46 problems

Let $u$ solve … where $V$ is a real-valued bounded potential. Landis' conjecture. If, for some $C>0$ and any $\epsilon>0$, … then $u$ is trivial. The conjecture concerns quantitati…

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…

Contractive chain-dynamics conjecture. The exact quantization conditions should specify reduced chain dynamics that can be stable (contractive) and thereby determine the exact quan…

Let $f(t)$ be the quasi-periodic function in the preceding perturbative construction, let $M(f)$ denote its mean, and let $g$ be defined by the expansion … with…

Let $u(x,t)$ solve the harmonic-oscillator Schrödinger Cauchy problem … with $u_0\in E^1(\tanh 2\alpha,\tanh 2\alpha)$ for some $\alpha>0$. Vemuri's endpoint conjecture. The evolut…

Fix $b1 eq 0$?

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

Uniform observability conjecture. For every $T>0$, there exists a constant $C=C(X,S,T)>0$ such that the observability inequality holds. Moreover, the constant can be chosen to depe…

For each $d\geq 1$, let $Y$ be the normed function space introduced in the paper for Schrödinger-wave observability, and let $Y_0$ denote the closure of continuous functions in $Y$…

Let $d\geq 1$, let $\mathbb{T}^d=\mathbb{R}^d/(2\pi\mathbb{Z})^d$, and let $V\in L^\infty_x(\mathbb{T}^d)$ be a bounded potential. For $u_0\in L^2_x(\mathbb{T}^d)$, write…

Let $V$ be a bounded potential on $\mathbb{R}^d$, and let $u$ be a nontrivial solution of … Suppose that $C$ and $\epsilon$ are positive constants. Kondrat'ev–Landis conjecture. Su…

Small-mass negativity conjecture. In this range, there exists $c_0=c(N,p,q,b)>0$ such that

Let $N$, $q$, $b$, and $p$ be parameters as in the paper, and let obreak 2q^ and $(N-q)^+$ denote the corresponding critical exponents and positive part. For $c0$, let…

Harmonic-oscillator phase-retrieval conjecture. If $|u(t,x)|=|v(t,x)|$ for every $t\in\mathbb{R}$ and every $x\in\mathbb{R}^d$, then there exists $c\in\mathbb{R}$ such that…

Let $f\rho L^2(\mathbb{R}^d)$, and let $\widehat{f}$ denote its Fourier transform. A third unitary operator is a unitary map $U:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$. Wright's co…

Let $H_i(t)$, for $i=1,\ldots,n$, be mutually commuting Hamiltonians associated with a Landau-Zener model, and let $\Psi(\vec{z})$ be a wavefunction depending on multi-time variabl…

Let $\tilde{\Delta}:=\frac{1}{2\pi}(\partial_{11}+\alpha\partial_{22})$ on $\mathbb{T}^2$, where $\alpha<0$ is irrational, and let $e^{it\tilde{\Delta}}f$ denote the solution to…

Periodic Schrödinger maximal-function incidence conjecture. Under these assumptions,

Let $N$ be a positive integer, let $(a_n)_{n=1}^N$ satisfy … and let $\lambda$ satisfy $N^{\frac14}\leq\lambda\leq N^{\frac12}$. Periodic Schrödinger maximal-function level set con…

Let $E(t,x,y)$ be the fundamental solution of a Schrödinger equation with a super-quadratic potential $V$ in spatial dimension $d$, and suppose that the one-dimensional result of Y…

Continuity conjecture. The correction term $w$ is continuous beyond the threshold

Revival prevalence conjecture. The revival effect is prevalent in a large class of quantum systems with discrete spectra whenever the eigenpairs' asymptotics support it.

The paper considers normalized solutions of Schrödinger equations on large bounded domains … is not essential for the results. This suggests that the existence and qualitative conc…

Consider the free Schrödinger evolution on $\mathbb{R}^n$, given by … For $1<p<\infty$, let $W^{\alpha,p}(\mathbb{R}^n)$ denote the Sobolev space. Euclidean Schrödinger smoothing c…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, and consider the fourth-order nonlinear Schrödinger equation with insensitizing controls in the $N$-dimensional setting. The Ca…