arXiv subjects

Alix Deleporte

Publications and source records attributed to Alix Deleporte.

18 recordsLinked to original sources

arXiv2026

Metric-uniform spectral inequality for the Laplacian on manifolds with bounded sectional curvature

Given a Riemannian manifold $M$ endowed with a smooth metric $g$ satisfying upper and lower sectional curvature bounds, we show an equivalence property between the $\mathrm{L}^2$ norm on $M$ and the $\mathrm{L}^2$ norm on subsets $\omega$ satisfying a thickness condition, for functions in the range of a spectral projector. The thickness condition is known to be optimal in this setting. The constant appearing in the equivalence of norms property depends only on the dimension of the manifold, curvature bounds, and frequency threshold of the spectral cutoff, but, crucially, not on the injectivity radius.

math.AP↗

arXiv2025

Fluctuations of two-dimensional determinantal processes associated with Berezin--Toeplitz operators

We consider a new class of determinantal point processes in the complex plane coming from the ground state of free fermions associated with Berezin--Toeplitz operators. These processes generalize the Ginibre ensemble from random matrix theory. We prove a two-term Szegő-type asymptotic expansion for the Laplace transform of smooth linear statistics. This implies a law of large number and central limit theorem for the empirical field. The limiting variance includes both contributions from the bulk and boundary of the droplet. The boundary fluctuations depend on the Hamiltonian dynamics associated with the underlying operator and, generally, are not conformally invariant.

math.PR↗

arXiv2025

Regular Bohr-Sommerfeld rules for non-self-adjoint Berezin--Toeplitz operators and complex Lagrangian states

We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin--Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy levels. These results form a complex version of the Bohr-Sommerfeld quantization conditions; they hold under a hypothesis that the skew-adjoint part is small but can be of principal order with respect to the semiclassical parameter. To this end, we develop a calculus of Fourier Integral Operators and Lagrangian states associated with complex Lagrangians; these tools are of independent interest.

math.SP↗

arXiv2024

Spectral estimates on hyperbolic surfaces and a necessary condition for observability of the heat semigroup on manifolds

This article is a continuation of arXiv:2401.14977. We study the concentration properties of spectral projectors on manifolds, in connection with the uncertainty principle. In arXiv:2401.14977, the second author proved an optimal uncertainty principle for the spectral projector of the Laplacian on the hyperbolic half-plane. The aim of the present work is to generalize this condition to surfaces with hyperbolic ends. In particular, we tackle the case of cusps, in which the volume of balls of fixed radius is not bounded from below. We establish that spectral estimates hold from sets satisfying a thickness condition, with a proof based on propagation of smallness estimates of Carleman and Logunov--Malinnikova type. We also prove the converse, namely the necessary character of the thickness condition, on any smooth manifold with Ricci curvature bounded from below.

math.AP↗

arXiv2024

Widom's conjecture: variance asymptotics and entropy bounds for counting statistics of free fermions

We obtain a central limit theorem for bulk counting statistics of free fermions in smooth domains of $\mathbb{R}^n$ with an explicit description of the covariance structure. This amounts to a study of the asymptotics of norms of commutators between spectral projectors of semiclassical Schrödinger operators and indicator functions supported in the bulk. In the spirit of the Widom conjecture, we show that the squared Hilbert-Schmidt norm of these commutators is of order $\hbar^{-n+1}\log(\hbar)$ as the semiclassical parameter $\hbar$ tends to $0$. We also give a new upper bound on the trace norm of these commutators and applications to estimations of the entanglement entropy for free fermions.

math.SP↗

arXiv2024

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

We study local asymptotics for the spectral projector associated to a Schrödinger operator $-\hbar^2Δ+V$ on $\mathbb{R}^n$ in the semiclassical limit as $\hbar\to0$. We prove local uniform convergence of the rescaled integral kernel of this projector towards a universal model, inside the classically allowed region as well as on its boundary. This implies universality of microscopic fluctuations for the corresponding free fermions (determinantal) point processes, both in the bulk and around regular boundary points. Our results apply for a general class of smooth potentials in arbitrary dimension $n\ge 1$. These results are complemented by studying both macroscopic and mesoscopic fluctuations of the point process. We obtain tail bounds for macroscopic linear statistics and, provided $n\geq 2$, a central limit theorem for both macroscopic and mesoscopic linear statistics in the bulk.

math-ph↗

arXiv2024

Real-analytic geodesics in the Mabuchi space of Kähler metrics and quantization

We prove the convergence of quantized Bergman geodesics to the Mabuchi geodesics for the initial value problem, in the case of real-analytic initial data and in short time. This partially solves a conjecture of Y. Rubinstein and the last author. We also argue against the existence of a solution to the boundary value problem, generically in real-analytic regularity.

