the need for a flat higher gauge structure to describe a berry phase associated with some resonance phenomena in the presence of a resonance crossing producing splitting of the base manifold for example a circle crossing in a plane we show that the rigorous geometrical structure within which the berry phase arises may be a 2 bundle a structure related to gerbes and to category theory rather than a fiber bundle the bloch wave operator plays an important role in the associated theory

phase space green s functions for modeling time harmonic scattering from smooth inhomogeneous objects the paper deals with inhomogeneous medium green s functions in the phase space domain by which the phase space local spectral distributions of the field scattered by a high contrast object due a genetic time harmonic incidence are evaluated two forms of phase space green s functions are considered one that links induced sources in the configuration space to phase space distributions of the scattered field while the other one directly links the phase space distribution of the incident field to phase space distributions of the scattered field the scattering mechanism is described in terms of local samplings of the object function which are localized in the object domain according to the scattered and incidence processing parameters applications in the field of inverse scattering may be expected to yield fast and efficient algorithms due to the capability of analytically evaluating forward scattering green s functions

general adiabatic evolution with a gap condition we consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity away from the rest of the spectrum the restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable

construction of high order force gradient algorithms for integration of motion in classical and quantum systems a consequent approach is proposed to construct symplectic force gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations within this approach the basic algorithms are first derived up to the eighth order by direct decompositions of exponential propagators and further collected using an advanced composition scheme to obtain the algorithms of higher orders contrary to the scheme proposed by chin and kidwell phys rev e 62 8746 2000 where high order algorithms are introduced by standard iterations of a force gradient integrator of order four the present method allows one to reduce the total number of expensive force and its gradient evaluations to a minimum at the same time the precision of the integration increases significantly especially with increasing the order of the generated schemes the algorithms are tested in molecular dynamics and celestial mechanics simulations it is shown in particular that the efficiency of the advanced fourth order based algorithms is better approximately in factors 5 to 1000 for orders 4 to 12 respectively the results corresponding to sixth and eighth order based composition schemes are also presented up to the sixteenth order for orders 14 and 16 such highly precise schemes at considerably smaller computational costs allow to reduce unphysical deviations in the total energy up in 100000 times with respect to those of the standard fourth order based iteration approach

non orthogonal domain parabolic equation and its tilted gaussian beam solutions a non orthogonal coordinate system which is a priori matched to localized initial field distributions for time harmonic wave propagation is presented applying in addition a rigorous paraxial asymptotic approximation results in a novel parabolic wave equation for beam type field propagation in 3d homogeneous media localized solutions to this equation that exactly match linearly phased gaussian aperture distributions are termed tilted gaussian beams these beams serve as the building blocks for various beam type expansion schemes application of the scalar waveobjects to electromagnetic field beam type expansion as well as reflection and transmission of these waveobjects by planar velocity dielectric discontinuity are presented a numerical example which demonstrates the enhanced accuracy of the tilted gaussian beams over the conventional ones concludes the paper

on the adiabatic theorem for nonself adjoint hamiltonians the authors consider the evolution of a two level system driven by a nonself adjoint hamiltonian h in t and treat the adiabatic limit in to 0 while adiabatic theorem like results do not hold true in general for this case they prove that they are still valid for the subspace corresponding to the eigenvalue having the largest imaginary part least dissipative eigenvalue the theory gives the full asymptotic expansion of the evolution restricted to this subspace the first correction beyond berry s phase is to their best knowledge given explicitly for the first time

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

spectral methods in time for parabolic problems a pseudospectral explicit scheme for solving linear periodic parabolic problems is described it has infinite accuracy both in time and in space the high accuracy is achieved while the time resolution parameter m m o 1 delta t for time marching algorithm and the space resolution parameter n n o 1 delta x must satisfy m o n 1 varepsilon varepsilon 0 compared to the common stability condition m o n 2 which must be satisfied in any explicit finite order time algorithm

improving the accuracy of the avf method the average vector field avf method is a b series scheme of the second order as a discrete gradient method it preserves exactly the energy integral for any canonical hamiltonian system we present and discuss two locally exact and energy preserving modifications of the avf method avf lex of the third order and avf slex of the fourth order applications to spherically symmetric potentials are given including a compact explicit expression for the avf scheme for the coulomb kepler problem

hamiltonian conserving discrete canonical equations based on variational difference quotients abstract new discrete mechanics based on the assumption of the discrete time is proposed the discrete mechanics does not contain any continuous differentiation but contains only difference quotients resulting discrete hamiltonian s canonical equations are single time step difference equations and exactly conserve the hamiltonian the canonical equations give the numerical results more accurately than the heun scheme and the 4th order runge kutta scheme

