42 problems

Let $u$ solve the one-dimensional fourth-order mixed-dispersion nonlinear Schrödinger equation, with power $alpha$ and ground-state solutions of the associated stationary equation.…

Consider the two-dimensional Ising model on the periodic torus … with nearest-neighbor ferromagnetic coupling $J$, Hamiltonian $\mathcal H_L$, periodic striped configurations…

Consider the spin-$J$ ferromagnetic XXZ chain in finite volume or infinite volume, whose frustration-free ground states minimize every nearest-neighbor interaction. Gottstein and W…

For the dodecahedron and icosahedron, the coplanar minimum energies satisfy … and … respectively. Conjecture. The dodecahedron and icosahedron have no coplanar ground states; conse…

Let $v\in{\mathfrak V}_{{\mathsf r}{\mathsf e}{\mathsf g}}\setminus{\mathfrak V}_{{\mathsf U}{\mathsf V}}$, so that the ultraviolet cutoff is removed, and let…

Conjecture. The bound (5.7) could remain valid assuming only the finiteness of $\tau(e^{-\beta h})$. This concerns extending the stated estimate beyond the hypotheses used to deriv…

Let $u_0$ be initial data for the one-dimensional fourth-order mixed-dispersion nonlinear Schrödinger equation, with power $\alpha<8$ and parameter satisfying $a<\sqrt{b}$. Let…

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 $\infty$.

Periodic Motzkin ground-state conjecture. The periodic Motzkin chain with $N$ sites has $2N+1$ degenerate ground states with zero eigenvalue, with independent states…

Let $u\in H^{1,a}(\mathbb{R}^d)\backslash\{0\}$ be a ground state for the degenerate nonlinear Schrödinger equation, meaning that … where … and … is the associated Nehari manifold.…

A ground state is a configuration of interacting particles that minimizes the system's energy, and a periodic lattice is a particle arrangement invariant under translations by the…

Let $w$ be a positive-definite interaction potential, and let $\mu$ be a sufficiently small positive density parameter. Consider infinite ground states of the Gross–Pitaevskii equa…

Positive ground-state uniqueness conjecture. The positive ground state of $(4.55)$ is unique in $H_{V,\mathrm{rad}}^{1}$ for the fixed parameter $\lambda<0$.

Small-mass nonexistence conjecture. In general, $m_c$ has no ground state optimizers when $c$ is sufficiently small.

Large-coupling absence conjecture. There exists $\lambda_{\mathsf c}>0$ such that $H(\lambda,0)$ has no ground state for

Let $V$ be the attractive inverse power potential considered in the paper, and let $\phi_\omega$ be the unique positive least action ground state. Here $\omega_0$ is the lower endp…

Let RSB hold in some dimension, let $\kappa_J$ be the zero-temperature periodic-boundary-condition metastate in that dimension, and let GSP mean ground state pair; two GSP's are in…

Let $N=2$ and consider Equation, with action value $S\in\mathbb{R}$ and ground state solutions understood in the variational setting of the equation. Positive-action ground-state c…

Let the ground state be the ground-state eigenfunction of a relativistic Schrödinger operator with a compactly supported potential, and let the potential well denote the region whe…

Let $\Omega\subset\mathbb R^N$ be the domain, let $s\in(0,1)$ and $p>1$, and let problem (5) denote the fractional critical-growth problem in the source. Let a ground state solutio…

Let $\Omega\subset\mathbb R^N$ be the domain, let $p>1$, let $\lambda>0$, and let $\sigma(-\Delta_p)$ denote the spectrum of the Dirichlet $p$-Laplacian. Theorems 1 and 2 assert gr…

Let $\mathcal G$ be the necklace graph, a periodic quantum graph consisting of alternating loops and single edges. At sufficiently small frequency $\omega$, there are two symmetric…

Let $\mathcal G$ be a classical tadpole graph, consisting of a ring of perimeter $2L$ and a semi-infinite tail attached at a vertex with Kirchhoff conditions. For the nonlinear Sch…

Consider a cubic-quintic nonlinear Schrödinger ground state $Q_\omega$ on the unstable branch and a perturbation of the corresponding initial data. Suppose the perturbed solution e…

For $d=2$, let $Q_\omega$ denote a cubic-quintic ground state solution at frequency $\omega\in(0,\tfrac{3}{16})$. Orbital stability conjecture. All cubic-quintic ground state solut…