Bose-Einstein Condensation in Continuum Models

“Prove that Bose-Einstein condensation occurs in a continuum model of a weakly interacting Bose gas, or determine whether the long-held assertion fails.” The question does not specify a particular continuum Hamiltonian, scaling limit, or class of states. A standard mathematical criterion in a specified sequence of bosonic states is expressed using the one-particle reduced density matrix: for normalized symmetric wave functions $\psi_N\in L^2_s(\mathbb{R}^{3N})$, let

$$\gamma_N(x;y)=N\int_{\mathbb{R}^{3(N-1)}}\psi_N(x,X)\overline{\psi_N(y,X)}\,dX.$$

Condensation in an orbital $\varphi$ means that $\langle\varphi,\gamma_N\varphi\rangle$ is of order $N$; complete condensation means that $\gamma_N/N$ tends to the rank-one projection onto $\varphi$. This criterion makes the phenomenon precise once the states and limit are specified, but the question itself does not select them. Bose gases in the Gross-Pitaevskii limit: a survey of some rigorous results (2022) Lieb, Bose-Einstein Condensation (1998)

Background

The problem was posed by Elliott H. Lieb in 1998 against the backdrop of experiments on dilute atomic gases and a long-standing expectation that weakly interacting bosons condense. The question then distinguished continuum gases from a special lattice result: Lieb, Kennedy, and Shastry had proved condensation for a particular lattice model. Later accounts describe the thermodynamic-limit lattice result of Dyson, Lieb, and Simon for three-dimensional hard-core bosons at half-filling, obtained by mapping to a spin system and using reflection positivity; subsequent work extended this approach to other lattice settings. Lieb, Bose-Einstein Condensation (1998) Bose gases in the Gross-Pitaevskii limit: a survey of some rigorous results (2022)

A crucial distinction is between a trapped gas in a Gross–Pitaevskii (GP) scaling and a homogeneous gas in the thermodynamic limit. In the GP regime, the particle number grows while the trap remains confining and the interaction scattering length is scaled with particle number. This is a dilute-gas limit, but it is not the fixed-density thermodynamic limit. Thus a continuum proof in a trap establishes condensation in a meaningful continuum model without settling the broader fixed-density question. Bose-Einstein Condensation of Dilute Gases in Traps (2002) Bose gases in the Gross-Pitaevskii limit: a survey of some rigorous results (2022)

Known results

Lieb, Seiringer, and Yngvason proved complete condensation for ground states in a continuum trapped-gas GP limit. In their setting, the trap potential is locally bounded and tends to infinity at spatial infinity; the pair interaction is nonnegative, radial, and of finite range. With interaction scaled so its scattering length $a$ satisfies $g=4\pi Na$ for fixed $g$, the normalized one-particle density matrix converges in trace-class norm to the projection onto the GP minimizer. This is a rigorous continuum result for the stated trapped scaling, not a proof in the thermodynamic limit. Bose-Einstein Condensation of Dilute Gases in Traps (2002)

A 2022 survey also reports condensation in the GP limit for approximate ground states: sequences whose energy per particle approaches the GP minimum have asymptotically all particles in the GP minimizer. For the thermodynamic limit of interacting gases, that survey identifies the rigorous lattice result as the available proof and describes progress for more general gases through renormalization-group methods. A 2025 preprint calls establishing condensation for dilute gases in this limit the most challenging mathematical problem, noting that results exist for confined gases but not, aside from a few cases described as not dilute, for the large-volume regime beyond the healing length. These dated statements distinguish established trapped-gas theorems from the unresolved broader direction; they do not by themselves establish its status after those publications. Bose gases in the Gross-Pitaevskii limit: a survey of some rigorous results (2022) Mathematical physics of dilute Bose gases (2025 preprint)