The spectral shift function for planar obstacle scattering at low energy

平面障碍物的低能散射渐近性

Low energy scattering asymptotics for planar obstacles

该研究计算平面障碍物预解式的低能渐近，推导散射矩阵等的渐近性，借助两个恒等式建立关联，指出主导奇点由对数容量或Robin常数表示，且结果有望推广至拉普拉斯算子的一般紧支集扰动。

Equidistribution of Phase Shifts in Obstacle Scattering

For scattering off a smooth, strictly convex obstacle $Ω\subset \mathbb{R}^d$ with positive curvature, we show that the eigenvalues of the scattering matrix -- the phase shifts -- equidistribute on the unit circle as the frequency $k \to \infty$ at a rate proportional to $k^{d - 1}$, under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector $S \subset \mathbb{S}^1$ not containing $1$, there are $c_d |S| \mathrm{Vol}(\partial Ω)\ k^{d - 1} + o(k^{d-1})$ eigenvalues for $k$ large, where $c_d$ is a constant depending only on the dimension. U…

A relative trace formula for obstacle scattering

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$. In this setting the absolutely continuous part of the Laplace operator $Δ$ with Dirichlet boundary conditions and the free Laplace operator $Δ_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(Δ)-g(Δ_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative…

Approximation and equidistribution of phase shifts: spherical symmetry

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} Δ+ V - E \end{equation*} where $h > 0$ is a semiclassical parameter, $Δ$ is the positive Laplacian on $\mathbb{R}^{d}$, $V$ is a smooth, compactly supported central potential function and $E > 0$ is an energy level. In this setting the scattering matrix $S_h(E)$ is a unitary operator on $L^2(\mathbb{S}^{d-1})$, hence with spectrum lying on the unit circle; moreover, the spectrum is discrete except at $1$. We show under certain additional assumptions on the potential that the eigenvalues of $S_h(E)$ can be divided into two…

Trace singularities in obstacle scattering and the Poisson relation for the relative trace

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$ for the Laplace operator $Δ$ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $Δ_1$ and $Δ_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative trace operator $g(Δ) - g(Δ_1) - g(Δ_2) + g(Δ_0)$ was introduced in [18] and shown to be trace-class for a large class of functions $g$, including certrain functions of polynomial growth. When $g$ is sufficiently regular at zero and fast decaying at infi…

Inverse scattering at fixed energy for radial magnetic Schr{ö}dinger operators with obstacle in dimension two

We study an inverse scattering problem at fixed energy for radial magnetic Schr{ö}dinger operators on R^2 \ B(0, r\_0), where r\_0 is a positive and arbitrarily small radius. We assume that the magnetic potential A satisfies a gauge condition and we consider the class C of smooth, radial and compactly supported electric potentials and magnetic fields denoted by V and B respectively. If (V, B) and (\tilde{V} , \tilde{B}) are two couples belonging to C, we then show that if the corresponding phase shifts $δ$\_l and \tilde{$δ$}\_l (i.e. the scattering data at fixed energy) coincide for all l $\in…