corner detection on digital curves based on local symmetry of the shape abstract the detection of corners on digital curves is a problem of important in shape analysis this paper describes a new method for detecting corners based on local symmetry of the shape the proposed method consists of two steps the first is the computation of symmetry measure for symmetry relative to a line at every point on a digital curve the second is the location of reasonable local maxima of the symmetry measure as corners the effectiveness of the new method is illustrated with the results of experiments for a number of different types of digital curves

point wise error estimate of a conservative difference scheme for the fractional schrodinger equation in this paper a new conservative difference scheme is proposed for solving the nonlinear space fractional schrodinger equation the riesz space fractional derivative is approximated by the second order accurate weighted and shifted grunwald difference operator building on a careful analysis of this operator the conservation properties including the mass and energy are analyzed and some norm equivalences are established then the numerical solution is shown to be bounded in various norms uniformly with respect to the discretization parameters and optimal order bounds on the global error of the scheme in the l 2 norm semi h 2 norm and l norm are derived the uniform bounds on the numerical solutions as well as the error bounds hold unconditionally in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed numerical tests are performed to support our theoretical results

existence of the global solution for fractional logarithmic schrodinger equation abstract in this paper we consider the initial boundary value problem for a class of fractional logarithmic schrodinger equation by using the fractional logarithmic sobolev inequality and introducing a family of potential wells we give some properties of the family of potential wells and obtain existence of global solution

local symmetries of digital contours from their chain codes in this work symmetry is evaluated as a numeric feature for each point of a contour using only the positions of a local vicinity of points a measurement is defined named as local symmetric deficiency lsd so that the lower this quantity is the higher the symmetry will be in the local region considered this approach is very simple and it is based on a suitable manipulation of the chain code of the curve its computational cost is very low and it has the advantages of a parallel algorithm since values for lsd can be computed for each point independently

the global analysis on the spectral collocation method for time fractional schrodinger equation abstract in this paper a spectral collocation method is proposed and analyzed for solving the time fractional schrodinger equation the space derivative is discretized using the collocation method and the time fractional derivative using grunwald letnikov formulation the stability and convergence of the full discretization scheme are analyzed based on the z transform the global behavior of the finite difference spectral collocation method is derived numerical examples show a good agreement with the theoretical analysis

the global existence and time decay for the solutions of the fractional pseudo parabolic equation abstract we consider the cauchy problem of fractional pseudo parabolic equation on the whole space r n n 1 here the fractional order is related to the diffusion type source term behaving as the usual diffusion term on the high frequency part it has a feature of regularity gain and regularity loss for 1 and 0 1 respectively we establish the global existence and time decay rates for small amplitude solutions to the cauchy problem for 0 in the case that 0 1 we introduce the time weighted energy method to overcome the weakly dissipative property of the equation

on the nonlocality of the fractional schrodinger equation a number of papers over the past eight years have claimed to solve the fractional schrodinger equation for systems ranging from the one dimensional infinite square well to the coulomb potential to one dimensional scattering with a rectangular barrier however some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal preventing the fractional schrodinger equation from being solved in the usual piecewise fashion we focus on the one dimensional infinite square well and show that the purported ground state which is based on a piecewise approach is definitely not a solution of the fractional schrodinger equation for the general fractional parameter on a more positive note we present a solution to the fractional schrodinger equation for the one dimensional harmonic oscillator with 1

neumann problem for nonlinear schrodinger equation with the riesz fractional derivative operator abstract we consider the initial boundary value problem for the fractional schrodinger equation posed on positive half line x 0 u t i u x x i u 2 u x 1 2 u 0 t 0 x 0 u x 0 u 0 x x 0 u x 0 t h t t 0 where x 1 2 is the fractional derivative operator defined by the riesz potential x 1 2 1 2 0 sign x y x y u y y d y we study the global existence in time and asymptotics of solutions to the initial boundary value problem

