158 problems

Absolute convergence conjecture. The integral

Lax–Phillips conjecture. There exists a sequence of resonances $\mu_j$ such that

Minimal-mass classification conjecture. Up to the symmetries of the equation, the only non-scattering solutions are the soliton $e^{it}Q$ and its pseudoconformal transformation.

Okada et al.'s conjecture. The pole distributions of the scattering matrices in the exteriors of isospectral domains in $\mathbb{R}^{2}$ are different.

Let ${\rvert E,b^\pm\rangle}={\rvert E^\pm\rangle}\in\Phi^\times_\mp$ be the Lippmann–Schwinger kets, and let the corresponding energy wave functions be … They are smooth functions…

Rigged Hilbert space conjecture. For empirical reasons, scattering phenomena are described by a pair of Rigged Hilbert Spaces.

Solenoid-limit conjecture. In some well-defined limit, largely independent of the details of the approximating potentials, the limiting situation should be described by a singular…

Scattering conjecture for critical NLS. There exist functions

Scattering conjecture for critical gKdV. There exist functions

The preceding construction involves operators, kernels, and generalized Marchenko equations, including the relation between $B$, $\tilde{B}$, and $\check{B}$, as well as the associ…

Let $H=-\Delta+V$ be the Schrödinger operator on $\mathbb{R}^d$, where $V$ is a slowly decaying potential. Simon's conjecture. If $V(x)$ is such that... The supplied candidate span…

Asymptotic completeness conjecture. The relation above, and its generalizations to other conserved quantities attributable to particles, should hold in all theories admitting a loc…

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 $\Gamma$ be an infinite asymptotically straight curve without self-intersections, with two leads. Suppose that the distance between points of $\Gamma$ becomes small somewhere,…

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

The one-dimensional scattering matrix for the delta-prime perturbation of the free Schrödinger operator on ${\mathbb R}$ is considered. A geometric realization means obtaining this…

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…

The calculation uses only the bootstrap property of the S-matrix, so let a diagonal scattering theory have this property. Polynomial recursion conjecture. The problem of calculatin…

Affine Toda field theories (ATFTs) are integrable scalar field theories associated with root systems of affine Lie algebras; the method in question uses elasticity, meaning absence…

More precise conjecture. To get rid of infrared divergences in the inclusive cross section of some process, one should choose the numerical part of the current to coincide with the…

Petkov's conjecture. The operator $G$ has no eigenvalues for every strictly convex obstacle $K$.

Uniform boundedness conjecture. There is a universal constant $B_{V_0,R_0}$ such that

Consider the focusing nonlinear Schrödinger equation with general harmonic trapping … for $z=(x,y)\in\mathbb R^d\times\mathbb R^N$. Non-scattering conjecture. Scattering does not o…

Let $\gamma_0$ denote the portion of $\gamma$ lying in the strip $[-d/2,d/2) \times \mathbb{R}$ and define … These are the quasi-periodic Hilbert transform and Laplace double-layer…

Let $\mathcal{H}$ and $\mathcal{I}$ denote the horizon and null infinity, respectively. Consider smooth scattering data settling down to a possibly extremal Kerr solution on…