discrete approximations on functional classes for the integrable nonlinear schrodinger dynamical system a symplectic finite dimensional reduction approach abstract we investigate discretizations of the integrable discrete nonlinear schrodinger dynamical system and related symplectic structures we develop an effective scheme of invariant reducing the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter we construct a finite set of recurrent algebraic regular relations allowing to generate solutions of the discrete nonlinear schrodinger dynamical system and we discuss the related functional spaces of solutions finally we discuss the fourier transform approach to studying the solution set of the discrete nonlinear schrodinger dynamical system and its functional analytical aspects

an improved stormer verlet method based on exact discretization for nonlinear oscillators abstract motivated by the advantage of exact discretization of a linear differential equation and the importance of symplectic numerical methods for conservative nonlinear oscillators a modified stormer verlet method relying on a parameter is proposed the main idea is firstly based on some analytical approximation strategies relating a linear equation with the corresponding nonlinear equation such that linear equation s frequency approximates the exact frequency of the nonlinear equation secondly forcing the modified stormer verlet method to solve the related linear equation exactly the convergence symplectic and symmetric properties of the new method are analyzed for numerical implementation the cubic duffing equation and the simple pendulum are solved by the new method with some approximate frequencies as the parameter respectively numerical results show that the new method is much more accurate than its classical partner

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

energy preserving numerical schemes of high accuracy for one dimensional hamiltonian systems we present a class of non standard numerical schemes which are modifications of the discrete gradient method they preserve the energy integral exactly up to the round off error the considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order in numerical experiments we compare our integrators with some other numerical schemes including the standard discrete gradient method the leap frog scheme and a symplectic scheme of fourth order we study the error accumulation for a very long time and the conservation of the energy integral

locally exact modifications of numerical schemes we present a new class of exponential integrators for ordinary differential equations locally exact modifications of known numerical schemes local exactness means that they preserve the linearization of the original system at every point in particular locally exact integrators preserve all fixed points and are a stable we apply this approach to popular schemes including euler schemes the implicit midpoint rule and the trapezoidal rule we found locally exact modifications of discrete gradient schemes for symmetric discrete gradients and coordinate increment discrete gradients preserving their main geometric property exact conservation of the energy integral for arbitrary multidimensional hamiltonian systems in canonical coordinates numerical experiments for a two dimensional anharmonic oscillator show that locally exact schemes have very good accuracy in the neighbourhood of stable equilibrium much higher than suggested by the order of new schemes locally exact modification sometimes increases the order but in many cases leaves it unchanged

new definitions of exponential hyperbolic and trigonometric functions on time scales abstract we propose two new definitions of the exponential function on time scales the first definition is based on the cayley transformation while the second one is a natural extension of exact discretizations our exponential functions map the imaginary axis into the unit circle therefore it is possible to define hyperbolic and trigonometric functions on time scales in a standard way the resulting functions preserve most of the qualitative properties of the corresponding continuous functions in particular pythagorean trigonometric identities hold exactly on any time scale dynamic equations satisfied by cayley motivated functions have a natural similarity to the corresponding differential equations the exact discretization is less convenient as far as dynamic equations and differentiation are concerned

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

the role of the geometric phases in adiabatic population tracking for non hermitian hamiltonians the definition of instantaneous eigenstate populations for a dynamical non self adjoint system is not obvious the nadirect extension of the definition used for the self adjoint case leads to inconsistencies the resulting artifacts can induce a false inversion of population or a false adiabaticity we show that the inconsistency can be avoided by introducing geometric phases in another possible definition of populations an example is given which demonstrates both the anomalous effects and their removal by our approach

unified approach to hamiltonian systems poisson systems gradient systems and systems with lyapunov functions or first integrals systems with a first integral i e constant of motion or a lyapunov function can be written as linear gradient systems dot x l x nabla v x for an appropriate matrix function l with a generalization to several integrals or lyapunov functions the discrete time analogue delta x delta t l bar nabla v where bar nabla is a discrete gradient preserves v as an integral or lyapunov function respectively

