26 problems

Let $B$ be a magnetic field satisfying the force-balance equation referenced in the source, on a general toroidal domain. Grad's conjecture. Smooth solutions to the force-balance e…

Let $M_Fcsubset\mathbb T^3$ be a connected smooth two-manifold homologous to zero, and let a magnetic field determine trajectories on it. For a smooth one-parameter family…

In a periodic energy surface, let the level-set trajectories induced by a magnetic field include strongly chaotic trajectories, meaning trajectories whose lifts to the plane have n…

Let $A^*A$ be the operator whose eigensubspaces vary with the magnetic parameters, and consider the localization mechanism producing infinitely degenerate eigenvalues and continuou…

Colin de Verdière's conjecture. The total flux gives an upper bound on the multiplicity of the magnetic ground state. The conjecture proposes a geometric restriction on ground-stat…

Upper-bound conjecture. There should exist absolute constants $a$ and $b$, independent of $B$, such that

Let the three-dimensional toroidal-coil scattering problem be described by the scattering matrix and cross-sections introduced above. Generality conjecture. The situation described…

Thouless's conjecture. For $\lambda=1$,

A quadratically integrable system with integrals of nonstandard, nonseparable type may be related to a separable system only in a trivial manner. Non-subgroup-type integrals conjec…

Logarithmically enhanced area-law conjecture. As $L\to\infty$ and $B\to0$,

The systems under consideration are integrable and superintegrable systems on three-dimensional Euclidean space with magnetic fields that possess quadratic integrals of motion. The…

Existence conjecture. Hamada or Boozer coordinates should exist near a hyperbolic magnetic axis.

The Hartree equation is studied with initial data $Q_0(x,y)$ satisfying … where $H_x$ and $\bar{H}_y$ are the magnetic Hamiltonians acting in the $x$- and $y$-variables, respective…

One-dimensional reduction conjecture. Under these assumptions,

Under either of the paper's non-crossing hypotheses, let $a_k<b_k$ be the spectral-island endpoints supplied by the main theorem, with…

Nonaccumulation criterion. There is no accumulation of complex eigenvalues of $P_3(b,V)$ near zero if and only if $\Re(V)>0$.

Let $b_1$ and $b_2$ be the magnetic-field parameters, and let $\lambda_{p_n}$ and $\lambda_{p_n+1}$ be the band functions associated with the splitting case in which an energy leve…

Conjecture on Theorem 5.11. Theorem 5.11 should remain valid for perturbations by small constant fields $B$.

Kernel boundedness conjecture. The condition

Let $V\in L^{5/2}({\mathbb R}^3)$, let $h>0$ tend to zero, and let $E^{\rm S}(\beta,h,V)$ denote the Schrödinger energy in the self-generated magnetic field model, with…

Let $\Bbb D$ be the self-adjoint Dirac operator on $L^2(\Bbb R^2,\Bbb C)\oplus L^2(\Bbb R^2,\Bbb C)$ defined initially on…

Let $u_\parallel$ denote the parallel component of the velocity $u$, determined by an elliptic equation posed along magnetic field lines. Suppose the magnetic field lines are dense…

Consider a particle rotating in a magnetic field, with three-momentum parameterized by $gammaup$ and critical value $\lvert \gamma \rvert=\alpha/2$. Let $\vec n$ be the unit three-…

Let $\lambda_0(\alpha)$ be the ground state energy of the operator defined in the paper, and let $\hat{\nu}$ be its minimum value. A minimizer is a point…

Let $B\in C(\mathbb R^2)$ be a continuous magnetic field, and let $\mathcal E_B$ denote the energy space associated with $B$. The notation $B=\operatorname{curl} A'$ means that the…