21 problems

Let $B$ be a constant magnetic field and let $a:\mathbb{R}\to\mathbb{R}$ be a nontrivial function. Consider the Iwatsuka-type magnetic perturbation whose vector potential is…

Ivrii's conjecture. The main part of the spectral asymptotics is given by ${\mathcal E}^{\rm {MW}}$, while: (i) for fixed $(g^{jk})$ and general $(V_j),V$, the remainder is…

Let a finite, smooth, non-closed curve have arclength parameter $s$ and curvature $k(s)$. Define the comparison operator … with Dirichlet boundary conditions at the curve endpoints…

Zero-splitting and hopping conjecture. The potential $v_0$ can be chosen so that the associated double well Hamiltonian has zero eigenvalue splitting for infinitely many values of…

Let $H_V=H_0+V$ with $V(x,y)=v(x)$, and let the spectral bands be indexed by $n$. Band-width conjecture. The width of each band decreases as $n$ grows, but does not ultimately tend…

Let $H_L$ be the Landau Hamiltonian and let $V(x,y)=v(x)$ be a translationally invariant regular potential with $v:\mathbb{R}\to\mathbb{R}$. Consider the perturbed operator $H_L+V$…

Let $\mathcal{S}_{\beta}$ be an infinite sector of opening angle $\beta\in(0,\pi)$, and let $\mu_{\beta}$ denote the bottom of the spectrum of the Neumann magnetic Schrödinger oper…

Shen's conjecture. Some condition on $\nabla {\bf B}$ is necessary to prove Theorem 0.11 of Shen.

Purely absolutely continuous spectrum conjecture. The spectrum of $H_0$ should be purely absolutely continuous as long as $b$ is non-constant.

Assume Hypothesis … . For each sufficiently small $\epsilon$, let the spectral bands $[a_k,b_k]$ below $E$ be those in Theorem … should remain valid with bands of size … for some…

Let $H^{\epsilon,\kappa}$ be the magnetic Schrödinger operator considered in the paper, with constant background magnetic field $\epsilon B_0$ and slowly varying perturbation deter…

Magnetic gap-labelling conjecture for strongly minimal actions. If the action is strongly minimal, then the containment in the magnetic gap-labelling conjecture for minimal actions…

Magnetic gap-labelling conjecture. The magnetic gap-labelling group is contained in the subgroup of $\mathbb R$ given by

Ground-state asymptotic conjecture. One has

Bon's conjecture. The ground state energy, meaning the bottom of the spectrum of $P_{\Omega_\alpha}$, is constant and equal to $\Theta_0$ when

The eigenvalue lower-bound conjecture. For every natural $m$, the stated lower bound holds for sufficiently small $h>0$.

Ground-state spectral asymptotics conjecture. Under these assumptions,

Let $B$ be a magnetic field, and let $\beta$ and $\tilde{u}$ be the parameters and rescaled heat evolution appearing in Theorem; denote by $\gamma_B$ the decay exponent in that the…

For each $n$, let $\lambda_{{\mathsf{D}},n}(\eta)$ and $\lambda_{{\mathsf{N}},n}(\eta)$ denote the Dirichlet and Neumann eigenvalue branches of the boundary model operator as funct…

Near a boundary, consider the eigenvalues of the boundary model operator, with $u={mathsf{D}}$ or ${mathsf{N}}$ denoting Dirichlet or Neumann boundary conditions, and interpret…

Let $H$ be the unperturbed magnetic Schrödinger operator, let $V$ be the perturbation, and let $\xi(\cdot;H \pm V,H)$ denote the spectral shift function (SSF) for the pair…