Long-Time Dynamics of Variable Coefficient mKdV Solitary Waves

We study the Korteweg-de Vries-type equation dt u=-dx(dx^2 u+f(u)-B(t,x)u), where B is a small and bounded, slowly varying function and f is a nonlinearity. Many variable coefficient KdV-type equations can be rescaled into this equation. We study the long time behaviour of solutions with initial conditions close to a stable, B=0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave, whose centre and scale evolve according to a certain dynamical law involving the function B(t,x), plus an H^1-small fluctuation.Comment: 19 page

Review of finite fields: Applications to discrete Fourier, transforms and Reed-Solomon coding

An attempt is made to provide a step-by-step approach to the subject of finite fields. Rigorous proofs and highly theoretical materials are avoided. The simple concepts of groups, rings, and fields are discussed and developed more or less heuristically. Examples are used liberally to illustrate the meaning of definitions and theories. Applications include discrete Fourier transforms and Reed-Solomon coding

Generalizing Tsirelson's bound on Bell inequalities using a min-max principle

Bounds on the norm of quantum operators associated with classical Bell-type inequalities can be derived from their maximal eigenvalues. This quantitative method enables detailed predictions of the maximal violations of Bell-type inequalities.Comment: 4 pages, 2 figures, RevTeX4, replaced with published versio

Rigorous Dynamics and Radiation Theory for a Pauli-Fierz Model in the Ultraviolet Limit

The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behaviour of the survival and transition amplitudes for the process of decay of the excited states of the charged particle, and the emission of photons in the decay process. In particular, for the survival amplitude the exact time behaviour is found. This is completely determined by the resonances of the systems plus a tail term prevailing in the asymptotic, long time regime. Moreover, the survival amplitude exhibites in a fairly clear way the Lamb shift correction to the unperturbed frequencies of the oscillator.Comment: Shortened version. To appear in J. Math. Phy

Wave operator bounds for 1-dimensional Schr\"odinger operators with singular potentials and applications

Boundedness of wave operators for Schr\"odinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.Comment: 16 pages, 0 figure

Initial Value Problems and Signature Change

We make a rigorous study of classical field equations on a 2-dimensional signature changing spacetime using the techniques of operator theory. Boundary conditions at the surface of signature change are determined by forming self-adjoint extensions of the Schr\"odinger Hamiltonian. We show that the initial value problem for the Klein--Gordon equation on this spacetime is ill-posed in the sense that its solutions are unstable. Furthermore, if the initial data is smooth and compactly supported away from the surface of signature change, the solution has divergent $L^2$-norm after finite time.Comment: 33 pages, LaTeX The introduction has been altered, and new work (relating our previous results to continuous signature change) has been include

Quantum transport in a resonant tunnel junction coupled to a nanomechanical oscillator

We discuss the quantum transport of electrons through a resonant tunnel junction coupled to a nanomechanical oscillator at zero temperature. By using the Green's function technique we calculate the transport properties of electrons through a single dot strongly coupled to a single oscillator. We consider a finite chemical potential difference between the right and left leads. In addition to the main resonant peak of electrons on the dot, we find satellite peaks due to the creation of phonons. These satellite peaks become sharper and more significant with increasing coupling strength between the electrons and the oscillator. We also consider the energy transferred from the electrons to the oscillator.Comment: Updated in response to referees' comments. Section IV amended including figure

Divergences in the Effective Action for Acausal Spacetimes

The 1--loop effective Lagrangian for a massive scalar field on an arbitrary causality violating spacetime is calculated using the methods of Euclidean quantum field theory in curved spacetime. Fields of spin 1/2, spin 1 and twisted field configurations are also considered. In general, we find that the Lagrangian diverges to minus infinity at each of the nth polarised hypersurfaces of the spacetime with a structure governed by a DeWitt-Schwinger type expansion.Comment: 17 pages, Late

Scattering theory for lattice operators in dimension d≥3d\geq 3

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension $d\geq 3$ the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in presence of embedded eigenvalues and threshold singularities.Comment: Minor errors and misprints corrected; new result on absense of embedded eigenvalues for potential scattering; to appear in RM

Klein-Gordon Solutions on Non-Globally Hyperbolic Standard Static Spacetimes

We construct a class of solutions to the Cauchy problem of the Klein-Gordon equation on any standard static spacetime. Specifically, we have constructed solutions to the Cauchy problem based on any self-adjoint extension (satisfying a technical condition: "acceptability") of (some variant of) the Laplace-Beltrami operator defined on test functions in an $L^2$-space of the static hypersurface. The proof of the existence of this construction completes and extends work originally done by Wald. Further results include the uniqueness of these solutions, their support properties, the construction of the space of solutions and the energy and symplectic form on this space, an analysis of certain symmetries on the space of solutions and of various examples of this method, including the construction of a non-bounded below acceptable self-adjoint extension generating the dynamics