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We prove that semitoric systems are completely spectrally determined : from the joint spectrum of a quantum semitoric system, in the semiclassical limit, one can recover all symplectic invariants of the underlying classical semitoric system, and hence the isomorphism class of the system. Moreover, by using specific algorithms and formulas, we obtain a constructive way of computing the invariants from the spectrum.
This article is devoted to the description of the eigenvalues and eigenfunctions of the magnetic Laplacian in the semiclassical limit via the complex WKB method. Under the assumption that the magnetic field has a unique and non-degenerate minimum, we construct the local complex WKB approximations for eigenfunctions on a general surface. Furthermore, in the case of the Euclidean plane, with a radially symmetric magnetic field, the eigenfunctions are approximated in an exponentially weighted space.
We consider semiclassical self-adjoint operators whose symbol, defined on a two-dimensional symplectic manifold, reaches a non-degenerate minimum $b_0$ on a closed curve. We derive a classical and quantum normal form which gives uniform eigenvalue asymptotics in a window $(−∞, b_0 + \varepsilon)$ for $\varepsilon>0$ independent on the semiclassical parameter. These asymptotics are obtained in two complementary settings: either an approximate invariance of the system under translation along the curve, which produces oscillating eigenvalues, or a Morse hypothesis reminiscent of Helffer-Sjöstrand’s “miniwell” situation.
We present a new continuous automata based computer simulation of virus propagation in human populations, and apply it to the Covid-19 outbreak, in various scales and situations. We also take the opportunity to propose various mathematical questions, and ask about their biological relevance.
We discuss the problem of recovering geometric objects from the spectrum of a quantum integrable system. In the case of one degree of freedom, precise results exist. In the general case, we report on the recent notion of good labellings of asymptotic lattices.
We establish a magnetic Agmon estimate in the case of a purely magnetic single non-degenerate well, by means of the Fourier-Bros-Iagolnitzer transform and microlocal exponential estimates à la Martinez-Sjöstrand.
We explain why Theorem B in the original article does not follow from the main result of this paper (Theorem A). While we conjecture that Theorem B should nevertheless be true, in this erratum we prove a slightly weaker version of it.
For a two degree of freedom quantum integrable system, a new spectral quantity is defined, the quantum rotation number. In the semiclassical limit, the quantum rotation number can be detected on a joint spectrum and is shown to converge to the well-known classical rotation number. The proof requires not only semiclassical analysis (including Bohr-Sommerfeld quantization rules) but also a detailed study on how quantum labels can be assigned to the joint spectrum in a smooth way. This leads to the definition and analysis of asymptotic lattices. The general results are applied to the semitoric case where formulas become particularly natural.
Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\CM^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.
We prove that the Schwartz class is stable under the magnetic Schrödinger flow when the magnetic 2-form is non-degenerate and does not oscillate too much at infinity.
We consider a charged particle on a plane, subject to a strong, purely magnetic external field. It is well known that the quantum evolution closely follows the classical dynamics for short periods of time, while for times larger than $\ln \frac{1}{\hbar}$, where $\hbar$ is Planck’s constant, purely quantum phenomena are expected to happen. In this paper we investigate the Schrödinger evolution of generalized coherent states for times of order $1/\hbar$. We prove that, when the initial energy is low, the initial states splits into multiple wavepackets, each one following the average dynamics of the guiding center motion but at its own speed.
We study the Hamiltonian dynamics of a charged particle submitted to a pure magnetic field in a two-dimensional domain. We provide conditions on the magnetic field in a neighbourhood of the boundary to ensure the confinement of the particle. We also prove a formula for the scattering angle in the case of radial magnetic fields.
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 counterpart are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system?
This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms *à la Birkhoff*. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional $\hbar$-pseudo-differential operator whose Weyl’s symbol admits an asymptotic expansion in powers of $\hbar^{\frac1 2}$.
Using an abstract notion of semiclassical quantization for self-adjoint operators, we prove that the joint spectrum of a collection of commuting semiclassical self-adjoint operators converges to the classical spectrum given by the joint image of the principal symbols, in the semiclassical limit. This includes Berezin-Toeplitz quantization and certain cases of ℏ-pseudodifferential quantization, for instance when the symbols are uniformly bounded, and extends a result by L. Polterovich and the authors. In the last part of the paper we review the recent solution to the inverse problem for quantum integrable systems with periodic Hamiltonians, and explain how it also follows from the main result in this paper.
Quantum semitoric systems form a large class of quantum Hamiltonian integrable systems with circular symmetry which has received great attention in the past decade. They include systems of high interest to physicists and mathematicians such as the Jaynes-Cummings model (1963), which describes a two-level atom interacting with a quantized mode of an optical cavity, and more generally the so-called systems of Jaynes-Cummings type. In this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of Jaynes-Cummings type. We prove, assuming the Bohr-Sommerfeld rules hold, that if the joint spectrum of two of these systems coincide up to O(ℏ²), then the systems are isomorphic.
This paper is devoted to the classical mechanics and spectral analysis of a pure magnetic Hamiltonian in $\mathbb{R}^2$. It is established that both the dynamics and the semiclassical spectral theory can be treated through a Birkhoff normal form and reduced to the study of a family of one dimensional Hamiltonians. As a corollary, recent results by Helffer-Kordyukov are extended to higher energies.
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 $\hbar$-pseudodifferential operators on cotangent bundles and Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.