arXiv research

We create a polynomial with knot-like nodal lines.

Algorithm constructs polynomials with specific nodal sets.

New bounds found for nodal sets on special manifolds.

Study of loops in sums of Laplace eigenfunctions on surfaces.

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

New chaos formula simplifies variance calculation for Gaussian nodal volumes.

It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in $\mathbb R^3$ vanishes on a real analytically ruled two-dimensional surface $S \subset \mathbb R^3$ then $S$ is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial…

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.

We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.

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

Study conformal invariants from nodal sets on manifolds with boundary.

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

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 consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the…

Quantum ergodic eigenfunctions improve inner radius bounds of nodal domains.

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.

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 $mN$.

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 …

We use a modified Bochner technique to derive an inequality relating the nodal set of eigenspinors to eigenvalues of the Dirac operator on closed surfaces. In addition, we apply this technique to solutions of similar spinorial equations.

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…

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

We study the asymptotic properties of the conormal cycle of nodal sets associated to a random superposition of eigenfunctions of the Laplacian on a smooth compact Riemannian manifold without boundary. In the case where the dimension is odd, we show that the expectation of the corresponding current of integration equidi…

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…

The paper proves that Gaussian field critical points have finite moments.

Sharp uncertainty principle for nodal sets in singular spaces.