Semiclassical Dynamics¶with Exponentially Small Error Estimates

We construct approximate solutions to the time–dependent Schr?dinger?equation

Spacing Between Phase Shifts in a Simple¶Scattering Problem

We prove a scattering theoretical version of the Berry–Tabor conjecture: for an almost every surface in a class of cylindrical surfaces of revolution, the large energy limit of the pair correlation measure of the quantum phase shifts is Poisson, that is, it is given by the uniform measure. Received: 15 August 1998 / Accepted: 15 February 1999     

Localization in One Dimensional Random Media:¶A Scattering Theoretic Approach

We use scattering theoretic methods to prove exponential localization for random displacement models in one dimension. The operators we consider model both quantum and classical wave propagation. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations. We show that randomly displaced, non-reflectionless single sites lead to localization. Received: 23 September 1999 / Accepted: 13 March 2000     

Asymptotically Flat Initial Data¶with Prescribed Regularity at Infinity

We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate. Received: 12 February 2001 / Accepted: 28 May 2001     

Decay Rates and Probability Estimates¶for Massive Dirac Particles¶in the Kerr–Newman Black Hole Geometry

The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in at least at the rate t ^−5/6. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p = 0, 1 or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator constructed in [4]. Received: 20 August 2001 / Accepted: 22 January 2002 RID="*" ID="*"Present address: NWF I – Mathematik, Universit?t Regensburg, 93040 Regensburg, Germany.?E-mail: felix.finster@mathematik.uni-regensburg.de RID="**" ID="**"Research supported by NSERC grant # RGPIN 105490-1998. RID="***" ID="***"Research supported in part by the NSF, Grant No. DMS-0103998. RID="****" ID="****"Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

Scattering Theory for Quantum Fields¶with Indefinite Metric

In this work, we discuss the scattering theory of local, relativistic quantum fields with indefinite metric. Since the results of Haag–Ruelle theory do not carry over to the case of indefinite metric [4], we propose an axiomatic framework for the construction of in- and out-states, such that the LSZ asymptotic condition can be derived from the assumptions. The central mathematical object for this construction is the collection of mixed vacuum expectation values of local, in- and out-fields, called the “form factor functional”, which is required to fulfill a Hilbert space structure condition. Given a scattering matrix with polynomial transfer functions, we then construct interpolating, local, relativistic quantum fields with indefinite metric, which fit into the given scattering framework. Received: 13 September 1999/ Accepted: 1 August 2000     

Global Uniqueness in the Inverse Scattering Problem¶for the Schrödinger Operator¶with External Yang–Mills Potentials

We consider the Schr?dinger equation in R ⁿ , n≥ 3, with external Yang–Mills potentials having compact supports. We prove the uniqueness modulo a gauge transformation of the solution of the inverse boundary value problem in a bounded convex domain. A similar uniqueness result holds for the inverse scattering problem at a fixed energy. Received: 11 August 2000 / Accepted: 24 May 2001     

Resonances in Positron-Mg Scattering

We present the results for resonances in positron-Mg scattering at low impact energy （0-5.0eV） by using the momentum space coupled-channel optical （CCO） method in momentum space. The S-partial wave resonance at 3.880eV, P-partial wave resonance at 4.020eV, and D-partial wave resonance at 4.267eV are found.     

Resonances in Positron-He Scattering

We present the results for resonances in positron-He scattering at low impact energy （19.3-24.0eV） by using the momentum space coupled-channel optical （CCO） method. The S-partial wave resonance at 20.16 eV is found for the first time.     

