54 problems

Let $l$ be a nonnegative integer, and consider an ellipsoid with semiaxes described by parameters $b1$, $b2$, and $b3$. Define the sequences $N_k$, $k=0\ldots 2l$, and $\hat{N}_k$,…

Pleijel's conjecture. The same asymptotic result should hold for a free membrane, that is, for Neumann boundary conditions.

Let a graph be close to a tree, and let the nodal count of its $n$-th eigenstate be the number of its nodal domains. Nodal count conjecture. The nodal count of the $n$-th eigenstat…

Let a chain domain consist of $k$ planar domains connected in a line, and consider its Neumann eigenfunctions ordered by increasing eigenvalue. Courant sharpness conjecture. The fi…

Let $\phi$ be a Maass cusp form on $\mathbb{X}$, let $\lambda_\phi$ be its Laplace eigenvalue, and let $N(\phi)$ denote the number of connected components of…

Consider the two-dimensional Gaussian random wave model and wave functions of chaotic billiards. Their nodal sets are the zero sets of the corresponding random waves or eigenfuncti…

Let $\Omega_\epsilon$ be the perturbed symmetric dumbbell domain, with $M$ the length parameter and $g$ the neck-defining function. Let $\{\mu_k\}_{k=1}^\infty$ denote the Neumann…

Conjecture on ternary strong nodal domains. For any eigenfunction $f$ of $H(n,3)$, $n\geq 1$, with eigenvalue $3n$ we have $\mathrm{SND}(f)\geq n+1$.

Bírógyiköğlü–Hordijk–Leydold–Pisanski–Stadler conjecture. For every $1\leq i\leq n-2$, there is an eigenfunction $f$ of $H(n,2)$ with eigenvalue $2i$ such that $\mathrm{SND}(f)=2$.

Let $cmathscr{C}_{_{!P}}$ be a cylinder of perimeter $P$, let $V$ be a nodal domain of an eigenfunction with eigenvalue $clambda$, and let $cmathbb{D}$ denote the unit disc with fi…

Nodal-domain count conjecture. There are constants $c_1(\epsilon,p,q)$ and $c_2(\epsilon,p,q)$ such that, with probability $1-o_N(1)$, the $i$th eigenvector $\bm{\varphi}$ satisfie…

Let $\Omega$ range over planar domains with regular boundary, and let $\operatorname{Pl}(\Delta^D_\Omega)$ and $\operatorname{Pl}(\Delta^N_\Omega)$ denote the Pleijel constants for…

Let the random plane wave (RPW) be the stationary Gaussian field in $\mathbb{R}^2$ considered in the source, let $\ell$ denote the level, let $R$ denote the linear size of a box, a…

Let $f$ be a centred stationary Gaussian field in $\mathbb{R}^2$ such that, with probability one, it is sufficiently smooth; the joint distribution of $f$ and its derivatives is no…

Let interacting spin-$\tfrac{1}{2}$ fermions have a spatial wave function, and consider its nodal set separating regions where the wave function has different signs. Nodal-domain c…

Full-support eigenfunction conjecture. There exists a choice of eigenfunctions $\psi_n$ forming an orthonormal basis of $L^2(\mathcal{G})$ and a subsequence…

Let $(M^n,g)$ be a closed Riemannian manifold, and let $u_k$ be a Laplace eigenfunction with eigenvalue $\lambda_k$. For a ball $B$ of radius comparable to $1/\sqrt{\lambda_k}$, co…

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with piecewise smooth boundary, or a compact Riemannian surface, with or without boundary. Let $\Delta$ be the Laplace–Beltram…

Small-parameter stability conjecture. There exists $h_0>0$ such that for $0<h\leq h_0$, the Courant-sharp cases for the Robin problem are the same, except the fifth one, as those f…

Let $M$ be a compact two-dimensional manifold and let $H=-\operatorname{div}(a(x)\cdot\nabla)$ be a uniformly elliptic operator of Laplacian type, with eigenfunctions $\phi_n$ form…

Let $\Gamma$ be a compact metric graph, let $\partial\Gamma$ denote its boundary, let $\beta$ be its first Betti number, and let $\omega_n$ denote the Neumann surplus associated wi…

Polterovich's conjecture. The optimal upper bound is

Let $D=[0,1]^2\subset\mathbb{R}^2$ be the unit square, let $\phi_j$ be a Dirichlet eigenfunction associated with the $j$-th eigenvalue, and let $N(\phi_j)$ denote the number of nod…

Assume $d\geqslant 2$, that the coefficient $a(t)$ or $A(s)$ is positive and locally Lipschitz continuous, and that $V(x)$ is a radial, locally bounded confining potential. Let…

Let $L+V(x)=\psi(-\Delta)+V(x)$ be a non-local Schrödinger operator with confining potential $V$, and let $f_n$ be its $n$-th eigenfunction. A nodal part is a connected component o…