Related papers: Sharp upper bounds on resonances for perturbations…

For various compactly supported perturbations of the Laplacian in odd dimensions $n$, we prove a sharp upper bound of the resonance counting function $N(r)$ of the type $N(r) \le A_n r^n(1+o(1))$ with an explicit constant $A_n$. In a few…

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or…

For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature.…

This paper proves sharp lower bounds on a resonance counting function for obstacle scattering in even-dimensional Euclidean space without a need for trapping assumptions. Similar lower bounds are proved for some other compactly supported…

On geometrically finite hyperbolic manifolds $\Gamma\backslash H^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R\to \infty$. In…

We prove sharp upper bounds for the number of resonances in boxes of size 1 at high frequency for the Laplacian on finite volume surfaces with hyperbolic cusps. As a corollary, we obtain a Weyl asymptotic for the number of resonances in…

We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to an abstractly defined class of black box perturbations of the Laplacian in $\mathbb{R}^n$ which can be analytically extended from…

We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound $\mathrm{Im}\,\lambda \leq -\frac{1}{2}$ which is optimal in dimension $2$. In odd dimensions we also show that…

Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove…

We prove some sharp upper bounds on the number of resonances associated with the Laplacian, or Laplacian plus potential, on a manifold with infinite cylidrical ends.

This paper is concerned with the numerical computation of scattering resonances of the Laplacian for Dirichlet obstacles with rough boundary. We prove that under mild geometric assumptions on the obstacle there exists an algorithm whose…

In this paper, we give a polynomial lower bound for the resonances of $-\Delta$ perturbed by an obstacle in even-dimensional Euclidean spaces, $n\geq4$. The proof is based on a Poisson Summation Formula which comes from the Hadamard…

We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in…

We prove a polynomial upper bound on the number of resonances in a disc whose radius tends to infinity for even asymptotically hyperbolic manifolds with real-analytic ends. Our analysis also gives a similar upper bound on the number of…

In even dimensional Euclidean scattering, the resonances lie on the logarithmic cover of the complex plane. This paper studies resonances for obstacle scattering in ${\mathbb R}^d$ with Dirchlet or admissable Robin boundary conditions, when…

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real-valued potentials. For $L^1$-potentials, we obtain optimal…

The resonances for the Dirichlet and Neumann Laplacian are studied on compactly perturbed waveguides. An upper bound on the number of resonances near the physical plane is proven. In the absence of resonances, an upper bound is proven for…

The study of wave propagation outside bounded obstacles uncovers the existence of resonances for the Laplace operator, which are complex-valued generalized eigenvalues, relevant to estimate the long time asymptotics of the wave. In order to…

In this note we explicitly compute the resonances on hyperbolic cones. These are hyperbolic manifolds with a conic singularity equipped with a warped product metric. The calculation is based on separation of variables and Kummer's…

As a consequence of a result of Cardoso and Vodev, we show that the resolvent of the Laplacian on asymptotically hyperbolic manifolds is analytic in an exponential neighbourhood of the critical line. The case of non-trapping metrics with…