15 problems

Let $x$) and $p$ be the canonical position and momentum operators, and let $E_0$ be the ground-state energy of the self-adjoint operator $|x|+|p|$, namely … The optimal constant…

Completeness conjecture. Every reflection-positive process admits a representation as a deep NN-QM whose inputs $y_t^{(i)}$ are symmetric Markov processes.

Let $N$ be the parameter appearing in the expanded quantum infinite well, let $\Lambda\in\mathbb{Q}$, and assume that there is no fragmentation. A plateau is a plateau of the proba…

Let $H$ be a Hamiltonian defining a quantum system, let $Z_H(t)$ be its partition function, and let $\widetilde{Z}_H(t)$ be its quasi-partition function as defined from the special…

Hohenberg–Kohn conjecture. For every $\rho\in\mathcal{D}^N$, either

Let $H_N$ be the $N$-electron atomic Hamiltonian with nuclear charge $Z\in\mathbb{N}$. Strong ionization conjecture. There is a universal constant $C>0$ such that, whenever $N>Z+C$…

Let $H_N$ be the $N$-electron atomic Hamiltonian with nuclear charge $Z\in\mathbb{N}$, and let $E_N=\inf\sigma(H_N)$. Denote by $N_c=N_c(Z)$ the largest number of electrons for whi…

Hamiltonian construction conjecture. Groups of this form may admit a construction of the Hamiltonian analogous to the construction of $H_{eff}$, such that passing from the larger g…

Let $J_x,J_y,J_z$ be the generators of an irreducible, unitary, $n$-dimensional representation of $SU(2)$, and set … Write $o(1)$ for a quantity tending to zero as $n\to\infty$. Th…

Cayley–Menger factorization conjecture. The determinant $\det g^{\mu\nu}$ is homogeneous of degree $M$ in the $M$ variables $\rho_{ij}$ and factors as

Algebraicity conjecture. All symmetry operators $K_{ij}$ are algebraic differential operators in the variables $\rho_{ij}$, with coefficients linear in the $\rho_{ij}$.

Let $\hat{x}(\circ,\bullet)$ be a D-progressive four-dimensional stochastic process as defined by the forward and backward stochastic differential equations, with diffusion paramet…

Let $F$ denote the $M\times M$ discrete fractional Fourier transform defined in Candan, Kutay, and Ozaktas (2000), and let $I$ be the $M\times M$ identity matrix. The columns of a…

Let $\Psi$ be a quantum state such that the Fermi equation $g_{\mathrm{F}}(x,p)$ defines a hypersurface in phase space bounding a compact set $\Omega_{\mathrm{F}}$. A quantum blob…

A quantum blob is a phase-space set denoted by $\mathcal{QB}^{2n}$. Let $f_t^H$ be the classical Hamilton flow associated with a Hamiltonian $H$, and let $t$ be the observation tim…