14 problems

Linear resolvent-growth conjecture. There exists a constant $C$ such that

Let $H_c$ be the complex Schrödinger operator considered above, and consider curves of the form … where $b>0$, $\eta>0$, and $p$ is independent of $\eta$. Boulton's critical-index…

Let $n\in\mathbb{N}$, let $a(x)=x^{2n}$, and consider the quadratic operator function … with level curves $\lambda_b=-c(b)+ib$, where $b\to+\infty$, and fix $\varepsilon>0$ by requ…

Sharp resolvent conjecture. If $(\frac1p,\frac1q)\in\mathcal R_0\setminus([B,E]\cup[B',E']\cup[D,C)\cup[D',C))$, then

Kwon–Lee resolvent conjecture. On the stripe $\mathcal{S}$, the estimate

Let $n\geqslant 2$, let $(\frac{1}{p},\frac{1}{q})$ lie in the previously defined region $\mathcal{R}_{1}\cup\mathcal{R}_{2}\cup\mathcal{R}_{3}\cup\mathcal{R}_{3}'$, and let…

All-$p$ resolvent estimate conjecture. The resolvent estimate should be valid for all $1<p<\infty$ if $\Omega$ is convex.

Optimal resolvent-region conjecture. For every $\varepsilon > 0$, there is a constant $C_{\varepsilon}$ such that, whenever $z \in \mathcal{R}_{\mathrm{opt}}$,

Fractional resolvent conjecture. There exists a constant $C$, depending only on $p$, $q$, $d$ and $s$, such that for every $z\in\mathbb C\setminus[0,\infty)$,

Resolvent estimate conjecture. There exists a constant $C$, depending only on $p$, $q$ and $d$, such that for every $z\in\mathbb C\setminus[0,\infty)$,

Zworski's conjecture. The estimate

Optimal dispersive estimate conjecture. In dimension $n=3$, the estimate

Transmission resolvent conjecture. The resolvent $\widetilde {\cal R}_0(\lambda)$ satisfies condition (1.12).

Let $S(z)$ be the scattering matrix, holomorphically extended to the strip … and suppose that $S(z)$ has no poles in $U_{\delta}$. Stoyanov–Petkov conjecture. For every…