40 problems

Single-measurement uniqueness conjecture. Uniqueness for Schiffer's problem should follow from a single far-field pattern.

Let $E_{\ell}^{(0)}$ denote the ground-state energy associated with the non-negative integer angular-momentum value $\ell$. Grosse–Martin conjecture. Knowledge of the ground-state…

Let $q(x)$ be a compactly supported potential, and let $a>0$ be its radius, defined for non-spherically symmetric potentials in the same way as for spherically symmetric potentials…

Let $q_1$ and $q_2$ be two positive spherically symmetric potentials, and consider their fixed-energy phase shifts for all angular momenta. The positive-potential phase-shift conje…

Semiclassical asymptotics of the ILW scattering data. The eigenvalues satisfy the ILW Weyl law

Consider convolutional neural networks used in inverse-scattering reconstruction, with convolution blocks providing local operations, and consider scatterers outside the training d…

Let CNNB denote the neural-network inverse-scattering scheme that incorporates Born inversion after a convolutional neural network, and consider reconstruction from noisy data. CNN…

Let $m$ be the number of soliton pairs, let $L$ denote the length, let $\breve{k}$ be the distinguished point associated with the branch-cut configuration, and let $I_j$ be the act…

Let $d=2$ or $d=3$, let $B_2$ be the ball of radius $2$, and let … A Born transmission-eigenvalue strip-finiteness conjecture. For any $C>0$, the strip … contains at most finitely…

Let $d=2$ or $d=3$, let $B_2$ be the ball of radius $2$, and let $m$ be the refractive-index function … A Born transmission-eigenvalue strip-finiteness conjecture. Any strip parall…

Let $d=2$ or $d=3$, let $k$ be the wave number, and let $k_\epsilon>0$ be the resonant frequency in the Drude–Lorentz term … A complex transmission-eigenvalue accumulation conjectu…

Inverse determination conjecture. The wavenumber resonances together with the no-mode-conversion wavenumbers determine the reflection matrix, and the reflection matrix determines t…

Let $\Delta_h=h_{{\rm opt},B}(+\infty,t)-h_{{\rm opt},B}(-\infty,t)$ be the height difference for the optimal profile with Brownian initial condition, and let $\kappa_0$,…

For a screen with complex-valued $csigma$, suppose that Steklov eigenvalues exist. Using a complex Steklov parameter $clambda$, and extending the search domain to a region of the c…

Let $G_k=[g_k^{(1)},g_k^{(2)},\ldots,g_k^{(K)}]$ be a sequence of input sets whose ranges approximate surface distributions at the observable boundary $S\subset\partial\Omega$ as…

Let $\Sigma\subset\mathbb{C}$ be the set of spectral points, and let $V_0$ and $Q_0$ be the solution basis and Lanczos matrix corresponding to the background $q_0=0$. Let $T$ be th…

Corner-singularity convergence conjecture. If $D$ is a polygonal scatterer, the convergence rate should rely on the singular behavior of the scattered field near corner points, and…

Consider the scattering problem with $V\in L^\infty(\Omega)$ and a nontrivial entire solution $u^i$ of … in $\mathbb{R}^n$. Non-invisibility conjecture. In general, the correspondi…

Let $\theta_V$ be the dataset of far-field patterns generated by a fixed incident direction $d\in\mathbb{S}^{n-1}$ and all positive frequencies and observation directions, as defin…

Let $\Gamma$ be the curve appearing in Lemmas … , and suppose that $\Gamma$ contains a single non-analytical point. Single-point non-analyticity conjecture. Lemmas … remain valid u…

Intertwining-property conjecture. For more general regular scattering functions $S\in\mathcal{S}_0$, the intertwining property holds on these tubes. The construction is known for a…

Let $\eta_{\it \Phi}(i)$ denote the reconstruction error at iteration $i$. Conjecture on error monitoring. Monitoring $\eta_{\it \Phi}(i)$ is a useful approach to deciding when the…

Let $\eta_{\it \Phi}(i)$ denote the error monitored at iteration $i$ in the T-matrix completion reconstruction algorithm. In the strong-nonlinearity regime, suppose some noninterac…

Let … should also hold far beyond $H^{-3/4}(\mathbb{R})$ and could be achieved by a suitable extension of the inverse scattering transform method for $\mathbb{L}_{q}$. This would e…

Let $q$ and $\tilde q$ be potentials for the non-selfadjoint radial equation considered in the paper, with $\tilde q-C(\lambda^2+\frac{1}{4})$ satisfying the isospectrality conditi…