arXiv subjects

Charlotte Dietze

Publications and source records attributed to Charlotte Dietze.

15 recordsLinked to original sources

arXiv2026

Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential

We consider a gas of weakly interacting bosons in three dimensions subject to an external potential in the mean field regime. Assuming that the initial state of our system is a product state, we show that in the trace topology of one-body density matrices, the dynamics of the system can be described by the solution to the corresponding Hartree type equation. Using a dispersive estimate for the Hartree type equation, we obtain an error term that is uniform in time. Moreover, the dependence of the error term on the particle number is optimal.

math-ph↗

arXiv2026

Weyl formulae for some singular metrics with application to acoustic modes in gas giants

This paper is motivated by recent works on inverse problems for acoustic wave propagation in the interior of gas giant planets. In such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with boundary whose metric blows up near the boundary. Here, the spectral analysis of the corresponding Laplace-Beltrami operator is presented and the Weyl law is derived. The involved exponents depend on the Hausdorff dimension which, in the supercritical case, is larger than the topological dimension.

math.AP↗

arXiv2026

Weyl asymptotics for singular metrics with a variable boundary degeneracy exponent

We consider a compact smooth manifold $X$ of dimension $n+1$ with boundary $M=\partial X$. In a collar neighborhood of $M$, we assume that the metric has the form $g=u^{-α}\bar g$, where $u$ is a boundary defining function, $α\in C^1(M;[0,2))$ and $\bar g$ is a $C^1$ Riemannian metric up to $M$. Since $α<2$, the boundary lies at finite $g$-distance and $(X,g)$ is a singular metric space. We study the Weyl asymptotics of the Friedrichs Laplacian $\triangle\_g$ when the degeneracy exponent $α$ varies along $M$. If the maximum $α\_{\mathrm{max}}$ of $α$ on $M$ is strictly larger than the critical value $α\_c=\frac{2}{n+1}$, then we prove that the points where $α$ is close to $α\_{\mathrm{max}}$ govern the leading term in the Weyl asymptotics. If $α\_{\mathrm{max}}\leqα\_c$, then the leading term is governed by the truncated volume $\vol\_g(\{\dist(\cdot,M)>λ^{-1/2}\})$. When the maximum set of $α$ is Morse-Bott, we compute the associated constants and the logarithmic corrections. To the best of our knowledge, this is the first Weyl law in this setting with a boundary-dependent degeneracy exponent. The results highlight a sharp transition at $α\_c$ between a boundary-dominated non-classical regime and a truncated-volume regime.

math.SP↗

arXiv2026

Convergence speed for the average density of eigenfunctions for singular Riemannian manifolds

We consider a class of singular Riemannian metrics on a compact Riemannian manifold with boundary and the eigenfunctions of the corresponding Laplace-Beltrami operator. In our setting, the average density of eigenfunctions with eigenvalue less than $λ$ converges weakly to the uniform normalised measure on the boundary as $λ\to\infty$. In this work, we show a quantitative estimate on the speed of this convergence in the Wasserstein-sense in the transverse coordinate to the boundary.

math.AP↗

arXiv2025

The critical case for the concentration of eigenfunctions on singular Riemannian manifolds

We consider a compact Riemannian manifold with boundary with a certain class of critical singular Riemannian metrics that are singular at the boundary. The corresponding Laplace-Beltrami operator can be seen as a Grushin-type operator plus a potential. We show in the critical case that the average density of eigenfunctions for the Laplace-Beltrami operator with eigenvalues below $λ>0$ is distributed over all length scales between $λ^{-1/2}$ and $1$ near the boundary. We give a precise description of this distribution as $λ\to\infty$.

math.SP↗

arXiv2025

Minimizing sequences of Sobolev inequalities revisited

We give a new proof of the compactness of minimizing sequences of the Sobolev inequalities in the critical case. Our approach relies on a simplified version of the concentration-compactness principle, which does not require any refinement of the Sobolev embedding theorem.

math.AP↗

arXiv2024

Hardy-Sobolev interpolation inequalities

We derive a family of interpolation estimates which improve Hardy's inequality and cover the Sobolev critical exponent. We also determine all optimizers among radial functions in the endpoint case and discuss open questions on nonrestricted optimizers.

