185 problems

Exact-solvability conjecture. All maximally superintegrable systems are exactly solvable.

A quantum system with $n$ degrees of freedom is called maximally superintegrable when it possesses $2n-1$ integrals of motion, the maximum possible number. An exactly solvable quan…

Let $\mathcal{X}$ denote the completely disconnected space modelling physical space at the Planck length, and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Volovich's conjec…

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…

Let $M$ be a dimension, and let $U_1$, $U_2$, and $U_3$ be unitary matrices. In phase retrieval, an intensity-measurement operator is formed as $\Phi=[U_1~U_2~U_3]$. Wright's conje…

Let a negative atomic ion be described by a non-relativistic many-particle Hamiltonian whose nuclear charge varies as a parameter, and let the nuclear charge be critical when the l…

The Schrödinger equation leads to the eigenvalue problem … For a finite-energy solution, the proposed interpretation of a Bohr transition is the long-time asymptotics … with…

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…

TTW quantum-superintegrability conjecture. The TTW system is quantum superintegrable for all rational values of $k$.

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…

Accardi's characterization conjecture. This property characterizes the pair $Q$ and $P$ acting on $L^{2}(\mathbb{R},m_0)$.

Convergence conjecture. In some suitable topology, each sequence $\{f_k^{[n]}\}_{n\geq 0}$ converges to $f$, for every $k=0,1,2,\dots$.

Let ${\cal S}$ be the entropy functional defined on the odd subspace ${\cal H}_-$ of $L^2(\mathbb{R})$. The family $(\Phi_a)$ is a minimizing sequence, with…

Let a reformulation be an alternative formulation of quantum theory, including formulations that introduce hidden variables or other unobserved properties, and let quantum nonlocal…

Let a local algorithm be an algorithmic representation of a theory that is local in the sense that all of its irreducible elements are local. Let QM denote quantum mechanics and it…

Let $\widetilde{E}_{\gamma_i}$ denote the perturbed energy appearing in the replacement … with … Although the upper bound of the summation index $a$ may differ from the approximati…

Factor conjecture. A quantum mechanical system bounded in space and isolated from outside influences can be mathematically described as a bounded quantum system in the sense of Def…

An atom is being considered in the context of early quantum theory, where observed spectral lines are associated with energy quanta and transitions between atomic states. Bohr's co…

Let a Schrödinger Hamiltonian be invariant under the joint action of parity (P) and time reversal (T) transformations. Bender's conjecture. Under this less restrictive condition th…

Let $Q$ and $D$ be the basic affine coherent-state operators, and let ${\cal H}$ be a self-adjoint Hamiltonian operator. For $0<\beta\leq 1/2$, suppose the propagator and the assoc…

Let a four-body system be a quantum-mechanical system of four constituents, and call a subsystem stable if it is bound. A four-body system has a stable three-body subsystem when at…

Let an external potential be switched on and off adiabatically, and consider the eigenvalues of the corresponding Dirac operator emerging from its negative and positive spectra. Ne…

The state space of a quantum system is a Hilbert space, and bound electron states in a molecule may have an intrinsic two-valuedness. Different electrons, and later other fermions,…

Symplectic camel conjecture. The property of the symplectic camel is the key to a better understanding of not only quantum mechanics, but also of classical phenomena.

Interpretive conjecture. A viable interpretation of EQM may be that, while linear QM is the fundamental theory of simple systems, EQM describes relatively isolated systems in this…