32 problems

Let $c_{\rm LO}(1,3)$ be the smallest constant for which the Lieb–Oxford inequality is valid in dimension $d=3$ with Coulomb interaction $w(\mathbf r)=|\mathbf r|^{-1}$. Lieb–Oxfor…

Exit statistics theorem. Neglecting the possibility of clustering, the flux-across-surfaces theorem for potential scattering should generalize to this $N$-particle setting: for eve…

Fix a radius $R_0>0$. The localization theorem provides balls $B_1,\dots,B_n$ in $\mathbb{R}^3$, each of radius $R_0$, and a normalized vector $\Psi$ in the physical Hilbert space…

Let E^{\rm QM}_{\rm bose}(N,\mathord{\text{\boldmath Omega}},g) denote the bosonic ground state energy, and let the bounds in equation (upbos) give its asymptotic upper and low…

Collective-field approximation conjecture. For the more general problems considered in this paper, $F_{\phi}(v)$ is close to $F_\infty(v)$.

Pure-state density representability conjecture. One may naturally conjecture that

The ground state energy and reduced density matrices admit a perturbative expansion whose first terms are denoted by $\gamma_0$ and $\gamma_1$. On the torus, translation invariance…

Nam's coefficient conjecture. The main part of the conjecture is that

Let $E(N,lambda v)$ and $E(N-1,lambda v)$ denote the ground-state energies of the trapped Bose gas with interaction scaling $lambda$, and let $T$ and $v$ satisfy appropriate condit…

Let $\rho_Z$ be the one-body density of a ground state of the many-body atomic Hamiltonian with nuclear charge $Z$ and $N=N_c(Z)$ electrons. Universality conjecture. Along subseque…

Let $b^*(k)$ and $b(k)$ be the particle-hole creation and annihilation operators, and let $\Omega$ denote the vacuum vector. The operators $b^*(k)$ commute pairwise, and the vacuum…

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…

Degeneracy-index conjecture. For every case considered,

Nonvanishing-minimizer conjecture. Any minimizer $\psi$ of $F^{\boldsymbol{\alpha},(0)}(r)$ satisfies

Arbitrary-particle-number representability conjecture. The pure excited $v$-representability theorem stated for $N=1$ holds for any $N$, with a sufficient condition that minimizers…

Lundholm–Nam–Portmann conjecture. In the strong-coupling limit,

Radial Schrödinger-operator conjecture. The operator ${\cal H}^{(n)}_{\rm rad}$ is equivalent to a $\frac{n(n-1)}{2}$-dimensional radial Schrödinger operator.

Calogero's conjecture. There should exist a nontrivial relation between the multiparticle Calogero model and the system of decoupled oscillators.

Calogero-inspired integrability conjecture. The integrability of the nonlinear Schrödinger equation is a consequence of the exact solvability of the underlying Lieb–Liniger model;…

Dipolar Gross–Pitaevskii convergence conjecture. Similar results to the paper's derivation of the dipolar Gross–Pitaevskii energy should hold for

Let $H_{\rm BH}$ be the two-mode Bose–Hubbard Hamiltonian, let $\Psi_N$ be its ground state, and let $\gamma_{\Psi_N}^{(1)}$ denote the one-body density matrix of $\Psi_N$. Let…

Let a system with initial wave function $\psi_0$ evolve under interaction potential $w$ and external potential $v$. For a different interaction potential $w'$, there should be a un…

Let $H=T+V_{\mathrm{ee}}+V_{\mathrm{ext}}$ be the Hamiltonian of a bounded many-electron system, with external potential $v$ and one-particle density $n$ in a non-degenerate ground…

Tracer-particle reduction conjecture. All the physics boils down to the motion of the tracer particles, with the tracer wave function governed by the effective Hamiltonian…