90 problems

Let a quantum system arise as the quantization of a classically chaotic system, and consider its eigenvalues and their mean spacing. Random Matrix conjecture. Generically, the eige…

Spectral mapping conjecture. For the operator $\boldsymbol{A}_k$, the spectral mapping theorem should hold. This would imply estimate (6.2) in the cited stability analysis. The con…

Self-intersection expansion conjecture. If the terms containing $\tau^m$ are generated by diagrams with $n=m-1$ self-intersections, then

Let $A^*A$ be the operator whose eigensubspaces vary with the magnetic parameters, and consider the localization mechanism producing infinitely degenerate eigenvalues and continuou…

Let $\Gamma$ be an infinite asymptotically straight curve without self-intersections, with two leads. Suppose that the distance between points of $\Gamma$ becomes small somewhere,…

Let a finite, smooth, non-closed curve have arclength parameter $s$ and curvature $k(s)$. Define the comparison operator … with Dirichlet boundary conditions at the curve endpoints…

Consider periodic systems whose components have different dimensions, in particular spheres joined by linear segments, and let the gap-to-band width ratio refer to the ratio of the…

Let $\psi_k^{(0)}\rangle$ be the basis of eigenfunctions of the unperturbed system, and let $A_i(n,{\mathbf L};v)$ denote the normalized coefficient quantity for the corresponding…

A Šeba-class system is an integrable system perturbed by a point singularity or in an analogous manner; the relevant spectral and eigenfunction quantities are those studied for qua…

The one-dimensional scattering matrix for the delta-prime perturbation of the free Schrödinger operator on ${\mathbb R}$ is considered. A geometric realization means obtaining this…

Consider two waveguides of widths $d_1$ and $d_2$, let … and define the waveguide-asymmetry parameter … Let $E(\ell)$ denote the eigenvalue associated with a narrow window of width…

Asymptotic combinatorial substantiation conjecture. This agreement might be rigorously substantiated by asymptotic combinatorial theory.

In a quantum graph, consider a group of isometric periodic orbits and the code words that encode the orbits in that group. Permutation-and-symmetry conjecture. The code words of th…

Nodal universality conjecture. The variance has linear growth

Let a quantum graph be associated with a transition matrix having a spectral gap, and let RMT-like behavior mean spectral statistics described by the appropriate random-matrix ense…

For positive integers $n$ and $m$ with $m\geq n/4$, let $T^\downarrow(n,m)$ denote the lower quantum Turán number, the smallest integer $d$ such that there exists an operator syste…

Let a quantum graph have suitable mild assumptions on its graph structure and edge lengths, and let the zeros of its $k$-th eigenfunction be counted. Smilansky–Gnutzmann–Weber conj…

Shape conjecture. The action ground state $u=(u_c,u_h)$ has one of the shapes given by Proposition, moving from case $(iv)$ to case $(i)$ as $L$ increases from $0$ to $\infty$.

Finite quantum group realization conjecture. Every finite quantum group arises as the quantum automorphism group of a finite-dimensional quantum graph.

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…

Let $\Gamma$ be a metric graph with $V$ vertices, $E$ edges, edge lengths $(\ell_e)_{e\in\mathcal{E}}$, total length … and let $[T_\Gamma]$ denote the integer associated with…

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…

Consider metric graphs whose edge lengths occur on two scales, with vertex conditions depending on the shrinking-edge parameter as in the two-edge model discussed above. The negati…

Let $bb_1$ and $bb_2$ denote the first two positive eigenvalues of a graph in the class observed in the experiment, namely graphs (often complete graphs) joined by one edge, also k…