geometrical phase in the cyclic evolution of non hermitian systems the authors derive by a biorthonormal state approach the analogy of berry s phase factor for open non conservative systems for both adiabatic and non adiabatic evolution in the latter case a non unitary evolution operator method is exploited an application is given to the optical supermode propagation in the free electron laser

berry s phase and the planar three body problem berry s angular 2 form in seen as a correction to the symplectic structure in a separation of variables type scheme where the variables are canonically non separable this view is applied to the planar three body problem where the rotational and vibrational motions are not separable it is shown that the corrected symplectic structure gives the correct quantisation

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

extended numerov method for computing eigenvalues of specific schrodinger equations numerov s method and an extended version of it are introduced for computing eigenvalues of schrodinger equations with potentials v x which are even functions with respect to x furthermore it is assumed that the wavefunctions tend to zero for x to or infinity the derived results are compared with previously derived numerical data and with available exact values

a spectral solution of the sturm liouville equation comparison of classical and nonclassical basis sets abstract this paper considers a spectral method of solution of the sturm liouville equation and the associated schroedinger equation the main objective is to develop a collocation method based on quadrature collocation points generated from nonclassical polynomials the polynomials and associated quadrature points are calculated with gautschi s stieltjes procedure from some specified weight function the particular spectral method used here with nonclassical basis sets is referred to as the quadrature discretization method qdm the qdm and a related weighted qdm are applied to several sturm liouville and schroedinger equations and the results are compared with the traditional spectral methods based on chebyshev and legendre quadrature points the results are also compared with the results of other workers wherever available the qdm was found to give the most rapid convergence relative to other methods for the problems studied

generalised mersenne numbers revisited generalised mersenne numbers gmns were defined by solinas in 1999 and feature in the nist fips 186 2 and secg standards for use in elliptic curve cryptography their form is such that modular reduction is extremely efficient thus making them an attractive choice for modular multiplication implementation however the issue of residue multiplication efficiency seems to have been overlooked asymptotically using a cyclic rather than a linear convolution residue multiplication modulo a mersenne number is twice as fast as integer multiplication this property does not hold for prime gmns unless they are of mersenne s form in this work we exploit an alternative generalisation of mersenne numbers for which an analogue of the above property and hence the same efficiency ratio holds even at bitlengths for which schoolbook multiplication is optimal while also maintaining very efficient reduction moreover our proposed primes are abundant at any bitlength whereas gmns are extremely rare our multiplication and reduction algorithms can also be easily parallelised making our arithmetic particularly suitable for hardware implementation furthermore the field representation we propose also naturally protects against side channel attacks including timing attacks simple power analysis and differential power analysis which is essential in many cryptographic scenarios in constrast to gmns

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

a gauge theory for a quantum system with isospin a quantum system with external symmetry is made into a principal fibre bundle with a nonabelian lie group as the structure group and is equipped with a connection a gauge theory for a nonrigid molecule on the basis of the observation that the vector bundle associated with the principal fibre bundle provides a setting for quantum mechanics of the internal molecular motion

weights of the exponential fitting multistep algorithms for first order odes abstract we describe a numerical method for the calculation of the weights of the linear multistep algorithms for solving first order differential equations the main novelties are that i we admit nonequidistant mesh points in the partition and ii the weights are determined on the basis of the exponential functions exp i x i 1 2 3 rather than on the power function set as it is done for the classical weights in this way the method allows computing not only the weights of the well established algorithms but also those of new ones another novelty consists in the construction of a general scheme for the error analysis of this kind of algorithms some relevant numerical illustrations are given

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

on the quantization of the modified poschl teller potential it is shown that the connection one form on the prequantization line bundle that generates the correct energy spectrum for the modified poschl teller potential consists of the canonical one form on the corresponding cotangent bundle together with an adjustment term that arises from the berry s connection

a sixth order exponentially fitted method for the numerical solution of the radial abstract a new sixth order method of runge kutta type is developed for the numerical integration of the single channel radial schrodinger equation the formula derived contains certain free parameters which allows it to be fitted automatically to exponential functions extensive numerical testing on the resonance problem and on the bound states problem indicates that this new method is generally more efficient than other previously developed finite difference methods

