15 problems

Let the magnetic Dirichlet-to-Neumann operator be the operator associated with a magnetic field of strength tending to infinity, and let its ground state energy denote its lowest e…

Let $\Gamma$ be the relevant group, let $\sigma$ be a multiplier on $\Gamma$ with $\delta(\sigma)=0$, and let … be the pairing of the cohomology class of $\sigma$ with the fundamen…

Complete Hessian degeneration conjecture. For some values of $t$, the bottom of the spectrum of $M_t$ exhibits complete degeneration, in the sense that its Hessian vanishes complet…

Let $\mathscr{L}$ be the magnetic operator considered in the paper, and let $h$ be the semiclassical parameter. Under the assumptions defining the two-dimensional purely magnetic-w…

Let $\Omega\subset\mathbb{R}^d$ be any bounded set, and let the magnetic field be constant. Denote by $\lambda^{\rm DN}(A,\Omega)$ the ground-state energy of the magnetic Dirichlet…

Let $\Omega$ be a regular domain in $\mathbb{R}^2$, let $A_0$ be a magnetic potential whose magnetic field is constant and has norm $1$, and let $\lambda^{\rm DN}(bA_0,\Omega)$ den…

Let $\Omega$ be a regular domain in $\mathbb{R}^2$, and let $A_0$ be a magnetic potential whose magnetic field is constant and has norm $1$. Let $\Lambda_{bA_0}$ be the associated…

Let $D(0,1)\subset\mathbb{R}^2$ be the unit disk, let $\Lambda^{DN}(b)$ be the magnetic Dirichlet-to-Neumann map for a constant magnetic field of strength $2b$, and let…

For a finite connected graph $G$, let the nodal surplus distribution be the distribution of values of $\phi(h',k)-(k-1)$, averaged over all signings $h'$ of a discrete Schrödinger…

Duclos–Exner conjecture. The discrete spectrum of this operator is empty when the magnetic field is sufficiently strong and uniform.

Let $\mu_{1}$ be the first band function, let $\xi_{0}$ denote its minimum point, and write $\mu_{1}^{(3)}(\xi_{0})$ for its third derivative there. Third-derivative conjecture. ……

Let $W=\Phi V$ satisfy Assumption (A1), where $\Phi\in\mathbb{C}\setminus\mathbb{R}e^{ik\frac{\pi}{2}}$ for some $k\in\mathbb{Z}$, and let $V:\mathbb{R}^3\longrightarrow\mathbb{R}$…

Let $\zeta_{1}(\tau)$ be the lowest band function associated with the three-dimensional magnetic Hamiltonian with axisymmetric potential. The preceding analysis shows that…

Resonance-accumulation conjecture. The singularities of the SSF at the higher Landau levels are related to the accumulation of resonances of $H$ at these levels.

Let $\Gamma$ be a graph, let $\Gamma_{\gamma_1,\ldots,\gamma_\beta}^{\mathrm{cut}}$ be its cut graph, and let $\lambda_n(\Gamma_{\mathrm{mag}}^{\alpha_1,\ldots,\alpha_\beta})$ deno…