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San Vũ Ngoc

Publications and source records attributed to San Vũ Ngoc.

12 recordsLinked to original sources
arXiv2026

WKB for semiclassical operators: How to fly over caustics (and more)

The method initiated by Wentzel, Kramers, and Brillouin to find approximate solutions to the Schrödinger equation lies at the origin of the spectacular development of microlocal and semiclassical analysis. When used naively, the approach appears to break down at caustics, but Maslov showed how a simple generalization could overcome this difficulty. In this paper, after a partial historical review, we take advantage of more recent advances in microlocal analysis to present a unified treatment of this generalized Maslov-WKB method, using a microlocal sheaf-theoretic approach. This framework provides a rigorous proof of the Bohr Sommerfeld Einstein Brillouin Keller quantization conditions for the eigenvalues of general semiclassical operators (pseudodifferential and Berezin Toeplitz) in one degree of freedom. We also review some applications and extensions.

math.SP↗
arXiv2024

On the symplectic geometry of $A_k$ singularities

This paper presents a complete symplectic classification of $A_k$ Hamiltonians on $\mathbb R^2$, in the analytic and smooth categories. Precisely, consider the pair $(H, ω)$ consisting of a Hamiltonian and a symplectic structure on $\mathbb R^2$ such that $H$ has an $A_{k-1}$ singularity at the origin with $k\geq 2$. We classify such pairs near the origin, up to fiberwise symplectomorphisms, and up to $H$-preserving symplectomorphisms. The classification is obtained by bringing the pair $(H, ω)$ to a symplectic normal form $$\big(H = ξ^2 \pm x^k, \ ω= d (f d ξ)\big), \quad f = \sum_{i=1}^{k-1} x^i f_i(x^k),$$ modulo some relations which are explicitly given. We also show that the group of $H$-preserving symplectomorphisms of an $A_{k-1}$ singularity for $k$ odd consists of symplectomorphisms that can be included into a $C^\infty$-smooth (resp., real-analytic) $H$-preserving flow, whereas for $k$ even with $k \ge 4$ the same is true modulo the $\mathbb Z_2$-subgroup generated by the involution $Inv(x,ξ) = (-x,-ξ)$. The paper is concluded with a brief discussion of the conjecture that the symplectic invariants of $A_{k-1}$ singularities are spectrally determined.

math.SG↗
arXiv2024

The inverse spectral problem for quantum semitoric systems

Given a quantum semitoric system composed of pseudodifferential operators, Berezin-Toeplitz operators, or a combination of both, we obtain explicit formulas for recovering, from the semiclassical asymptotics of the joint spectrum, all symplectic invariants of the underlying classical semitoric system. Our formulas are based on the possibility to obtain good quantum numbers for joint eigenvalues from the bare data of the joint spectrum. In the spectral region corresponding to regular values of the momentum map, the algorithms developed by Dauge, Hall and the second author [27] produce such labellings. In our proof, it was crucial to extend these algorithms to the boundary of the spectrum, which led to the new notion of asymptotic half-lattices, and to globalize the resulting labellings. Using the construction given by Pelayo and the second author in [79], our results prove that semitoric systems are completely spectrally determined in an algorithmic way~: from the joint spectrum of a quantum semitoric system one can construct a representative of the isomorphism class of the underlying classical semitoric system. In particular, this recovers the uniqueness result obtained by Pelayo and the authors in [62,61], and completes it with the explicit computation of all invariants, including the twisting index. In the cases of the spin-oscillator and the coupled angular momenta, we implement the algorithms and illustrate numerically the computation of the invariants from the joint spectrum.

math-ph↗
arXiv2020

Magnetic WKB constructions on surfaces

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.

math.SP↗
arXiv2019

Exponential localization in 2D pure magnetic wells

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-Sj{ö}strand.

math.AP↗
arXiv2015

Spectral limits of semiclassical commuting self-adjoint operators

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 $\hbar$-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.

math.SP↗
arXiv2013

The affine invariant of generalized semitoric systems

A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which correspond to fibers of F which are topologically multipinched tori. The image F(M) is a singular affine manifold which contains a distinguished set of isolated points in its interior: the focus-focus values {(x_i,y_i)} of F. By performing a vertical cutting procedure along the lines {x:=x_i}, we construct a homeomorphism f : F(M) --> f(F(M)), which restricts to an affine diffeomorphism away from these vertical lines, and generalizes a construction of Vu Ngoc. The set Δ:=f(F(M)) in R^2 is a symplectic invariant of (M,ω,F), which encodes the affine structure of F. Moreover, Δmay be described as a countable union of planar regions of four distinct types, where each type is defined as the region bounded between the graphs of two functions with various properties (piecewise linear, continuous, convex, etc). If F is a toric system, Δis a convex polygon (as proven by Atiyah and Guillemin-Sternberg) and f is the identity.

math.SG↗
arXiv2013

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.

math.DS↗
arXiv2013

First steps in symplectic and spectral theory of integrable systems

The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best approach to such a large classification task would be, it is possible to single out some promising directions and preliminary problems. This paper discusses them and hints at a possible path, still loosely defined, to arrive at a classification. It mainly relies on recent progress concerning integrable systems with only non-hyperbolic and non-degenerate singularities. This work originated in an attempt to develop a theory aimed at answering some questions in quantum spectroscopy. Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time. The development of semiclassical analysis with microlocal techniques in the last forty years now permits a constant interplay between spectral theory and symplectic geometry. A main goal of this paper is to emphasize the symplectic issues that are relevant to quantum mechanical integrable systems, and to propose a strategy to solve them.

math.DS↗
arXiv2013

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 critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.

math-ph↗
arXiv2013

Semiclassical quantization and spectral limits of h-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.

math-ph↗