Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials

We prove global smoothing and Strichartz estimates for the Schrödinger, wave, Klein–Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.     

Global small solutions to the critical radial Dirac equation with potential

We solve globally a radial cubic Dirac equation perturbed with a small potential, with data of small critical norm H¹. The main tool is a new endpoint estimate of the perturbed Dirac flow for a class of radial-type initial data.     

Rank one perturbations of not semibounded operators

Rank one perturbations of selfadjoint operators which are not necessarily semibounded are studied in the present paper. It is proven that such perturbations are uniquely defined, if they are bounded in the sense of forms. We also show that form unbounded rank one perturbations can be uniquely defined if the original operator and the perturbation are homogeneous with respect to a certain one parameter semigroup. The perturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Krein's formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense without renormalization of the coupling constant only if the original operator is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimension.     

扰动色谱方程黎曼解的极限-delta激波的行成和转换

本文主要讨论扰动色谱方程delta激波解的行成和转换,并讨论上述方程的黎曼问题.当扰动参数趋于零时,通过研究黎曼解的极限,我们可以观察到如下两个重要现象:激波和接触间断重合行成delta激波,一类激波(一个变量含有delta函数).     

Algebro-geometric solutions of the Dirac hierarchy

We introduce a Lenard equation and present two special solutions of it. We use one solution to derive an extended Dirac hierarchy and the other to construct the generating function. The generating function yields conserved integrals of the Dirac Hamiltonian system and defines an algebraic curve. Based on the theory of algebraic curves, we prove that the Dirac Hamiltonian system is integrable and obtain algebro-geometric solutions of the Dirac hierarchy.     

Solution of the inverse quasiperiodic problem for the Dirac system

We present a complete solution of the inverse problem of spectral analysis for the Dirac operator with quasiperiodic boundary conditions. We prove a uniqueness theorem for the solution of the inverse problem and obtain necessary and sufficient conditions for a sequence of real numbers to be the spectrum of a quasiperiodic Dirac problem.     

Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits

In this paper we prove several results for the scattering phase (spectral shift function) related with perturbations of the electromagnetic field for the Dirac operator in the Euclidean space. Many accurate results are now available for perturbations of the Schrödinger operator, in the high energy regime or in the semi-classical regime. Here we extend these results to the Dirac operator. There are several technical problems to overcome because the Dirac operator is a system, its symbol is a 4×4 matrix, and its continuous spectrum has positive and negative values. We show that we can separate positive and negative energies to prove high energy asymptotic expansion and we construct a semi-classical Foldy-Wouthuysen transformation in the semi-classical case. We also prove an asymptotic expansion for the scattering phase when the speed of light tends to infinity (non-relativistic limit).     

Quasiconvex functions, SO(n) and two elastic wells

We use W^1,∞ approximations of minimizing sequences to study the growth of some quasiconvex functions near their zero sets. We show that for SO(n), the quasiconvexification of the distance function dist²(·, SO(n)) can be bounded below by the distance function itself. In certain cases of the incompatible two elastic well structure, we establish a similar result. We also prove that for small Lipschitz perturbations of SO(n) and of the two well structure, the Young measure limits of gradients supported on these perturbed sets are Dirac masses.     

Transmission problems for Maxwell's equations with weakly Lipschitz interfaces

We prove sufficient conditions on material constants, frequency and Lipschitz regularity of interface for well posedness of a generalized Maxwell transmission problem in finite energy norms. This is done by embedding Maxwell's equations in an elliptic Dirac equation, by constructing the natural trace space for the transmission problem and using Hodge decompositions for operators d and δ on weakly Lipschitz domains to prove stability. We also obtain results for boundary value problems and transmission problems for the Hodge–Dirac equation and prove spectral estimates for boundary singular integral operators related to double layer potentials. Copyright © 2005 John Wiley & Sons, Ltd.     

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

We extend classical results of Kostant et al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.     

Spectral properties of Dirac operators with singular potentials

We investigate the spectral properties of Dirac operators with singular potentials which are constructed by means of a cut-off procedure. We prove the invariance of the essential spectrum, establish norm resolvent convergence of the cut-off operators, and prove spectral gap formulas.     

Radiative transport limit of Dirac equations with random electromagnetic field

Abstract

This paper concerns the kinetic limit of the Dirac equation with a random electromagnetic field. We give a detailed mathematical analysis of the radiative transport limit for the phase space energy density of solutions to the Dirac equation. Our derivation is based on a martingale method and a perturbed test function expansion. This requires the electromagnetic field to be a Markovian space-time random field. The main mathematical tool in the derivation of the kinetic limit is the matrix-valued Wigner transform of the vector-valued Dirac solution. The major novelty compared with the scalar (Schrödinger) case is the proof of the weak convergence of cross modes to zero. The propagating modes are shown to converge in an appropriate probabilistic sense to their deterministic limit.     

Eigenvalue estimates for the Dirac operator on quaternionic K?hler manifolds

We consider the Dirac operator on compact quaternionic K?hler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space. Received April 21, 1998; in final form June 16, 1998     

Interactions of delta shock waves for a class of nonstrictly hyperbolic system of conservation laws

In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter $\varepsilon$ tend to zero, we analyze the stability of Riemann solutions.     

On the Absolutely Continuous Spectrum of Dirac Operator

Abstract

We prove that the massless Dirac operator in ?³ with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.     

Eigenvalue estimates for Dirac operators coupled to instantons

We prove a sharp lower bound for the first positive eigenvalue of Dirac operators coupled to instantons and discuss the limit case.     

Dirac Operators on Hermitian Spin Surfaces

We prove the conformal invariance of the dimension of thekernel of any of the self-adjoint Dirac operators associated to thecanonical Hermitian connections on Hermitian spin surface. In the caseof a surface of nonnegative conformal scalar curvature we estimate thefirst eigenvalue of the self-adjoint Dirac operator associated to theChern connection and list the surfaces on which its kernel isnontrivial.     

A Sobolev-like inequality for the Dirac operator

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.     

Nonexistence of time‐periodic solutions of the Dirac equation in an axisymmetric black hole geometry

We prove that in the nonextreme Kerr‐Newman black hole geometry, the Dirac equation has no normalizable, time‐periodic solutions. A key tool is Chan‐drasekhar's separation of the Dirac equation in this geometry. A similar nonexistence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast to the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity. © 2000 John Wiley & Sons, Inc.     

Some remarks on strong unique continuation for the laplace operator and its powers

ABSTRACT

Some spectral properties of magnetic Schrödinger and Dirac operators perturbed by long range magnetic fields are investigated. If the intensity of the field is small enough, a better location of the perturbed spectrum is given. In particular, if the unperturbed spectrum is discrete, we show that the perturbed eigenvalues are given in terms of an absolutely convergent series with respect to a magnetic parameter, from which the usual asymptotic expansion can be derived.