Semiclassical Analysis for Magnetic Scattering by Two Solenoidal Fields: Total Cross Sections

The fact that vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the Aharonov–Bohm effect (A–B effect). We study this quantum effect through the semiclassical analysis on total cross sections in the magnetic scattering by two solenoidal (point-like) fields with total flux vanishing in two dimensions. We derive the asymptotic formula with first three terms. The system with two parallel fields seems to be important in practical aspects as well as in theoretical aspects, because it may be thought of a toroidal solenoid with zero cross section in three dimensions under the idealization that the two fields connect at infinity in their direction. The corresponding classical mechanical system has the trajectory oscillating between two centers of fields. The special emphasis is placed on analyzing how the trapping effect from classical mechanics is related to the A–B quantum effect in the semiclassical asymptotic formula. Submitted: September 3, 2006. Accepted: January 10, 2007.     

Resonances in scattering by two magnetic fields at large separation and a complex scaling method

We study the quantum resonances in magnetic scattering in two dimensions. The scattering system consists of two obstacles by which the magnetic fields are completely shielded. The trajectories trapped between the two obstacles are shown to generate the resonances near the positive real axis, when the distance between the obstacles goes to infinity. The location is described in terms of the backward amplitudes for scattering by each obstacle. A difficulty arises from the fact that even if the supports of the magnetic fields are largely separated from each other, the corresponding vector potentials are not expected to be well separated. To overcome this, we make use of a gauge transformation and develop a new type of complex scaling method. We can cover the scattering by two solenoids at large separation as a special case. The obtained result heavily depends on the magnetic fluxes of the solenoids. This indicates that the Aharonov–Bohm effect influences the location of resonances.     

Gamow vectors for an unstable relativistic quantum field

We study a model of cubic interaction between two scalar fields with a scattering resonance. The resonance manifests as two poles of the analytic continuations of the Green function with respect to energy. The Gamow vectors associated to these resonances acquire meaning in suitable rigged Fock spaces. Finally, we discuss some properties of the S-matrix for unstable fields.     

Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications

We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows.

     

A spin integral equation for electromagnetic and acoustic scattering

We present a new integral equation for solving the Maxwell scattering problem against a perfect conductor. The very same algorithm also applies to sound-soft as well as sound-hard Helmholtz scattering, and in fact the latter two can be solved in parallel in three dimensions. Our integral equation does not break down at interior spurious resonances, and uses spaces of functions without any algebraic or differential constraints. The operator to invert at the boundary involves a singular integral operator closely related to the three-dimensional Cauchy singular integral, and is bounded on natural function spaces and depend analytically on the wave number. Our operators act on functions with pairs of complex two-by-two matrices as values, using a spin representation of the fields.     

A numerical study of divergence-free kernel approximations

Approximation properties of divergence-free vector fields by global and local solenoidal bases are studied. A comparison between interpolants generated with radial kernels and multivariate polynomials is presented. Numerical results show higher rates of convergence for derivatives of the vector field being approximated in directions enforced by the divergence operator when a rectangular grid is used. We also compute the growth of Lebesgue constants for uniform and clustered nodes and study the flat limit of divergence-free interpolants based on radial kernels. Numerical results are presented for two- and three-dimensional vector fields.     

Tensor tomography on surfaces

We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.     

Aharonov-Bohm Effect in Scattering by Point-like Magnetic Fields at Large Separation

The aim is to study the Aharonov–Bohm e.ect in the scattering by two point–like magnetic .elds at large separation in two dimensions. We analyze the asymptotic behavior of scattering amplitude when the distance between the centers of two .elds goes to in.nity. The obtained result heavily depends on the .uxes of .elds and on incident and .nal directions.     

Aharonov�CBohm Effect in Resonances of Magnetic Schr?dinger Operators with Potentials with Supports at Large Separation

Vector potentials are known to have a direct significance to quantum particles moving in the magnetic field. This is called the Aharonov–Bohm effect and is known as one of the most remarkable quantum phenomena. Here we study this quantum effect through the resonance problem. We consider the scattering system consisting of two scalar potentials and one magnetic field with supports at large separation in two dimensions. The system has trajectories oscillating between these supports. We give a sharp lower bound on the resonance widths as the distances between the three supports go to infinity. The bound is described in terms of the backward amplitude for scattering by each of the scalar potentials and by the magnetic field, and it also depends heavily on the magnetic flux of the field.     

Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows

We consider a phase field model for the flow of two partly miscible incompressible, viscous fluids of non-Newtonian (power law) type. In the model it is assumed that the densities of the fluids are equal. We prove the existence of weak solutions for general initial data and arbitrarily large times with the aid of a parabolic Lipschitz truncation method, which preserves solenoidal velocity fields and was recently developed by Breit, Diening, and Schwarzacher.     

A note on finite-sheeted covering maps from 2-dimensional compact Abelian groups

We consider finite-sheeted covering maps from 2-dimensional compact connected abelian groups to Klein bottle weak solenoidal spaces, metric continua which are not groups. We show that whenever a group covers a Klein bottle weak solenoidal space it covers groups as well, moreover it covers the product of two solenoids. The converse is not true, we give an example of group which covers groups with any finite number of sheets, but does not cover any Klein bottle weak solenoidal space.     

Reconstruction of the solenoidal part of a three-dimensional vector field by its ray transforms along straight lines parallel to coordinate planes

A numerical solution to a vector field reconstruction problem is proposed. It is assumed that the field is given in a unit sphere. The approximation of the solenoidal part of the vector field is constructed from ray transforms known over all straight lines parallel to one of the coordinate planes. Numerical simulations confirm that the proposed method yields good results of reconstruction of solenoidal vector fields.     

Regularity of ghosts in tensor tomography

We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L²-orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If. We give an explicit procedure for recovering the wave front set.     

Quantum decay rates in chaotic scattering

We study quantum scattering on manifolds equivalent to the Euclidean space near infinity, in the semiclassical regime. We assume that the corresponding classical flow admits a non-trivial trapped set, and that the dynamics on this set is of Axiom A type (uniformly hyperbolic). We are interested in the distribution of quantum resonances near the real axis. In two dimensions, we prove that, if the trapped set is sufficiently “thin”, then there exists a gap between the resonances and the real axis (that is, quantum decay rates are bounded from below). In higher dimension, the condition for this gap is given in terms of a certain topological pressure associated with the classical flow. Under the same assumption, we also prove a resolvent estimate with a logarithmic loss compared to non-trapping situations.     

Direct separation of two-dimensional vector fields into irrotational and solenoidal parts

An iterative technique is given for separating a two-dimensional vector into irrotational and solenoidal parts. In contrast to classical methods, the procedure operates directly on the orthogonal scalar components of the vector field, and gives the two separate fields as the result of a single sequence of operations. The manipulations are somewhat similar to a relaxation process. An example of the decomposition of a meteorological wind field is given.     

Multilevel methods for mixed finite elements in three dimensions

In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as s of the –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof. Received November 4, 1996 / Revised version received February 2, 1998     

Compton scattering in intense magnetic fields

Compton scattering in intense magnetic fields in the general frame of reference is studied with the help of the QED perturbation theory in the incoming interaction picture. A general expression for the cross section is derived which reduces naturally to the one in the electron-rest frame of reference. This expression can be approximately simplified for the scattering of a high-energy electron with a low-frequency photon. Based on this simplified expreaaion, spectrum functions, as well as power spectra of scattered photons with high energies resulting from the inverse Compton scattering are calculated which manifest clearly a feature of resonances. Project supported by the National Natural Science Foundation of China (Grant No. 19573008) and the Science Research Division of Shanghai Jiaotong University.     

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.