Uniform restriction lower bounds and nongeodesic common nodal curves in integrable systems

Xianchao Wu; Xiao Xiao

Home/Archive/arXiv

arXiv · 2610.23836

Uniform restriction lower bounds and nongeodesic common nodal curves in integrable systems

Xianchao Wu·Xiao Xiao

View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct a family of real analytic metrics on the two-torus, for which a fixed embedded nongeodesic circle is a nodal component of infinitely many real Laplace eigenfunctions. The examples address a question asked by Bourgain-Rudnick \cite{BR11}, and give a negative answer to the question asked by Toth-Zelditch \cite{TZ21} and Han \cite{Han}: whether a curve shared by infinitely many nodal sets must be geodesic. To explain the restriction obstruction, we prove uniform $L^2$ lower bounds for joint eigenfunctions of quantum completely integrable (QCI) systems under transverse crossing and isolation of a tangential characteristic branch, then extend to multi-branch setting. The results are used to characterize the possible joint nodal curves in the Liouville tori, narrowing down the choices. Finally, the counterexample realizes one possibility for which cancellation happens exactly on a nongeodesic curve.

Explore related subjects

math.AP math-ph math.DG math.MP math.SP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Xianchao Wu, Xiao Xiao. 2026-10-16. Uniform restriction lower bounds and nongeodesic common nodal curves in integrable systems. https://arxiv.org/abs/2610.23836

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantitative Schauder estimates for hypoelliptic equations

We derive Schauder estimates using ideas from Campanato's approach for a general class of local hypoelliptic operators and non-local kinetic equations. The method covers equations in divergence and non-divergence form. In particular our results are applicable to the inhomogeneous Landau and to the non-cutoff Boltzmann equation. The paper is self-contained.

math.AP↗

arXiv2026

Scaling-Critical Theory for the Boltzmann and Landau Equations

This paper develops a scaling-critical theory for the spatially inhomogeneous non-cutoff Boltzmann and Landau equations with very soft potentials. For sufficiently small initial perturbations in a localized, weighted, anisotropic Riesz-potential norm, we prove global well-posedness near a Maxwellian in the whole space. This critical phase-space norm captures the Boltzmann--Landau scaling, the velocity-dependent anisotropy, and the hypoelliptic transport structure. The proof combines frozen-operator estimates, a critical fixed-point argument, and weighted hypocoercive energy estimates. We also develop a short-time pointwise Green-function theory for variable-coefficient kinetic equations with a nonnegative Hölder background. We first construct the small-jump Green function by freezing coefficients along kinetic characteristics and then recover the full kernel through a convergent parametrix expansion. The resulting bounds capture the fractional Kolmogorov geometry near characteristics and rapid decay away from them. These estimates form the analytic core of the scaling-critical theory and provide a robust pointwise framework for further regularity analysis in the soft-potential regime.

math.AP↗

arXiv2026

Nonlinear effects of incoming time-periodic boundary conditions on shock waves for scalar conservation laws

This paper is concerned with the large-time behavior of inviscid and viscous shocks for scalar conservation laws on the half-line $(-\infty,0)$ under an incoming \emph{time-periodic} boundary condition. This problem arises naturally from classical piston problems in fluid mechanics. Even for arbitrarily small periodic oscillations of the boundary data, the effect of the boundary manifests itself through a persistent time-periodic component near the boundary, which behaves as a \emph{boundary wave}. For inviscid shocks moving away from the boundary ($s<0$), this boundary wave repeatedly interacts with the shock over infinitely many periods. We show that the repeated interactions do not generate secular growth, but instead produce a finite \emph{cumulative} correction to the asymptotic shock location. For the inviscid stationary case $s=0$, the time-periodic boundary \emph{modulates} the background stationary shock into a genuinely time-periodic shock configuration, to which the solution converges. In the viscous case with $s<0$, the boundary wave couples with the viscous shock through a transitional region induced by diffusion, and the solution converges asymptotically to their superposition. Our results show that for scalar conservation laws, the persistent boundary wave modifies the shock without producing resonance.

math.AP↗