transmission through locally periodic potentials in space fractional quantum mechanics abstract we study transmission through locally periodic potentials in the framework of space fractional quantum mechanics sfqm in particular we calculate the transmission probabilities t n for a particle in a chain of n dirac barriers and for the other case a chain of n square barriers we use a transfer matrix m matrix approach in the context of sfqm to obtain equations for t n for both systems bandlike structures emerge even for a small number of barriers of as low as n 4 and are more apparent for lower values of levy index considered

quasi periodic solutions of a fractional nonlinear schrodinger equation in the present paper it is proved that there are many quasi periodic solutions of a class of space fractional nonlinear schrodinger equations with the riesz fractional derivative by means of kam kolmogorov arnold moser theorem in the present paper it is proved that there are many quasi periodic solutions of a class of space fractional nonlinear schrodinger equations with the riesz fractional derivative by means of kam kolmogorov arnold moser theorem

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

collocation method for fractional quantum mechanics we show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional laplacian using a collocation approach based on little sinc functions which discretizes the schrodinger equation on a uniform grid the different boundary conditions are naturally implemented using sets of functions with the appropriate behavior good convergence properties are observed a comparison with results based on a wentzel kramers brillouin analysis is performed

on the l convergence of a difference scheme for coupled nonlinear schrodinger equations in this article a finite difference scheme for coupled nonlinear schrodinger equations is studied the existence of the difference solution is proved by brouwer fixed point theorem with the aid of the fact that the difference solution satisfies two conservation laws the finite difference solution is proved to be bounded in the discrete l norm then the difference solution is shown to be unique and second order convergent in the discrete l norm finally a convergent iterative algorithm is presented

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

profile decompositions and blowup phenomena of mass critical fractional schrodinger equations abstract we study under the radial symmetry assumption the solutions to the fractional schrodinger equations of critical nonlinearity in r 1 d d 2 with levy index 2 d 2 d 1 2 we first prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with mass critical hartree type nonlinearity

approximate inverse circulant plus diagonal preconditioners for toeplitz plus diagonal matrices we consider the solutions of hermitian positive definite toeplitz plus diagonal systems t d x b where t is a toeplitz matrix and d is diagonal and positive however unlike the case of toeplitz systems no fast direct solvers have been developed for solving them in this paper we employ the preconditioned conjugate gradient method with approximate inverse circulant plus diagonal preconditioners to solving such systems the proposed preconditioner can be constructed and implemented efficiently using fast fourier transforms we show that if the entries of t decay away exponentially from the main diagonals the preconditioned conjugate gradient method applied to the preconditioned system converges very quickly numerical examples including spatial regularization for image deconvolution application are given to illustrate the effectiveness of the proposed preconditioner

analysis of time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise abstract in this paper we establish the existence and uniqueness of the mild solution and the holder continuity for the time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise a full discrete scheme with spectral galerkin method in space and exponential method in time is proposed it is rigorously proved that the numerical scheme admits an unique solution finally we present the optimal error estimates of the proposed scheme based on the regularity analysis

crank nicolson difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative in this paper the crank nicolson cn difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative is studied the existence of this difference solution is proved by the brouwer fixed point theorem the stability and convergence of the cn scheme are discussed in the l 2 norm when the fractional order is two all those results are in accord with the difference scheme developed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also presented

random attractors for the stochastic coupled fractional ginzburg landau equation with additive noise this paper is concerned with the stochastic coupled fractional ginzburg landau equation with additive noise we first transform the stochastic coupled fractional ginzburg landau equation into random equations whose solutions generate a random dynamical system then we prove the existence of random attractor for random dynamical system

differentiation to fractional orders and the fractional telegraph equation using methods of differential and integral calculus we present and discuss the calculation of a fractional green function associated with the one dimensional case of the so called general fractional telegraph equation with one space variable this is a fractional partial differential equation with constant coefficients the equation is solved by means of juxtaposition of transforms i e we introduce the laplace transform in the time variable and the fourier transform in the space variable several particular cases are discussed in terms of the parameters some known results are recovered as a by product of our main result we obtain two new relations involving the two parameter mittag leffler function

