31 problems

Consider a dilute homogeneous Bose gas with particle density $\rho$, interaction potential $v$, and positive scattering length $a>0$. Let the leading-order ground-state energy asym…

Excitation-spectrum conjecture. The low-lying spectrum of $H_N^{\rm GP}$ is given by

Bounded-coefficient conjecture. This representation holds with coefficients satisfying

Let $n$-vortex states $\phi$ be considered for a harmonic potential, with rotation frequency $\Omega\geq 0$ and interaction parameter $a>0$. Instability conjecture. An $n$-vortex w…

Let $v$ be an interaction potential with positive scattering length $a>0$, and suppose that the corresponding system has no $N$-body bound states for any $N$. Let … should hold whe…

Low-density Bose-gas conjecture. The formula above requires only a positive scattering length and the absence of any many-body, negative-energy bound state.

Polynomial structure conjecture. The displayed polynomial structure of the small-$cL$ expansion of $g_N(x,0)$ continues for all $N$.

Polynomial structure conjecture. This pattern holds for all $N\geq 2$: for every $p\geq 2$, the coefficient of $(N-1)(cL)^p$ is a polynomial of degree $p-2$ in $N-1$.

Sütő's cycle-percolation conjecture. Cycle percolation occurs if and only if there is Bose–Einstein condensation. The conjecture proposes an equivalence between a nonzero limiting…

Let $N$ interacting bosons occupy a cube $\Lambda$ of side $L$ in $d$-dimensional Euclidean space, with canonical permutation-cycle probabilities $P_{\Lambda,N}$ and cycle length…

Let $X$ and $Y$ be configurations in a lattice system, with amplitudes $a_X$ and $a_Y$, boundary sizes $|\partial X|$ and $|\partial Y|$, and graph distance $d_G(X,Y)$. Let…

NRT3's conjecture. The subleading term in the asymptotic expansion of $E_N$ is proportional to $\sqrt{N}$.

Consider the partially interacting Bose-gas model defined by the source's cycle representation, with condensation characterized by finiteness of the corresponding critical Green's…

Let $e_{\rm 3B}$ denote the ground-state energy per unit volume in the thermodynamic limit, let $\rho$ be the particle density, and let $b_{\mathcal M}(V)$ be the three-body scatte…

Let $H_N^{\rm Comb}$ be the Hamiltonian with two-body potential $W$ at scaling parameter $\beta=1$ and three-body potential $V$. Let $e^{\rm Comb}_{\rm GP}(b_1,b_2)$ be the minimum…

Let $H_N^{\rm GP}$ be the Gross–Pitaevskii Hamiltonian, let $E_{\rm GP}(N)$ denote its ground-state energy, and let $e_{\rm GP}$ be the minimum of the corresponding Gross–Pitaevski…

Let $s$ be the spatial dimension, let $\mathfrak R(L^2(\mathbf O))$ be the resolvent algebra associated with a bounded region $\mathbf O$, and consider the thermodynamic-limit stat…

The simplified approach concerns the ground-state energy, condensate fraction, and correlation function of a repulsive Bose gas, and is considered for repulsive potentials of posit…

Let $e(\rho)$ denote the energy per particle as a function of the gas density $\rho$, obtained from the simple equation for the ground state of an interacting Bose gas. The paper e…

Let $e(\rho)$ be the energy per particle as a function of the gas density $\rho$, obtained from the simple equation for the ground state of an interacting Bose gas. Convexity conje…

Let $\rho(e)$ denote the density as a function of the energy per particle $e$, obtained from solutions of the simple equation for the ground state of an interacting Bose gas. When…

Consider the infrared dynamics of the two-dimensional Bose gas, where the scaling exponents associated with persistent-homology observables form a continuous spectrum depending on…

In a two-dimensional Bose gas far from equilibrium, anomalous vortex kinetics refers to the altered universal scaling behavior caused by vorticial excitations interacting anomalous…

Let $\mathcal V\in L^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, with $p>\max\{\frac d2,1\}$, be non-negative. The density function $\rho(e)$ is the continuous function characterized b…

Emergence of the renormalized Gibbs measure. In the limit $\lambda=T^{-1}\to0$, one obtains the renormalized Gibbs measure in the sense that