14 problems

Let $H_0$ and $V$ be the operators specified in the preceding setup, and let an absorbing point mean a point of the resolvent set at which some coupling resonance function tends to…

In $\mathbb{R}^d$, where $d$ is an arbitrary positive integer, let $H=-\Delta+V$ be a Schrödinger operator with $V(\boldsymbol{x})$ of unbounded support. Suppose that $H$ has a bou…

Let $V$ be a super-exponentially decreasing potential, let $\widehat V$ have completely regular growth, and let $B(V)$ be its Breit–Wigner series. Assume that $V$ is not compactly…

Let $V$ be a super-exponentially decreasing potential, so that its Fourier–Laplace transform $\widehat V$ is entire, and let $F$ be its Froese function, … Assume that $\widehat V$…

Breit–Wigner convergence conjecture. The series $B(V)$ converges if and only if $V$ is compactly supported.

Let $b_1$ and $b_2$ be constants satisfying $0\le b_1<b_2<+\infty$. For each admissible frequency parameter $\alpha$, let $\beta(\alpha)$ denote the minimal decay rate among the co…

Generic resonance abundance conjecture. For an open and dense set of $g\in\mathcal{G}(M)$, there is an infinite number of resonances outside any such strip. The conjecture asserts…

Let $M$ be a cusp manifold and let $\mathcal{G}(M)$ denote the space of $C^\infty$ metrics on $M$ for which $(M,g)$ is a cusp manifold with negative sectional curvature. The theore…

Let $\Gamma_\lambda$ be the Hecke triangle group, let $\chi$ be a unitary finite-dimensional representation, and let $\mathcal L_{s,\chi}$ be the associated transfer operator. At a…

Let $\mathcal{R}_q$ be the resonance set associated with the congruence subgroup $\Gamma(q)$, and let $\delta$ denote the Hausdorff dimension of the limit set. Uniform essential sp…

Jakobson–Naud's essential spectral gap conjecture.

Let $\Gamma$ be a geometrically finite discrete group of isometries of $\mathbb{H}^{n+1}$ such that $X=\Gamma\backslash\mathbb{H}^{n+1}$ is a smooth noncompact manifold. Let $R(X)$…

Quantum ergodicity conjecture for resonant states. Under these assumptions, there exists a density-one subsequence of resonant states in every strip…

Essential spectral gap conjecture. One has