Related papers: Counting the Resonances in High and Even Dimension…
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…
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…
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.…
For certain compactly supported metric and/or potential perturbations of the Laplacian on $\mathbb{H}^{n+1}$, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the…
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…
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…
We show that all resonances in Dirichlet obstacle scattering (in $\mathbb{C}$ in odd dimensions and in the logarithmic cover of $\mathbb{C}\setminus\{0\}$ in even dimensions) are generically simple in the class of obstacles with $C^k$ (and…
We consider scattering by an abstract compactly supported perturbation in R^n. To include the traditional cases of potential, obstacle and metric scattering without going into their particular nature we adopt the "black box" formalism…
We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb{R}$. In the cases of two or three delta poles, we are able to show that resonances…
We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary…
The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions $d \geq 2$. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto…
The scalar scattering of a plane wave by a smooth obstacle with impedance boundary conditions is considered. Upper bounds for the Total Cross Section and for the absorbed power are presented.
Most particles in nature are unstable, manifesting as resonances in scattering processes. Using analyticity and unitarity, we show nonperturbatively that resonances, defined as poles on higher Riemann sheets of scattering amplitudes, share…
We consider scattering by an obstacle in $\Real^d$, $d\geq 3 $ odd. We show that for the Neumann Laplacian if an obstacle has the same resonances as the ball of radius $\rho$ does, then the obstacle is a ball of radius $\rho$. We give…
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…
The particle approach to one-dimensional potential scattering is applied to non relativistic tunnelling between two, three and four identical barriers. We demonstrate as expected that the infinite sum of particle contributions yield the…
We develop a multidimensional coupled channel method suitable for studying the interplay of bound state resonance and phonon assisted scattering of inert gas atoms from solid surfaces in one, two and three dimensions. This enables us to get…
The paper concerns scattering of plane waves by a bounded obstacle with complex valued impedance boundary conditions. We study the spectrum of the Neumann-to-Dirichlet operator for small wave numbers and long wave asymptotic behavior of the…
We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…
For semiclassical problems we establish upper bounds on the number of resonances in boxes of size $h$ along the real axis, in terms of the dimension of the set of trapped trajectories. The proof uses second microlocalization.