable symplectic manifold, which is ‘quantized’ by a unitary representation, how does the equivariant decomposition into isotypical components of the latter reflect the geometry of the action? We shall consider the asymptotic and local aspects of this relation, and our basic tool will be the description of Szegö kernels as Fourier integral operators, which is due to Boutet de Monvel and Sjöstraend. Thus our development will build on the approach to geometric quantization originally introduced by Boutet de Monvel and Guillemin, and more recently extended, and further unfolded and elaborated, principally, by Shiffman and Zelditch.
References
We’ll touch mainly on the content of the following papers, to which we refer for an adequate bibliography on this theme:
In the first lecture, I will present the general problem of quantization. Starting from a classical phase space (symplectic manifold), how to define a quantum space (Hilbert space) and transform Hamiltonians into linear operators? I will give details about the left, right, and Weyl quantization on ${\mathbb R}^{2n}$.
In the second lecture, I will show how to develop a full microlocal theory: how to localize quantum mechanics in phase space? Using the notion of semiclassical wavefront set, we obtain a very flexible calculus that can be used to prove Bohr-Sommerfeld rules at any order for a very accurate description of the spectrum of semiclassical operators. I will also give a quick overview of what can be done for non selfadjoint operators using Sjöstrand's theory of complexified phase spaces.
In this short course, we shall review some techniques and recent results regarding the equivariant asymptotics of the Szegö kernels associated to a compact quantizable manifold with a Hamiltonian action; while our focus will actually be on algebro-geometric Szegö kernels, much of what will be discussed admits a generalization to the symplectic almost complex setting.
The thread of the discussion will be offered by the following basic question: if we are given a Hamiltonian action on a quantizable symplectic manifold, which is ‘quantized’ by a unitary representation, how does the equivariant decomposition into isotypical components of the latter reflect the geometry of the action? We shall consider the asymptotic and local aspects of this relation, and our basic tool will be the description of Szegö kernels as Fourier integral operators, which is due to Boutet de Monvel and Sjöstraend. Thus our development will build on the approach to geometric quantization originally introduced by Boutet de Monvel and Guillemin, and more recently extended, and further unfolded and elaborated, principally, by Shiffman and Zelditch.
References
We’ll touch mainly on the content of the following papers, to which we refer for an adequate bibliography on this theme:
In this short course, we shall review some techniques and recent results regarding the equivariant asymptotics of the Szegö kernels associated to a compact quantizable manifold with a Hamiltonian action; while our focus will actually be on algebro-geometric Szegö kernels, much of what will be discussed admits a generalization to the symplectic almost complex setting.
The thread of the discussion will be offered by the following basic question: if we are given a Hamiltonian action on a quantizable symplectic manifold, which is ‘quantized’ by a unitary representation, how does the equivariant decomposition into isotypical components of the latter reflect the geometry of the action? We shall consider the asymptotic and local aspects of this relation, and our basic tool will be the description of Szegö kernels as Fourier integral operators, which is due to Boutet de Monvel and Sjöstraend. Thus our development will build on the approach to geometric quantization originally introduced by Boutet de Monvel and Guillemin, and more recently extended, and further unfolded and elaborated, principally, by Shiffman and Zelditch.
References
We’ll touch mainly on the content of the following papers, to which we refer for an adequate bibliography on this theme:
In the third lecture, I will concentrate on the applications of microlocal analysis to integrable systems. What is the quantum analogue of the classical singular lagrangian foliation? How to compute the joint spectrum of commuting operators? In the case of semitoric sytems, I will present recent results concerning the inverse question: can you recover the symplectic geometry from the quantum spectrum?
