19 problems

Higher-dimensional Thomas-effect conjecture. The absence of the Thomas effect in $s=2$ might also occur in other systems of greater physical relevance.

Let $y:\mathbb{R}\to\mathbb{R}^3$ be a curve and let $H_{\alpha,y}(t)=H_{\alpha,y(t)}$ be the corresponding time-dependent Hamiltonian, with quadratic-form domain $D(F_y(t))$ and p…

Let $N$ be the number of points and let $(P_m)$ denote the property that the maximizer represented by regular polygons has the specified geometric extremal behavior for $m$ points.…

Strong-coupling asymptotics conjecture. The asymptotics is in fact the same as in the previously stated energy asymptotics.

A repulsive $\delta'$ interaction in one dimension can be approximated by smooth potentials with a negative well, and the nonlinear energy admits ground states. Ground-state inheri…

Let $\beta$ be the expansion parameter, let $a$ be the regularization parameter, and let $\psi_{a,\beta}(x)$ denote the function given by Theorem. At fixed $a$ and $x$, consider an…

Stability-transition conjecture. The ground state $u_{\widetilde{T}}$ is stable up to a critical value of $\omega$ and becomes unstable thereafter.

Let $q$ be the strength of the Coulomb potential in the one-particle Dirac Hamiltonian … Here $c$ is the speed of light, $\hbar$ is the reduced Planck constant, $m$ is the particle…

Hexagonal-pattern conjecture. For large $N$ and strong point interactions, the optimal pattern should be approximately hexagonal; this is not expected to hold in the weak-coupling…

Ground-state optimization conjecture. The inequality

Let $Y=\{y_1,\ldots,y_N\}\subset\mathbb{R}^3$ be a configuration of $N$ interaction centers, let $\alpha=(\alpha_1,\ldots,\alpha_N)$ be the coupling parameters, and let…

Arbitrarily large resonance multiplicity conjecture. As $N$ increases, one can find $Y$ and $\alpha$ such that $r_{\alpha,Y}$ becomes arbitrarily large.

Non-degeneracy conjecture. No degeneracy is possible when $N\geq 5$.

Non-degeneracy conjecture. If a zero mode exists, then it is non-degenerate for every $N$.

Weighted Lorentz dispersive estimates conjecture. For every $\alpha\ne0$ and $y\in\mathbb{R}^3$, one has

Let $H_{\alpha,y}$ be the Schrödinger operator with a point interaction of strength $\alpha$ at $y\in\mathbb{R}^3$, let $P_{ac}$ denote the projection onto its absolutely continuou…

Zero-distance heat-trace conjecture. Taking the zero-distance limit in the double $\delta$-$\delta^\prime$ potential, the heat trace for the superposed potential will be of the for…

The distributional approach yields interaction distributions concentrated at the origin, and regularizations using $\delta'$-converging sequences must converge in the sense of dist…

Let $\Psi$ be a $\delta'$-like profile, and let $|T_\Psi(\alpha)|^2$ denote the squared transmission coefficient at resonant intensities $\alpha\in\Sigma_\Psi$. Rapid-transmission-…