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

classical non adiabatic angles if a family of tori in phase space is driven by a time dependent hamiltonian flow in such a way as to return after some time to the original family there generally results a shift in the angle variables one realisation of this process is in the cyclic adiabatic change of a classical hamiltonian and the angle change has previously been shown to separate naturally into a dynamical part and a geometrical part here the same geometrical angle change is extracted when the return is achieved non adiabatically and the dynamical remainder calculated two examples are given the precession of a spin and the rotation of phase space ellipses

numerov method maximally adapted to the schro dinger equation abstract we construct and investigate the two step scheme which is maximally specialized for the schrodinger equation the new scheme is found to preserve all attractive features of the standard numerov method ease in programming flexibility in application speed in run etc with the important advantage of being more efficient for problems involving high values of energy its efficiency is actually found to be close to that of the piecewise perturbation methods a fact which strongly recommends it for further applications

a new variable step method for the numerical integration of the one dimensional schro dinger equation the numerical integration of the one dimensional schrodinger equation is considered a new variable step method is presented within the canonical functions scheme characterized namely by i the replacement of the schrodinger equation by the equivalent integral equation ii the representation of the wavefunction by a superposition of simple integrals iii the definition of taylor series like difference equations at each interval an estimate of the step size h is determined in order to obtain the desired local accuracy for the practical case of a given potential function with polynomial interpolations the step size h is directly given by a simple formula depending on i the potential function in the interval ii the initial values of the wavefunction at the origin of the interval this variable step method is applied to the morse potential function used for the recent raptis and cash method comput phys commun 36 113 1985 for this application the present method presents some advantages over the previous one

explicit numerov type methods with reduced number of stages we present in this paper a new approach for the derivation of hybrid explicit numerov type methods the new methodology does not require the intermediate use of high accuracy inter polatory nodes since we only need the taylor expansion of the internal points as a consequence a sixth order method is produced at a cost of only four stages per step instead of six stages needed for the methods which have appeared in the literature until now numerical results over some well known problems in physics and mechanics indicate the superiority of the new method 2003 elsevier science ltd all rights reserved

on the topological phases in a planar n point vortex system a topological phase is given for the planar n point vortex problem this phase arises from an element in the pure braid group and is independent of the evolution equations it is deduced from a regular representation of the group of diffeomorphisms on the plane this resulting representation of the pure braid group is the monodromy of the hypergeometric functions in n variables the relations between this topological phase and the usual geometrical phase are discussed

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

an exponentially fitted eighth order method for the numerical solution of the schrodinger equation an eighth algebraic order exponentially fitted method is developed for the numerical integration of the schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions an comparative error analysis is also given numerical and theoretical results indicate that the new method is much more efficient than other classical and exponentially fitted methods

repunit r49081 is a probable prime the repunit r49081 1049081 1 9 is a probable prime in order to prove primality r49080 must be approximately 33 3 factored the status of this factorization is included

on computing eigenvalues of the schrodinger equation for symmetrical potentials the problem of the numerical determination of eigenvalues of the one dimensional schrodinger equation with symmetrical potential v x is considered the problem is reduced to the definition of an eigenvalue function g e well defined for the given potential v it is theoretically proved that the even parity eigenvalues are given by g e 0 and the odd parity eigenvalues are given by g e infinity the method is canonical in the sense that it is independent of the eigenfunction yet along with the eigenvalue it allows the determination of the eigenfunction initial values this method is applied to the potentials v x2 v x6 bx2 v x2 lambda x2 1 gx2 where exact eigenvalues are available the numerical results compared with the exact ones and with those of previous confirmed methods show that the present method is accurate and efficient

a four step method for the numerical solution of the schro dinger equation abstract a new four step exponentially fitted method is developed in this paper the expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue schrodinger equation i e equivalent to positive energy

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention