San Vũ Ngọc

Semitoric integrable systems on symplectic 4-manifolds

Let (M,ω) be a symplectic 4-manifold. A semitoric integrable system on (M,ω) is a pair of smooth functions J,H∈C ∞(M,ℝ) for which J generates a Hamiltonian S1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce.

Symplectic theory of completely integrable Hamiltonian systems

This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of singular affine structures. These developments make use of results obtained by many authors in the second half of the twentieth century, notably Arnold, Duistermaat and Eliasson, of which we also give a concise survey. As a motivation, we present a collection of remarkable results proven in the early and mid 1980s in the theory of Hamiltonian Lie group actions by Atiyah, Guillemin-Sternberg and Delzant among others, and which inspired many people, including the authors, to work on more general Hamiltonian systems. The paper concludes discussing a spectral conjecture for quantum integrable systems.

A singular Poincaré lemma

We prove a Poincare lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a Cartan subalgebra of sp(2r, R). This result has a natural interpretation in terms of the cohomology associated to the infinitesimal deformation of a completely integrable system.

Hofer's question on intermediate symplectic capacities

Roughly twenty five years ago Hofer asked: can the cylinder B^2(1) \times \mathbb{R}^{2(n-1)} be symplectically embedded into B^{2(n-1)}(R) \times \mathbb{R}^2 for some R>0? We show that this is the case if R \geq \sqrt{2^{n-1}+2^{n-2}-2}. We deduce that there are no intermediate capacities, between 1-capacities, first constructed by Gromov in 1985, and n-capacities, answering another question of Hofer. In 2008, Guth reached the same conclusion under the additional hypothesis that the intermediate capacities should satisfy the exhaustion property.

Hamiltonian Dynamics and Spectral Theory for Spin–Oscillators

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.

Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin–Toeplitz operators

We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system. This gives a quick alternative solution to the isospectrality problem for quantum toric systems. If the operators are uniformly bounded, the convergence is uniform. Analogous results hold for non-commuting operators.

Symplectic inverse spectral theory for pseudodifferential operators

We prove, under some generic assumptions, that the semiclassical spectrum modulo O(ћ 2) of a one-dimensional pseudodifferential operator completely determines the symplectic geometry of the underlying classical system. In particular, the spectrum determines the Hamiltonian dynamics of the principal symbol.

Integrable systems, symmetries and quantization

These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterparts are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system?

Semiclassical Inverse Spectral Theory for Singularities of Focus–Focus Type

We prove, assuming that the Bohr–Sommerfeld rules hold, that the joint spectrum near a focus–focus singular value of a quantum integrable system determines the classical Lagrangian foliation around the full focus–focus leaf. The result applies, for instance, to ħ-pseudodifferential operators on cotangent bundles and Berezin–Toeplitz operators on prequantizable compact symplectic manifolds.

Microlocal Normal Forms for the Magnetic Laplacian

We explore symplectic techniques to obtain long time estimates for a purely magnetic confinement in two degrees of freedom. Using pseudo-differential calculus, the same techniques lead to microlocal normal forms for the magnetic Laplacian. In the case of a strong magnetic field, we prove a reduction to a 1D semiclassical pseudo-differential operator. This can be used to derive precise asymptotic expansions for the eigenvalues at any order.