Related papers: On the positivity of scattering operators for Poin…

To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator $M$ along the trapped ray between the two obstacles.…

There is a class of Laplacian like conformally invariant differential operators on differential forms $L^\ell_k$ which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as…

For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We…

Let $g$ and $\tilde{g}$ be Riemannian metrics on a noncompact manifold $M$, which are conformally equivalent. We show that under a very mild \emph{first order} control on the conformal factor, the wave operators corresponding to the…

In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second…

For any positive real number $s$, we study the scattering theory in a unified way for the fractional Schr\"{o}dinger operator $H=H_0+V$, where $H_0=(-\Delta)^\frac s2$ and the real-valued potential $V$ satisfies short range condition. We…

Let $P$ be a Schr\"odinger operator $D_t+\Delta_g$ with metric and potential perturbation that are compactly supported in spacetime $\mathbb{R}^{n+1}$. Here $D_t = -i \partial_t$ and $\Delta_g$ is the positive Laplacian. We consider the…

For a compact PSC Riemannian $n$-manifold $(M,g)$, the metric constant $\mathrm {Riem}(g)\in (0, \binom{n}{2}]$ is defined to be the infinimum over $M$ of the spectral scalar curvature $\frac{\sum_{i=1}^N\lambda_i}{\lambda_{\rm max}}$ of…

We prove a Poincar\'e, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature…

A scattering process can be described by suitably closing the system and considering the first return map from the entrance onto itself. This scattering map may be singular and discontinuous, but it will be measure preserving as a…

Let $\alpha\in(0,2)$ and $X_t$ be a symmetric $\alpha$-stable process. We define the scattering length $\Gamma(v)$ of the positive potential $v$ and prove several of its basic properties. We use the scattering length to findestimates for…

For negatively curved symmetric spaces it is known from [Hansen-Hilgert-Parthasarathy,2019] that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the…

This is the first in a series of papers on scattering theory for one-dimensional Schr\"odinger operators with highly singular potentials $q\in H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive Schr\"odinger…

In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic St{\"a}ckel manifolds having the topology of toric cylinders and satisfying the Robertson condition. On these manifolds the…

We study Jacobi operators $J_{p}$, $p> -1$, whose eigenfunctions are Laguerre polynomials. All operators $J_{p}$ have absolutely continuous simple spectra coinciding with the positive half-axis. This fact, however, by no means imply that…

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier…

In the present article, we assume that the first approximation of the scattering operator is given and that it has the logarithmic divergence. This first approximation allows us to construct the so called deviation factor. Using the…

Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, $g.$ A model form is established for such metrics close to the boundary. It is shown that the scattering matrix at energy…

We develop direct and inverse scattering theory for Jacobi operators (doubly infinite second order difference operators) with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on…

In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincar\"{e} exponent less than…