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

a convergence analysis for nonsymmetric lanczos algorithms a convergence analysis for the nonsymmetric lanczos algorithm is presented by using a tridiagonal structure of the algorithm some identities concerning ritz values and ritz vectors are established and used to derive approximation bounds in particular the analysis implies the classical results for the symmetric lanczos algorithm

on the observability and asymmetry of adiabatic state flips generated by exceptional points in open quantum systems where the effective hamiltonian is not hermitian it is known that the adiabatic or instantaneous basis can be multivalued by adiabatically transporting an eigenstate along a closed loop in the parameter space of the hamiltonian it is possible to end up in an eigenstate different from the initial eigenstate this adiabatic flip effect is an outcome of the appearance of a degeneracy known as an exceptional point inside the loop we show that contrary to what is expected of the transport properties of the eigenstate basis the interplay between gain loss and non adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect

error analysis of krylov methods in a nutshell error and residual bounds for the matrix iteration methods bicg qmr fom and gmres are derived in a simple and unified way

improving the accuracy of the discrete gradient method in the one dimensional case we present two numerical schemes of high accuracy for one dimensional dynamical systems they are modifications of the discrete gradient method and keep its advantages including stability and conservation of the energy integral however their accuracy is higher by several orders of magnitude

discrete gradient methods for solving odes numerically while preserving a first integral we show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods using this framework we construct several new integral preserving schemes

slow non hermitian cycling exact solutions and the stokes phenomenon for non hermitian hamiltonians with an isolated degeneracy exceptional point a model for cycling around loops that enclose or exclude the degeneracy is solved exactly in terms of bessel functions floquet solutions returning exactly to their initial states up to a constant are found as well as exact expressions for the adiabatic multipliers when the evolving states are represented as a superposition of eigenstates of the instantaneous hamiltonian adiabatically i e for slow cycles the multipliers of exponentially subdominant eigenstates can vary wildly unlike those driven by hermitian operators which change little these variations are explained as an example of the stokes phenomenon of asymptotics improved superadiabatic approximations tame the variations of the multipliers but do not eliminate them

on semiiterative methods generated by faber polynomials consider a nonsingular system of linear algebraic equationsa x b or in fixed point formx tx c where the eigenvalues oft are contained in some compact subset of the complex plane using faber polynomials iteration methods of the form y m mu m 0 left c ty m 1 right mu m 1 y m 1 ldots mu m k y m k m 0 m j 1 for allm 1 are constructed which include the chebyshev iterative method as well as the stationaryk step methods when is a rectangle we also develop four step methods which yield fast convergence in cases when the speed of convergence of the optimized chebyshev method is poor

the topology of the adiabatic passage process for molecular photodissociative dynamics we study population transfer processes for photodissociation as described within the optical potential model in the context of adiabatic or quasi adiabatic dynamics by a reformulation of the adiabatic passage theory in terms of the fibre bundle topology we extend the analysis made by yatsenko et al 2002 phys rev a 65 043407 from conservative to dissipative systems we show that this topology is associated with the problem of eigenvalue labelling and hence with the continuous eigenvalue following process in the control parameter space problems involving adiabatic passage the direct chirping process and non adiabatic transitions are studied with particular regard to the presence of resonance states we also discuss the role played by the molecular continua

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

systematic study of fields due to extended apertures by gaussian beam discretization gaussian beams are used as basis elements in field representations to gain insight into how the choice of beam parameters affects the final representation a systematic study for the simple test case of a one dimensional linearly phased cosine aperture distribution has been undertaken by successively adding individual displaced and or tilted beams with large narrow or matched waists one can assess how the elements in various portions of the lattice contribute to the build up of the actual field in the aperture near zone and far zone adding enough beams always guarantees homing in on the exact solution as is verified here by independent comparison different beam choices imply different modeling of the radiation process the understanding gained thereby is helpful for selecting beam parameters in subsequent applications where it is necessary to balance requirements of good convergence ease of computation and ability to track the beams through perturbing environments like a radome indications are that the narrow beams provide the most robust and versatile formulation to deal with these generalized conditions

time dependent tilted pulsed beams and their properties novel time dependent wavepacket equation and its pulsed field solutions are obtained by utilizing a non orthogonal coordinate system which is a priori matched to the field s planar linearly delayed pulsed localized aperture distributions these waveobjects that serve as the building blocks for various time dependent beam expansion schemes are termed tilted pulsed beams iso axial pulsed beams are parameterized in term of beam widths waist locations collimation lengths wave front radii of curvature and other features emphasis is placed on a direct time domain derivation a numerical example is presented in which the enhanced accuracy of the tilted pulsed beams over the conventional orthogonal coordinates ones in the well collimated zone is demonstrated

long time behaviour of discretizations of the simple pendulum equation we compare several discretizations of the simple pendulum equation in a series of numerical experiments the stress is put on the long time behaviour the chosen numerical schemes are either symplectic maps or integrable energy preserving maps or both therefore they preserve qualitative features of solutions such as periodicity we describe characteristic periodic time dependences of numerical estimates of the period and the amplitude and explain them as systematic numerical by effects produced by any method finally we propose a new numerical scheme which is a modification of the discrete gradient method this modified discrete gradient method preserves almost exactly the period of small oscillations for any time step