Semiclassical Limit for the Schrödinger Equation¶with a Short Scale Periodic Potential

We consider the dynamics generated by the Schr?dinger operator H=−?Δ+V(x)+W(ɛx), where V is a lattice periodic potential and W an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit ɛ→ 0 the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics. Received: 7 February 2000 / Accepted: 7 July 2000     

Geometric Optics and Long Range Scattering¶for One-Dimensional Nonlinear Schrödinger Equations

With the methods of geometric optics used in [2], we provide a new proof of some results of [10], to construct modified wave operators for the one-dimensional cubic Schr?dinger equation. We improve the rate of convergence of the nonlinear solution towards the simplified evolution, and get better control of the loss of regularity in Sobolev spaces. In particular, using the results of [9], we deduce the existence of a modified scattering operator with small data in some Sobolev spaces. We show that in terms of geometric optics, this gives rise to a “random phase shift” at a caustic. Received: 23 May 2000 / Accepted: 8 January 2001     

Thomson Scattering Process in Laser-Produced Plasmas

We present the evolutions of the electron temperature and plasma expansion velocity with Thomson scattering experiment. The observed time-resolved ion-acoustic image is reproduced by a numerical code which couples the Thomson scattering theory with the output parameters of the one-dimensional hydrocode MEDUSA.     

The Fundamental Solution and Strichartz Estimates for the Schrödinger Equation on Flat Euclidean Cones

We study the Schrödinger equation on a flat euclidean cone ${\mathbb{R}_+ \times \mathbb{S}^1_\rho}We study the Schr?dinger equation on a flat euclidean cone \mathbbR₊ ×\mathbbS¹_r{\mathbb{R}_+ \times \mathbb{S}^1_\rho} of cross-sectional radius ρ > 0, developing asymptotics for the fundamental solution both in the regime near the cone point and at radial infinity. These asymptotic expansions remain uniform while approaching the intersection of the “geometric front,” the part of the solution coming from formal application of the method of images, and the “diffractive front” emerging from the cone tip. As an application, we prove Strichartz estimates for the Schr?dinger propagator on this class of cones.     

Euclidean non-vacuum wormholes in Brans－Dicke theory

The Brans－Dicke theory is investigated in which the Pauli metric is identified to be a physical spacetime metric. The solutions of a wormhole are obtained in Brans－Dicke theory with a relativistic radiation field for ω>-3/2. However, it is found that one cannot construct a wormhole in the presence of a 3-form axion field.     

Survival Probability¶in Rank-One Perturbation Problems

A finite complex Borel measure μ on the unit circle or on the real line is called Rajchman if its Fourier coefficients tend to 0 as n→∞. In quantum dynamics the self-adjoint operators (Hamiltonians) whose spectral measures are Rajchman correspond to the systems having certain scattering properties. In this paper we study how a small perturbation of the operator can affect the Rajchman property of its spectral measure. Our approach is based on the notion of the local symmetry of measures. Received: 17 April 2001 / Accepted: 18 June 2001     

Parity Violation in Proton—Deuteron Scattering

JETP Letters - Effects of parity violation in the interaction of relativistic polarized protons and deuterons are discussed. P-odd asymmetries in the total and elastic scattering cross sections, in...     

Solving the Bethe–Salpeter Equation in Euclidean Space

Different approaches to solve the spinor–spinor Bethe–Salpeter (BS) equation in Euclidean space are considered. It is argued that the complete set of Dirac matrices is the most appropriate basis to define the partial amplitudes and to solve numerically the resulting system of equations with realistic interaction kernels. Other representations can be obtained by performing proper unitary transformations. A generalization of the iteration method for finding the energy spectrum of the BS equation is discussed and examples of concrete calculations are presented. Comparison of relativistic calculations with available experimental data and with corresponding non relativistic results together with an analysis of the role of Lorentz boost effects and relativistic corrections are presented. A novel method related to the use of hyperspherical harmonics is considered for a representation of the vertex functions suitable for numerical calculations.     

Stimulated Raman Scattering in Bi-temperature Vlasov Plasma

We obtain the nonlinear coupling equations for describing stimulating Raman scattering (SRS) of ultra-intense laser beam starting from the relativistic Vlasov-Maxwell equations. It is found that the relativistic electrons play an important role in the inhibition of the SRS instability and the growth rate of backward SRS strongly decreases with the increasing number of relativistic electrons.