21 problems

Let $H_\varepsilon^\omega$ be the magnetic Schrödinger operator obtained by replacing the approximating graph by a network with fat edge width $\varepsilon$, replacing the $\delta$…

Nodal universality conjecture. The variance has linear growth

Let $G:X\to X$ and $H:A\to A$ be irreflexive quantum graphs. The quantum graph isomorphism game is … with … A quantum graph isomorphism from $G$ to $H$ is the corresponding structu…

Let $G:X\to X$ and $H:A\to A$ be quantum graphs. A quantum homomorphism from $G$ to $H$ is a quantum function $E:X\to_{\mathcal K}A$ satisfying the quantum graph homomorphism condi…

Generic non-optimality conjecture. The set of edge-length vectors and values $M>0$ for which the constant potential minimizes the spectral gap in $C_{T,M}$ is of the first Baire ca…

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 $\{\Gamma_{\vec{l}}^{(\beta)}\}_{\beta\nearrow\infty}$ be any sequence of standard graphs parameterized by their first Betti numbers, with each graph having rationally independ…

Let … be its nodal surplus random variable. Write $\operatorname{Var}(\sigma)$ for its variance. Al-Obeid–Band–Berkolaiko's conjecture. As $\to\infty$, … where the convergence is i…

Let a quantum graph be a metric graph with edge lengths, and let the nodal domains of an eigenfunction be the connected components of the complement of its zero set. Consider well-…

Pumpkin hot-spots conjecture. The set $M$ has exactly two points and

Close hot-spots conjecture. For every $\epsilon>0$, there exists a doubly connected graph $\gamma$ such that

Rationally independent edge-length conjecture. There exists a graph $\gamma$ whose edge lengths are pairwise rationally independent and for which $|M|=3$; more generally, for every…

Generic two-hot-spots conjecture. Generically, $mu_2(amma)$ is simple and the corresponding eigenfunction has exactly one minimum and one maximum, so that $|M|=2$.

Finite-component conjecture. The set $M$ always has finitely many connected components; in particular, it is either finite or uncountable.

Bögli–Kennedy–Lang conjecture. If $\operatorname{Re}\alpha \to -\infty$, then there exists an infinite family of analytic branches of absolutely divergent eigenvalues such that, if…

Neumann surplus central limit conjecture. Then

Let $\mathcal{G}$ be any connected, compact graph with total length $L$ and diameter $D$, and let $\mu_k(\mathcal{G})$ denote its $k$-th eigenvalue. Assume that $L/k>D/2$. Strict d…

Let $\Gamma$ be a graph with some semi-infinite, asymptotically straight edges, and suppose … Consider the renormalized operator $-\Delta_{\alpha,\Gamma}+\frac14\alpha^2$ as…

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…

Let $(u_{n,\omega}^{\pm},0)$ be the stationary standing-wave branch indexed by $n\in\mathbb{N}$, and let the spectral stability problem be the linearized eigenvalue problem for the…

Differentiability conjecture. If $athsf{E}_d\not=\emptyset$, then the semigroup is differentiable for every $t>T$; equivalently, for every $x\in X$, the map