47 problems

Let $P$ be the generator associated with an open hyperbolic system, and suppose that the trapped set in $p^{-1}(1)$ is $E_0^*\cap p^{-1}(1)$ and has dimension $\delta$. For any…

Jakobson–Naud's conjecture. They conjecture that

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

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…

For a symmetric quantum baker's map, let $B_h$ be its quantum baker relation, let $D$ be the baker parameter, let $N=(2\pi h)^{-1}$, and let $\operatorname{Spec}(B_h)$ denote the s…

Let $\Omega$ be the domain under consideration, let $N_{0}$ be its Newton potential operator, and let $\Delta_{\Omega}^{N}$ be the Neumann Laplacian on $\Omega$. A simple spectrum…

Let the initial state be the sharply peaked state $\Psi_{T=0}(z)=\delta(z-z_\mathrm{i})$, and consider its time evolution in a false-vacuum region (FV region), with the desired exc…

The scattered field $u(\mathbf{r},t)$ is considered in two dimensions, with complex resonances $\rho_n$ and coefficients $c_{\mathbf{p},n}(\mathbf{r})$ in the asymptotic expansion…

Let $X$ be a geometrically finite hyperbolic surface, and let the resonance-counting function and critical exponent be as in the paper. The paper establishes a main theorem and a t…

In the case of convex co-compact hyperbolic surfaces, let $N(r,\gamma)$ be the resonance counting function in a strip of depth $\gamma$, and let $\gamma_{cl}$ denote the classical…

Let $N(r,\gamma)$ denote the resonance counting function in a strip of depth $\gamma$, let $d_H$ be the fractal dimension parameter associated with the trapped set, and let…

Ikawa's conjecture. If the obstacles are trapping and $d$ is odd, then there exists $\alpha>0$ such that

Let $H_0$ and $H_1$ be the operators in the scattering framework, and let $V=H_1-H_0$ be a perturbation for which the usual spectral shift function need not exist. At a real number…

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…

Let $\gamma_{th}=\frac{1-\delta}{2}$ be the threshold decay rate, where $\delta\in(0,1)$ is the dimension of the limit set, and let $R(E,Ch,\gamma h)$ denote the resonance rectangl…

Finite-resonance case conjecture. The assumption that $\lambda_0$ is not in the essential spectra of the middle operators, but is an eigenvalue of at least one of the single-pertur…

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.

Quantized logarithmic resonance conjecture. At least generically, all resonances in any logarithmic neighborhood of the real axis lie on quantized logarithmic curves

Sharp resonance abundance conjecture. For every $\epsilon>0$, there are infinitely many resonances $s\in\mathcal{R}_X$ satisfying

Let $\mathcal O\subset\mathbb R^3$ be a non-trapping obstacle, and let $\operatorname{Res}(\mathcal O)$ denote its resonance set. Suppose that $\mathcal O\subset B(0,1)$, where…

Let $H_\epsilon=-\frac{\partial^2}{\partial x^2}+v(x)-\epsilon x$ act on $L^2(\mathbb R)$, where $v$ is a real analytic $1$-periodic function and $\epsilon>0$. Its resonances form…

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…