25 problems

Feynman's conjecture. BEC is signalled by the decisive appearance of a macroscopic amount of “infinite” cycles, that is, cycles whose lengths diverge with the number of particles.

Let $R_n(t)$ be defined by … Then … The largest eigenvalue, or zero-momentum occupation number, is … Here $A=e^{\frac{1}{12}}e^{-\zeta'(-1)}$ is Glaisher's constant. Dyson's conjec…

In the perfect Bose gas with repulsive impurities, type I Bose–Einstein condensation means that the condensate is macroscopically occupied only in the ground state. Kac and Lutting…

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 the Kac–Luttinger model be the random continuum model with hard-ball impurities, and let the one-particle ground state be the lowest eigenfunction of the Dirichlet Laplacian on…

Consider the Kac–Luttinger model of a non-interacting Bose gas in three dimensions, with impurities given by randomly distributed hard balls. A state is macroscopically occupied wh…

Interlacement-particle conjecture. The interlacement particles directly relate to, and possibly are equal to, the particles in the condensate.

ODLRO conjecture. ODLRO, and therefore BEC, is valid in dimensions $d\geq 3$ at sufficiently small temperature, but is not valid at other temperatures or in other dimensions.

Empirical-path convergence conjecture. The empirical path measure $L_N$ converges weakly to the solution of the diffusion defined by

Let $N$ bosons be confined to the Euclidean torus $\mathbb T_L=(-L/2,L/2)^3$ and interact through a repulsive two-particle interaction $V\geq 0$, with Hamiltonian … Let…

Let $\rho$ be the particle density and let $\rho_{\rm c}$ be the critical density, with $[x]_+=\max\{x,0\}$ denoting the positive part of $x$. A condensation phase transition occur…

Consider an interacting quantum Bose gas in a box at positive temperature, with particles represented through an ensemble of Brownian cycles. A cycle is microscopic when its length…

Condensate-density conjecture. The quantity

PMF condensate-density conjecture. The condensate density is

Let bosons interact through a spatially repulsive interaction, and consider condensation in momentum space. London's conjecture. Momentum-space condensation of bosons is enhanced b…

Let $\rho>0$ be the particle density, $\beta>0$ the inverse temperature, and let $\rho$-dependent systems have one-particle eigenvalues…

Let liquid helium undergo its superfluid transition, and let Bose–Einstein condensation (BEC) mean macroscopic occupation of a single-particle quantum state. London's conjecture. B…

Initial-state dependence conjecture. Clever choices of such initial density-matrix sequences lead, in the large $N$ limit, to one-particle effective dynamics different from the Gro…

Consider correlated electron pairs, which can be treated as composite bosonic objects, in a system exhibiting superconductivity. Schafroth–Butler–Blatt conjecture. Superconductivit…

Consider a three-dimensional noninteracting Bose system at temperature and density, with total momentum ${\bf P}$, particle number $N$, and occupation number $N_{\bf 0}$ of the zer…

Consider the elliptic system … Let $N\geq 2$, and let $(u,v)$ be a solution satisfying … with the limits uniform in $x'\in\mathbb{R}^{N-1}$. Berestycki–Lin–Wei–Zhao's Gibbons-type…

Kinetic generalized Bose–Einstein condensation refers to generalized condensation in the kinetic-energy, or momentum, eigenstates of the one-particle Schrödinger operator. Type III…

Fix $\beta>0$ and dimension $d\geq 3$, and let $\rho$ denote the particle density. Say that the condition under discussion is satisfied when the optimal $P$ in…

Fix $\beta,\rho>0$. Let $P$ be an optimal random shift-invariant marked point process in the variational problem defining $\chi^{(\leq)}(\beta,\rho)$, and let $N_U^{(\ell)}$ denote…