arXiv research

Lower bounds on nodal sets for biharmonic Steklov problems established.

New bounds found for nodal sets on special manifolds.

Estimates nodal sets of eigenspinors on closed surfaces.

Study finds nodal solutions of Yamabe equation constant along isoparametric levels.

The study examines nodal geometry on smooth manifolds, providing bounds and insights.

New estimates for nodal and singular sets of parabolic inequalities.

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator …

The paper studies nodal sets of solutions to parabolic equations, proving finiteness and monotonicity properties.

The paper studies eigenfunctions and nodal sets of the Witten-Laplacian.

The paper studies nodal sets of smooth functions on compact manifolds and proves new bounds.

Study compares nodal sets of solutions to the Allen-Cahn equation.

Study conformal invariants from nodal sets on manifolds with boundary.

Study on the distribution of nodal sets on random superpositions of eigenfunctions.

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.

Study on the nodal set of Dirac equation solutions on manifolds.

We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.

Study on points where random spherical harmonic nodal set meets tangent vector field.

Study geodesics and nodal sets on hyperbolic surfaces, proving finiteness results.

New harmonic functions show nodal sets can be topologically complex despite frequency and regularity constraints.

The paper proves nodal properties for quantum harmonic oscillator and related Schrödinger operators.

We create a polynomial with knot-like nodal lines.

Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.

We prove a natural inequality which implies the known lower bounds for the $(n-1)$-dimensional Hausdorff measure of nodal sets for smooth compact manifolds.

Eigenfunctions on manifolds concentrate near nodal sets.

Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for …

The study improves upper bounds on nodal sets using quantum ergodicity.

For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.

Random spherical harmonics on $S^3$ have a single nodal component with expected genus proportional to $mN$.

Eigenfunctions of Laplacian form knots in 3-manifolds.

We study the nodal sets of non-degenerate eigenfunctions of the Laplacian on fibre bundles $π{:}\, M\to B$ in the adiabatic limit. This limit consists in considering a family $G_\varepsilon$ of Riemannian metrics, that are close to Riemannian submersions, for which the ratio of the diameter of the fibres to that of the…

Eigenfunctions of the Dirac operator on spheres reveal complex nodal structures.

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σin a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is Σ. We present…

New upper bound found for nodal sets of Laplace eigenfunctions.

The paper estimates the measure of nodal sets for solutions to a specific type of Schrödinger equation.

Eigenfunctions on spheres and tori with complex nodal sets.

Formulas calculate zero-set volume on Riemannian manifolds.

Given a Laplace eigenfunction on a surface, we study the distribution of its extrema on the nodal domains. It is classically known that the absolute value of the eigenfunction is asymptotically bounded by the 4-th root of the eigenvalue. It turns out that the number of nodal domains where the eigenfunction has an extre…

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a …

Algorithm constructs polynomials with specific nodal sets.

Study of loops in sums of Laplace eigenfunctions on surfaces.

We investigate nodal sets of magnetic Schroedinger operators with zero magnetic field, acting on a non simply connected domain in $\r^2$. For the case of circulation 1/2 of the magnetic vector potential around each hole in the region, we obtain a charactisation of the nodal set, and use this to obtain bounds on the mul…

The study bounds the Hausdorff measure of nodal sets of Laplace eigenfunctions.

Study eigenvalues and nodal sets of twisted Dirac operators on surfaces.

We consider the problem of prescribing the nodal set of the first nontrivial eigenfunction of the Laplacian in a conformal class. Our main result is that, given a separating closed hypersurface $Σ$ in a compact Riemannian manifold $(M,g_0)$ of dimension $d \geq 3$, there is a metric $g$ on $M$ conformally equivalent to…

Sharp uncertainty principle for nodal sets in singular spaces.

We use the Dong-Sogge-Zelditch formula to obtain a lower bound for the volume of the nodal sets of eigenfunctions. Our result improves the recent results of Sogge-Zelditch and in dimensions n \leq 5 gives a new proof for the lower bounds of Colding-Minicozzi.

This paper is concerned with the location of nodal sets of eigenfunctions of the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the Euclidean space of arbitrary dimension. In the limit when the radius of the neighbourhood tends to zero, it is known that spectral properties of the Laplacian are a…

Study shows strong min-max principle for phase transitions.