F. Peherstorfer, A. Volberg, P. Yuditskii
In recent works we considered an asymptotic problem for orthogonal polynomials when a Szegö measure on the unit circumference is perturbed by an arbitrary Blaschke sequence of point masses outside the unit disk. In the current work we consider a similar problem in the scattering setting.
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We find the high energy asymptotics for the singular Weyl--Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schrödinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.
An original approach to the inverse scattering for Jacobi matrices was suggested in a recent paper by Volberg-Yuditskii. The authors considered quite sophisticated spectral sets (including Cantor sets of positive Lebesgue measure), however they did not take into account the mass point spectrum. This paper follows similar lines for the continuous setting with an absolutely continuous spectrum on the half-axis and a pure point spectrum on the negative half-axis satisfying the Blaschke condition. This leads us to the solution of the inverse scattering problem for a class of canonical systems that…
For a Riemannian manifold $(M,g)$ which is isometric to the Euclidean space outside of a compact set, and whose trapped set has Liouville measure zero, we prove Weyl type asymptotics for the scattering phase with remainder depending on the classical escape rate and the maximal expansion rate. For Axiom A geodesic flows, this gives a polynomial improvement over the known remainders. We also show that the remainder can be bounded above by the number of resonances in some neighbourhoods of the real axis, and provide similar asymptotics for hyperbolic quotients using the Selberg zeta function.
We review recent results obtained in the scattering theory of dissipative quantum systems representing the long-time evolution of a system $S$ interacting with another system $S'$ and susceptible of being absorbed by $S'$. The effective dynamics of $S$ is generated by an operator of the form $H = H_0 + V - \mathrm{i} C^* C$ on the Hilbert space of the pure states of $S$, where $H_0$ is the self-adjoint generator of the free dynamics of $S$, $V$ is symmetric and $C$ is bounded. The main example is a neutron interacting with a nucleus in the nuclear optical model. We recall the basic objects of …
In this paper, we consider a sequence of open quantum graphs, with uniformly bounded data, and we are interested in the asymptotic distribution of their scattering resonances. Supposing that the number of leads in our quantum graphs is small compared to the total number of edges, we show that most resonances are close to the real axis. More precisely, the asymptotic distribution of resonances of our open quantum graphs is the same as the asymptotic distribution of the square-root of the eigenvalues of the closed quantum graphs obtained by removing all the leads.
Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of resonances of -Delta+V with modulus less than r is approximatively equal to a constant times a^dr^d.
The purpose of this paper is to derive sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, and, in particular, to derive relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. Also we will prove that Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.
The fixed energy scattering matrix is defined on a perturbed stratified medium, and for a class of perturbations, its main part is shown to be a Fourier integral operator on the sphere at infinity. This is facilitated by developing a refined limiting absorption principle. The symbol of the scattering matrix is shown to determine the asymptotics of a large class of perturbations.
We study certain resonance-counting functions for potential scattering on infinite cylinders or half-cylinders. Under certain conditions on the potential, we obtain asymptotics of the counting functions, with an explicit formula for the constant appearing in the leading term.
Let H=Δ+\sum_{#a=2} V_a be a 3-body Hamiltonian, H_a the subsystem Hamiltonians, Δthe positive Laplacian of the Euclidean metric on X_0=R^n, V_a real-valued. Buslaev and Merkurev have shown that, if the pair potentials decay sufficiently fast, for ϕsmooth and compactly supported, the operator ϕ(H)-ϕ(H_0)-\sum_{#a=2}(ϕ(H_a)-ϕ(H_0)) is trace class. Hence, one can define a modified spectral shift function σ, as a distribution on R, by taking its trace. In this paper we show that if V_a are Schwartz, then σis in fact smooth away from the thresholds, and obtain its high energy asymptotics. In addit…
The problem of recovering the asymptotics of a short range perturbation of the Euclidean Laplacian on n dimensional Eudlidean space from fixed energy scattering data is studied. It is shown that for greater than or equal to three that a magnetic potential is determined, modulo Gauge invariance, by its scattering matrix at a fixed non-zero energy. This result also holds for a wide class of scattering manifolds.
The asymptotic structure of forward scattering
本文在渐近欧几里得情形下重新研究前向散射中受扰平面波的结构，通过双面紧化给出物理空间渐近的初等描述，证明了完全多齐次性，并得到两类模型问题。
Local decay and asymptotic profile for the damped wave equation in the asymptotically Euclidean setting
本文证明了渐近欧氏设置下波方程的局部衰减估计，在偶数维给出大时间渐近轮廓超越最优衰减，在奇数维改进已知最佳估计，结果适用于具短程吸收指数的阻尼波方程。
We consider an inner problem for whispering gallery high-frequency asymptotic mode's scattering by a boundary inflection. The related boundary-value problem for a Schrödinger equation on a half-line with a potential linear in both space and time appears fundamental for describing transitions from modal to scattered asymptotic patterns, and despite having been intensively studied over several decades remains largely unsolved. We prove that the solution past the inflection point has a ``searchlight'' asymptotics corresponding to a beam concentrated near the limit ray, and establish certain decay…
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。
On the interplay between inverse scattering for asymptotically hyperbolic manifolds and the Calderón problem for the Conformal Laplacian
本文通过渐近双曲流形与Conformal Laplacian的关系，证明了特定能量下的散射矩阵能确定边界度量的jet结构。
We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schrödinger operator $H_{0}$. Our goal is to construct various spectral quantities of the operator $H$, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair $H_{0}$, $H$, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials $P_{n}(z)$ associated to the Jacobi matrix $H $ as $n\to\infty$. In particular, we consider the case of $z$ inside the spe…
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