math.DG↗

arXiv2023

The Szegő kernel in analytic regularity and analytic Fourier Integral Operators

We build a general theory of microlocal (homogeneous) Fourier Integral Operators in real-analytic regularity, following the general construction in the smooth case by Hörmander and Duistermaat. In particular, we prove that the Boutet-Sjöstrand parametrix for the Szegő projector at the boundary of a strongly pseudo-convex real-analytic domain can be realised by an analytic Fourier Integral Operator. We then study some applications, such as FBI-type transforms on compact, real-analytic Riemannian manifolds and propagators of one-homogeneous (pseudo)differential operators.

math.SP↗

arXiv2023

Central limit theorem for smooth statistics of one-dimensional free fermions

We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on $\mathbb{R}$, with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő-type central limit theorem (CLT) for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalization to a Gaussian limit with a $H^{1/2}$-type variance, which depends on the potential. In the one-well (one-cut) case, using the quantum action-angle theorem and additional micro-local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi-cut case, we show that for generic potentials, a similar result holds and the contributions of each wells are independent in the limit.

math.SP↗

arXiv2022

WKB eigenmode construction for analytic Toeplitz operators

We provide almost eigenfunctions for Toeplitz operators with real-analytic symbols, at the bottom of non-degenerate wells. These almost eigenfunctions follow the WKB ansatz; the error is O(exp(--cN)), where c > 0 and N $\rightarrow$ +$\infty$ is the inverse semiclassical parameter.

math.AP↗

arXiv2020

A direct approach to the analytic Bergman projection

We develop a direct approach to the semiclassical asymptotics for Bergman projections in exponentially weighted spaces of holomorphic functions, with real analytic strictly plurisubharmonic weights. In particular, the approach does not rely upon the Kuranishi trick and it allows us to shorten and simplify proofs of a result due to Rouby-Sjöstrand-Vũ Ngoc and Deleporte, stating that in the analytic case, the amplitude of the asymptotic Bergman projection is a realization of a classical analytic symbol.

math.AP↗

arXiv2020

Toeplitz operators with analytic symbols

We provide asymptotic formulas for the Bergman projector and Berezin-Toeplitz operators on a compact K{ä}hler manifold. These objects depend on an integer N and we study, in the limit N $\rightarrow$ +$\infty$, situations in which one can control them up to an error O(e^{-cN}) for some c > 0. We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to K{ä}hler man-ifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to O(e^{-cN}) on any real-analytic K{ä}hler manifold as N $\rightarrow$ +$\infty$. We also prove that Toeplitz operators with analytic symbols can be composed and inverted up to O(e^{-cN}). As an application, we study eigenfunction concentration for Toeplitz operators if both the manifold and the symbol are real-analytic. In this case we prove exponential decay in the classically forbidden region.

math.SP↗

arXiv2020

Uniform spectral asymptotics for semiclassical wells on phase space loops

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 allows us, in addition to the complete integrability of the system, to obtain eigenvalue asymptotics in a window $(-\infty,b_0+ε]$ for $ε> 0$ independent on the semiclassical parameter. These asymptotics are obtained in two complementary settings: either a symmetry of the system under translation along the curve, or a Morse hypothesis reminiscent of Helffer-Sjöstrand's "miniwell" situation.

math.SP↗

arXiv2020

Fractional exponential decay in the forbidden region for Toeplitz operators

We prove several results of concentration for eigenfunctions in Toeplitz quantization. With mild assumptions on the regularity, we prove that eigenfunctions are $O(exp(-cN^δ))$ away from the corresponding level set of the symbol, where N is the inverse semiclassical parameter and $0 < δ< 1$ depends on the regularity. As an application, we prove a precise bound for the free energy of spin systems at high temperatures, sharpening a result of Lieb.

math.SP↗

arXiv2018

The Bergman kernel in constant curvature

We present an elementary proof for an approximate expression of the Bergman kernel on homogeneous spaces, and products of them. The error term is exponentially small with respect to the inverse semiclassical parameter.

math.AP↗

arXiv2018

Quantum selection for spin systems

We report mathematical results on the process by which quantum order by disorder takes place for spin systems. The selection rules follow the influence of several competing contributions. Moreover there is no link between quantum selection and thermal selection. We present work in the general setting as well as toy models and examples for which quantum selection has an interesting behaviour.

cond-mat.stat-mech↗

arXiv2018

Low-energy spectrum of Toeplitz operators with a miniwell

In the semiclassical limit, it is well-known that the first eigenvector of a Toeplitz operator concentrates on the minimal set of the symbol. In this paper, we give a more precise criterion for concentration in the case where the minimal set of the symbol is a submanifold, in the spirit of the "miniwell condition" of Helffer-Sj{ö}strand.

math.SP↗

arXiv2017

Low-energy spectrum of Toeplitz operators: the case of wells

In the 1980s, Helffer and Sjöstrand examined in a series of articles the concentration of the ground state of a Schrödinger operator in the semiclassical limit. In a similar spirit, and using the asymptotics for the Szegö kernel, we show a theorem about the localization properties of the ground state of a Toeplitz operator, when the minimal set of the symbol is a finite set of non-degenerate critical points. Under the same condition on the symbol, for any integer K we describe the first K eigenvalues of the operator.

math.SP↗