51 problems

Ill-posedness conjecture. For any $\gamma\in \mathbb{R}$, there exists $\eta \in L_x^1+L_x^{\infty}(\mathbb{R}^2)$ such that (eq:NLS) is ill-posed in $H_x^\gamma(\mathbb{R}^2)$.

Feynman path-integral representation conjecture. For suitably chosen $\psi_0$ and $V$, the action $F_{\mathcal{H}_{x,t}(M)}(f)$ provides a representation of the solution…

Planchon's space-time conjecture. One has

Let $d\geq 2$, $N\geq 1$, and define … … Let $\mathcal{M}$ be a real analytic hypersurface in $\mathbb{T}^d$ with nonnegative curvature and surface measure $\sigma$. Paraboloid res…

For an integer $n\geq 1$, set $p=2+4/n$. Let $f_*$ be the Gaussian … and let $u_*(t,x)=\mathrm{e}^{-\mathrm{i}t\Delta}f_*(x)$ be the corresponding solution of the Schrödinger equat…

For $\alpha\in\mathbb{R}$, let $H^\alpha(\mathbb{R}^n)$ denote the Sobolev space, and let $f\in H^\alpha(\mathbb{R}^n)$ be the initial datum for the linear Schrödinger equation … w…

The Strichartz conjecture on $\mathbb{R}\times\mathbb{S}^3$. One has

Let $N\geq 1$, let $\varphi$ be the time cutoff appearing in the estimate, and let $\phi\in L^2(\mathbb{R}\times\mathbb{T})$. For $\xi_0\in\mathbb{R}\times\mathbb{Z}$, write…

Let $n\geq2$, let $(1/p,1/q)$ lie on the line segment $(O,A]$, and set $s$ by … For an orthonormal family $\{f_j\}_{j\in J}\subset\dot{H}^{s}(\mathbb{R}^n)$ and coefficients…

Let $n\geq1$, and let $A=(1/p,1/q)=((n-1)/(n+1),n/(n+1))$. For an orthonormal family $\{f_j\}_{j\in J}\subset L^2(\mathbb{R}^n)$ and a sequence…

Let $n=1$. Let $A$ denote the endpoint in the $(1/p,1/q)$-region for the frequency-global estimate, and let $\alpha$ be the summability exponent for the orthonormal system. Endpoin…

Let $I$ be a compact interval. For $p,q\in[1,\infty]$, write $L^q_xL^\infty_t(\mathbb{R}\times I)$ for the corresponding mixed-norm space. For $\alpha\in\mathbb{R}$, let…

Generic Schrödinger control conjecture. Generically with respect to the lengths $(\ell_i)_{i=1,\dots,N}$, (GGCC) is a necessary and sufficient condition for the internal control of…

Let $C_S(T,L)$ denote the best constant in the observability inequality for the one-dimensional Schrödinger equation on $(0,L)$, and define … The known bounds are…

Bilinear space-time estimate conjecture. For $2\leq p,r<\infty$, the estimate

Lee–Rogers–Seeger local space-time conjecture. The estimate

Let $H$ be the Schrödinger operator on a finite metric graph with infinite ends, and suppose that the lower-bound assumption does not hold. Slower-decay conjecture. The…

Let $H$ be the Schrödinger operator on a finite metric graph with infinite ends, and let the initial datum be zero on the finite heads. Consider the third component of the correspo…

Let $N$ be a positive integer, let $\theta>0$, and let $\{b_n\}_{n=1}^N$ be a sequence of complex numbers. Define $e(y)=\exp(2\pi i y)$. Periodic Schrödinger maximal operator conje…

Let $\omega\subset\mathbb{R}^d$ be an observation set, let $B_R(\phi^t(\rho))$ denote the ball of radius $R$ centered at the point $\phi^t(\rho)$ along the relevant classical flow,…

Let $u$ solve the linear Schrödinger equation … with initial data $u(0,x)=f(x)\in L^2(\mathcal{M})$ on $\mathcal{M}=\mathbb{S}^d$ or $\mathbb{T}^d$, where $d\geq 2$. The higher-dim…

Let … where $a_m$ is supported in … For $\lambda>0$, define the superlevel set $U_\lambda(\nu)=\{(x,t):|\nu(x,t)|\geq\lambda\}$. Periodic restriction conjecture. If $|a_m|\leq 1$ f…

The $1/4$-regularity conjecture. The minimal value of $s$ for which this convergence holds for every $f\in H^s(\mathbb{R}^n)$ should be $s=1/4$ in all dimensions. This was a long-s…

For $f\in L^2(\mathbb{R}^k)$, let $u_f(x,t)$ solve the Schrödinger equation with initial data $f$, and let $p=2+4/k$. The Strichartz inequality is … Here $A$ is the best possible c…

Let $M$ be the quotient under consideration of a group of Heisenberg type, let $U\subset M$ be open, and let $T_{\rm GCC}(U)$ denote the geometric control time associated with $U$.…