math.CA↗

arXiv2024

Peculiar behavior of the principal Laplacian eigenvalue for large negative Robin parameters

Let $Ω\subset\mathbb{R}^n$ with $n\ge 2$ be a bounded Lipschitz domain with outer unit normal $ν$. For $α\in\mathbb{R}$ let $R_Ω^α$ be the Laplacian in $Ω$ with the Robin boundary condition $\partial_νu+αu=0$, and denote by $E(R^α_Ω)$ its principal eigenvalue. In 2017 Bucur, Freitas and Kennedy stated the following open question: Does the limit of the ratio $E(R_Ω^α)/ α^2$ for $α\to-\infty$ always exist? We give a negative answer.

math.SP↗

arXiv2024

Concentration of eigenfunctions on singular Riemannian manifolds

We consider a compact Riemannian manifold with boundary and a metric that is singular at the boundary. The associated Laplace-Beltrami operator is of the form of a Grushin operator plus a singular potential. In a supercritical parameter regime, we identify the rate of concentration and profile of the high-frequency eigenfunctions that accumulate at the boundary. We give an application to acoustic modes on gas planets.

math.AP↗

arXiv2023

Isoperimetric inequalities for inner parallel curves

We prove weighted isoperimetric inequalities for smooth, bounded, and simply connected domains. More precisely, we show that the moment of inertia of inner parallel curves for domains with fixed perimeter attains its maximum for a disk. This inequality, which was previously only known for convex domains, allows us to extend an isoperimetric inequality for the magnetic Robin Laplacian to non-convex centrally symmetric domains. Furthermore, we extend our isoperimetric inequality for moments of inertia, which are second moments, to $p$-th moments for all $p$ smaller than or equal to two. We also show that the disk is a strict local maximiser in the nearly circular, centrally symmetric case for all $p$ strictly less than three, and that the inequality fails for all $p$ strictly bigger than three.

math.AP↗

arXiv2023

Totally odd depth-graded multiple zeta values and period polynomials

Inspired by a paper of Tasaka, we study the relations between totally odd, motivic depth-graded multiple zeta values. Our main objective is to determine the rank of the matrix $C_{N,r}$ defined by Brown. We will give new proofs for (conjecturally optimal) upper bounds on the rank of $C_{N,3}$ and $C_{N,4}$, which were first obtained by Tasaka. Finally, we present a recursive approach to the general problem, which reduces evaluating the rank of $C_{N,r}$ to an isomorphism conjecture.

math.NT↗

arXiv2023

Weyl's law for Neumann Schrödinger operators on Hölder domains

We review recent results on the semiclassical behaviour of Schrödinger operators with Neumann boundary conditions. In this setting, the validity of Weyl's law requires additional conditions on the potential. We will explain the techniques needed to control the number of bound states near the boundary, thus leading to universal estimates on the number of bound states.

math-ph↗

arXiv2023

Focusing dynamics of 2D Bose gases in the instability regime

We consider the dynamics of a 2D Bose gas with an interaction potential of the form $N^{2β-1}w(N^β\cdot)$ for $β\in (0,3/2)$. The interaction may be chosen to be negative and large, leading to the instability regime where the corresponding focusing cubic nonlinear Schr{ö}dinger equation (NLS) may blow up in finite time. We show that to leading order, the $N$-body quantum dynamics can be effectively described by the NLS prior to the blow-up time. Moreover, we prove the validity of the Bogoliubov approximation, where the excitations from the condensate are captured in a norm approximation of the many-body dynamics.

math-ph↗

arXiv2023

Semiclassical estimates for Schrödinger operators with Neumann boundary conditions on Hölder domains

We prove a universal bound for the number of negative eigenvalues of Schrödinger operators with Neumann boundary conditions on bounded Hölder domains, under suitable assumptions on the Hölder exponent and the external potential. Our bound yields the same semiclassical behaviour as the Weyl asymptotics for smooth domains. We also discuss different cases where Weyl's law holds and fails.

math-ph↗

arXiv2021

Dispersive Estimates for Nonlinear Schrödinger Equations with External Potentials

We consider the long time dynamics of nonlinear Schrödinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate, which is comparable to the decay of free solutions. Our proof relies on good control on a high Sobolev norm of the solution to estimate the terms in Duhamel's formula.

math-ph↗