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Philippe Charron

Publications and source records attributed to Philippe Charron.

8 recordsLinked to original sources
arXiv2026

Capacity estimates and improved lower bounds for the inner radius of nodal domains

We show that for every closed, smooth manifold $(M,g)$ of dimension $d$, there exists $c(g)$ such that any nodal domain $\Omega_\lambda$ of a Laplace eigenfunction with eigenvalue $\lambda$ contains a geodesic ball of radius at least $c(g) \lambda^{-1/2} \log\log(\lambda)^{-1/2}$ if $d=3$ and $c(g) \lambda^{-1/2} \log(\lambda)^{-\frac{d-3}{2}}$ if $d >3$. This ball is centered at any point at which the eigenfunction attains its maximum in absolute value within the nodal domain. Furthermore, we show that for any $d \geq 3$, there exist sequences of $\lambda$-nodal domains on $\mathbb{T}^d$ whose inner radius is of order $o(\lambda^{-1/2})$.

math.SP↗
arXiv2024

On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds

On a closed analytic manifold $(M,g)$, let $\phi_i$ be the eigenfunctions of $\Delta_g$ with eigenvalues $\lambda_i^2$ and let $f:=\prod \phi_{k_j}$ be a finite product of Laplace-Beltrami eigenfunctions. We show that $\left\langle f, \phi_i \right\rangle_{L^2(M)}$ decays exponentially as soon as $\lambda_i > C \sum \lambda_{k_j}$ for some constant $C$ depending only on $M$. Moreover, by using a lower bound on $\| f \|_{L^2(M)} $, we show that $99\%$ of the $L^2$-mass of $f$ can be recovered using only finitely many Fourier coefficients.

math.AP↗
arXiv2023

Sturm-Hurwitz Theorem for quantum graphs

We prove upper and lower bounds for the number of zeroes of linear combinations of Schr\"odinger eigenfunctions on metric (quantum) graphs. These bounds are distinct from both the interval and manifolds. We complement these bounds by giving non-trivial examples for the lower bound as well as sharp examples for the upper bound. In particular, we show that even tree graphs differ from the interval with respect to the nodal count of linear combinations of eigenfunctions. This stands in distinction to previous results which show that all tree graphs have to same eigenfunction nodal count as the interval.

math-ph↗
arXiv2023

The inner radius of nodal domains in high dimensions

We prove that every nodal domain of an eigenfunction of the Laplacian of eigenvalue $\lambda$ on a $d$-dimensional closed Riemannian manifold contains a ball of radius $c\lambda^{-1/2}(\log\lambda)^{-(d-2)/2}$. This ball is centered at a point at which the eigenfunction attains its maximum in absolute value within the nodal domain.

math.AP↗
arXiv2021

Pleijel's theorem for Schr\"odinger operators

We are concerned in this paper with the real eigenfunctions of Schr\"odinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the inequality stated in Courant's theorem is strict, except for finitely many eigenvalues. Results of this type originated in 1956 with Pleijel's Theorem on the Dirichlet Laplacian and were obtained for some classes of Schr\"odinger operators by the first author, alone and in collaboration with B. Helffer and T. Hoffmann-Ostenhof. Using methods in part inspired by work of the second author on Neumann and Robin Laplacians, we greatly extend the scope of these previous results.

math.SP↗
arXiv2020

Non-boundedness of the number of super level domains of eigenfunctions

Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in this case, the first eigenfunction is constant, and looking at the level sets of $u$ amounts to looking at the nodal sets $\{u-a=0\}$, where $a$ is a real constant. In the first part of the paper, we prove that the Extended Courant property is false for the subequilateral triangle and for regular $N$-gons ($N$ large), with the Neumann boundary condition. More precisely, we prove that there exists a Neumann eigenfunction $u_k$ of the $N$-gon, with labelling $k$, $4 \le k \le 6$, such that the set $\{u_k \not = 1\}$ has $(N+1)$ connected components. In the second part, we prove that there exists a metric $g$ on $\mathbb{T}^2$ (resp. on $\mathbb{S}^2$), which can be chosen arbitrarily close to the flat metric (resp. round metric), and an eigenfunction $u$ of the associated Laplace-Beltrami operator, such that the set $\{u \not = 1\}$ has infinitely many connected components. In particular the Extended Courant property is false for these closed surfaces. These results are strongly motivated by a recent paper by Buhovsky, Logunov and Sodin. As for the positive direction, in Appendix~B, we prove that the Extended Courant property is true for the isotropic quantum harmonic oscillator in $\mathbb{R}^2$.

math.SP↗
arXiv2016

Pleijel's theorem for Schrödinger operators with radial potentials

In 1956 $Å$. Pleijel gave his celebrated theorem showing that the inequality in Courant's theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunctions as the eigenvalues tend to $+\infty$. A similar question occurs naturally for Schr"\odinger operators. The first significant result has been obtained recently by the first author for the harmonic oscillator. The purpose of this paper is to consider more general potentials which are radial. We will analyze the case when the potential tends to $+\infty$ and the case when the potential is negative and tends to zero, where the considered eigenfucntion are associated to the eigenvalues below the essential spectrum.

math.SP↗