19 problems

The paper discusses coupled Bose–Einstein systems with dissipative effects, described by effective Hamiltonians that need not be Hermitian. Simple effective-Hamiltonian conjecture.…

Consider the quasiperiodically forced system arising from a superlattice potential with incommensurate wave numbers $kappa_1$ and $kappa_2$ satisfying a Diophantine condition. Exis…

Consider the nonlocal two-component Bose–Einstein system involving the fractional Laplacian $(-\Delta)^s$, with exponent $s>0$, in Euclidean space $\mathbb{R}^{n}$. De Giorgi prope…

Limiting-distribution conjecture. The asymptotic behavior also holds for purely imaginary values of $\lambda$, and consequently $\mathcal{N}_+$ has an asymptotic limiting distribut…

Let $e^{\rm Ab}(1)$ be the thermodynamic energy parameter defined by … where $G>0$, $K_L=\left[-\frac{L}{2},\frac{L}{2}\right]^2$, and $E^{\rm GP}(\mathcal{D},M)$ is the Neumann Gr…

Hugenholtz's conjecture. Under the selection rule used in the weak-coupling expansion, only two-point correlations survive at higher orders in $lambda$.

Let the self-similar solution of the wave kinetic equation have frequency variable $omega$ and let $x^*$ denote the exponent determined by the associated nonlinear eigenvalue probl…

Let the nonlinear eigenvalue problem referred to as equation be the problem governing the excited states of the Bose–Einstein condensate. The conjecture concerns eigenfunctions of…

Let $d>1$, let $u_{\text{\tiny\rm gs}}$ be the ground state of the Gross–Pitaevskii eigenvalue problem (GPEVP), and let $V_\varepsilon$ be a disorder potential satisfying assumptio…

Let $d>1$, and let $u_{\text{\tiny\rm gs}}$ denote the ground state of the Gross–Pitaevskii eigenvalue problem (GPEVP). Higher-dimensional ground-state bound. The ground state sati…

Let $z\in\mathbb{H}$ describe the lattice shape, let $(a,b)$ describe the relative position of the two component lattices, and let … where … and $\alpha\in[-1,1]$ measures the inte…

Let $L$ denote the system size, and let the ground state be obtained through the double limit in equation … resembles, or coincides with, the large $L$ limit of…

Consider multi-vortex configurations of total charge two, including the secondary branch given by an asymmetric pair of charge-one vortices and the symmetric pair of two charge-one…

Fourier-splitting decomposition conjecture. Under mild assumptions, there exist imaginary coefficients $n_i,m_i,f_i,g_i,e_i,k_i$ such that the displayed factorization is the soluti…

Let $(u_h,lambda_h)$ be the fine-scale ground state, and let $(u_H^{\operatorname*{c}},\lambda_H^{\operatorname*{c}})$ and $(u_h^{\operatorname*{c}},\lambda_h^{\operatorname*{c}})$…

Controllability conjecture. The first generalization should be possible for “benign” cases such as certain power nonlinearities, and local exact controllability around excited stat…

Consider the classical Hamiltonian … with coupling parameters $\lambda=\frac{kN}{2\epsilon}$ and $\beta=\frac{\mu}{\epsilon}$. Its fixed points are determined by…

Let $\widehat{\tau}$, $\widehat{g}$, and $\tau$ denote the parameters introduced in the model, and let $\epsilon$ tend to zero. Asymptotic interaction-parameter conjecture. … This…

Let $V$ be the flat trapping potential defined by … and … Here $G$ denotes the interaction parameter and $\Omega$ the rotation speed. Transition-scale conjecture. In this flat-trap…