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Jean-Claude Cuenin

Publications and source records attributed to Jean-Claude Cuenin.

At least 19 recordsLinked to original sources
arXiv2026

Random Schrödinger operators on manifolds and abstract bounds for multiplier-type operators

We study random Schrödinger operators on closed Riemannian manifolds with Anderson-type potentials. We prove high-probability spectral inclusion bounds showing that eigenvalues remain close to those of the Laplacian, with deviations controlled by a norm of the potential coefficients. Compared with deterministic bounds, this yields a square-root cancellation gain. The proof is based on a general principle showing that randomisation improves operator norm bounds for multiplier-type operators, which we formulate in both discrete and continuous settings.

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arXiv2026

Orthonormal Spectral Cluster Bounds on Manifolds with Nonpositive Curvature

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold with nonpositive sectional curvature. We prove sharp, logarithmically improved spectral cluster bounds for orthonormal systems in the supercritical range. More precisely, for spectral windows of size $(\log λ)^{-1}$, we obtain the orthonormal analogue of the logarithmically improved $L^q$ estimates of Hassell-Tacy. Our argument combines the universal orthonormal spectral cluster bounds of Frank-Sabin with Bérard-type kernel estimates and a generalization of the Bourgain-Shao-Sogge-Yao multiplier estimate to the orthonormal setting.

math.AP↗
arXiv2025

Eigenvalue bounds for Schrödinger operators with complex potentials on compact manifolds

We prove eigenvalue bounds for Schrödinger operator $-Δ_g+V$ on compact manifolds with complex potentials $V$. The bounds depend only on an $L^q$-norm of the potential, and they are shown to be optimal, in a certain sense, on the round sphere and more general Zoll manifolds. These bounds are natural analogues of Frank's \cite{MR2820160} results in the Euclidean case.

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arXiv2025

Smoothing inequalities for transport metrics in compact spaces

We prove general upper estimates for the distance between two Borel probability measures in Wasserstein metric in terms of the Fourier transforms of the measures. We work in compact manifolds including the torus, the Euclidean unit sphere, compact Lie groups and compact homogeneous spaces, and treat the Wasserstein metric $W_p$ in the full range $1 \le p \le \infty$ for the first time. The proofs are based on a comparison between the Wasserstein metric and a dual Sobolev norm, Riesz transform estimates and Hausdorff--Young inequalities on compact manifolds. As an application, we show that spherical designs are optimally close to the uniform measure on the sphere in Wasserstein metric.

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arXiv2024

Effective upper bounds on the number of resonance in potential scattering

We prove upper bounds on the number of resonances and eigenvalues of Schrödinger operators $-Δ+V$ with complex-valued potentials, where $d\geq 3$ is odd. The novel feature of our upper bounds is that they are \emph{effective}, in the sense that they only depend on an exponentially weighted norm of V. Our main focus is on potentials in the Lorentz space $L^{(d+1)/2,1/2}$, but we also obtain new results for compactly supported or pointwise decaying potentials. The main technical innovation, possibly of independent interest, are singular value estimates for Fourier-extension type operators. The obtained upper bounds not only recover several known results in a unified way, they also provide new bounds for potentials which are not amenable to previous methods.

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arXiv2024

Bounds for Eigenvalue Sums of Schrödinger Operators with Complex Radial Potentials

We consider eigenvalue sums of Schrödinger operators $-Δ+V$ on $L^2(\R^d)$ with complex radial potentials $V\in L^q(\R^d)$, $q<d$. We prove quantitative bounds on the distribution of the eigenvlaues in terms of the $L^q$ norm of $V$. A consequence of our bounds is that, if the eigenvlaues $(z_j)$ accumulates to a point in $(0,\infty)$, then $(\im z_j)$ is summable. The key technical tools are resolvent estimates in Schatten spaces. We show that these resolvent estimates follow from spectral measure estimates by an epsilon removal argument.

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arXiv2023

From spectral cluster to uniform resolvent estimates on compact manifolds

It is well known that uniform resolvent estimates imply spectral cluster estimates. We show that the converse is also true in some cases. In particular, the universal spectral cluster estimates of Sogge \cite{MR930395} for the Laplace--Beltrami operator on compact Riemannian manifolds without boundary directly imply the uniform Sobolev inequality of Dos Santos Ferreira, Kenig and Salo \cite{MR3200351}, without any reference to parametrices. This observation also yields new resolvent estimates for manifolds with boundary or with nonsmooth metrics, based on spectral cluster bounds of Smith--Sogge \cite{MR2316270} and Smith, Koch and Tataru \cite{MR2443996}, respectively. We also convert the recent spectral cluster bounds of Canzani and Galkowski \cite{Canzani--Galkowski} to improved resolvent bounds. Moreover, we show that the resolvent estimates are stable under perturbations and use this to establish uniform Sobolev and spectral cluster inequalities for Schrödinger operators with singular potentials.

math.AP↗
arXiv2022

Random Schrödinger operators with complex decaying potentials

We prove that the eigenvalues of a continuum random Schrödinger operator $-Δ+ V_ω$ of Anderson type, with complex decaying potential, can be bounded (with high probability) in terms of an $L^q$ norm of the potential for all $q\leq d+1$. This shows that in the random setting, the exponent $q$ can be essentially doubled compared to the deterministic bounds of Frank (Bull. Lond. Math. Soc., 2011). This improvement is based on ideas of Bourgain (Discrete Contin. Dyn. Syst., 2002) related to almost sure scattering for lattice Schrödinger operators.

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arXiv2021

Sharp time decay estimates for the discrete Klein-Gordon equation

We establish sharp time decay estimates for the the Klein-Gordon equation on the cubic lattice in dimensions $d=2,3,4$. The $\ell^1\to\ell^{\infty}$ dispersive decay rate is $|t|^{-3/4}$ for $d=2$, $|t|^{-7/6}$ for $d=3$ and $|t|^{-3/2}\log|t|$ for $d=4$. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.

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arXiv2021

Counterexample to the Laptev--Safronov conjecture

We prove that the Laptev--Safronov conjecture (Comm. Math. Phys., 2009) is false in the range that is not covered by Frank's positive result (Bull. Lond. Math. Soc., 2011). The simple counterexample is adaptable to a large class of Schrödinger type operators, for which we also prove new sharp upper bounds.

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arXiv2021

Fourier transform of surface--carried measures of two-dimensional generic surfaces and applications

We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schrödinger operators on the cubic lattice.

math.AP↗
arXiv2021

Schrödinger operators with complex sparse potentials

We establish quantitative upper and lower bounds for Schrödinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S.\ Boegli (Comm. Math. Phys., 2017, 352, 629-639), of a Schrödinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull. Lond. Math. Soc., 2011, 43, 745-750 and Trans. Amer. Math. Soc., 2018, 370, 219-240) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann. Inst. H. Poincaré Sect. A (N.S.), 1983, 38, 7-13) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp.

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