continuity of the lyapunov exponents for quasiperiodic cocycles consider the banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a diophantine translation on the circle we prove a precise higher dimensional avalanche principle and use it in an inductive scheme to show that the lyapunov spectrum blocks associated to a gap pattern in the lyapunov spectrum of such a cocycle are locally holder continuous moreover we show that all lyapunov exponents are continuous everywhere in this banach manifold irrespective of any gap pattern in their spectra these results also hold for diophantine translations on higher dimensional tori albeit with a loss in the modulus of continuity of the lyapunov spectrum blocks

time fractional schrodinger equation fox s h functions and the effective potential after introducing the formalism of the general space and time fractional schrodinger equation we concentrate on the time fractional schrodinger equation and present new results via the elegant language of fox s h functions we show that the general time dependent part of the wave function for the separable solutions of the time fractional schrodinger equation is the mittag leffler function with an imaginary argument by two different methods after separating the mittag leffler function into its real and imaginary parts in contrast to existing works we show that the total probability is 1 and decays with time introducing the effective potential approach we also write the mittag leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts in the light of these we reconsider the simple box problem

on partially inexact hss iteration methods for the complex symmetric linear systems in space fractional cnls equations the space fractional coupled nonlinear schrdinger cnls equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix and a symmetric positive definite diagonal plus toeplitz matrix the hermitian and skew hermitian splitting hss method and the partially inexact hss pihss method are employed to solve the discretized linear system in the inner iteration processes of the hss method we only need to solve the linear sub systems associated with the hermitian part inexactly by the conjugate gradient cg method resulting in pihss iteration method theoretical analyses show that both hss and pihss methods are unconditionally convergent numerical examples are given to demonstrate the effectiveness of the hss iteration and the pihss iteration

fractional schrodinger operator with delta potential localized on circle we consider a system governed by the fractional schodinger operator with a delta potential supported by a circle in r2 we find out the function counting the number of bound states in particular we give the necessary and sufficient conditions for the absence of bound state in our system furthermore we reproduce the form of eigenfunctions and analyze the asymptotic behavior of eigenvalues for the strong coupling constant case

a linearly implicit conservative difference scheme for the space fractional coupled nonlinear schrodinger equations abstract in this paper a linearly implicit conservative difference scheme for the coupled nonlinear schrodinger equations with space fractional derivative is proposed this scheme conserves the mass and energy in the discrete level and only needs to solve a linear system at each step the existence and uniqueness of the difference solution are proved the stability and convergence of the scheme are discussed and it is shown to be convergent of order o 2 h 2 in the discrete l 2 norm with the time step and mesh size h when the fractional order is two all those results are in accord with the difference scheme proposed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also reported

parameter estimation for the fractional schrodinger equation using bayesian method in this paper the fractional schrodinger equation is studied the bayesian method is put forward to estimate some relevant parameters of the equation results show that the estimated values can fit well with the exact solution the varying initial values and maximum iterations have little effect on the estimated results it indicates that the bayesian method is efficient for the multi parameter estimation for the fractional schrodinger equation this method can also be used to estimate parameters for the fractional schrodinger equation in other potential field

applications of continuity and discontinuity of a fractional derivative of the wave functions to fractional quantum mechanics the space fractional schrodinger equation with a finite square potential periodic potential and delta function potential is studied in this paper we find that the continuity or discontinuity condition of a fractional derivative of the wave functions should be considered to solve the fractional schrodinger equation in fractional quantum mechanics more parity states than those given by standard quantum mechanics for the finite square potential well are obtained the corresponding energy equations are derived and then solved by graphical methods we show the validity of bloch s theorem and reveal the energy band structure for the periodic potential the jump discontinuity condition for the fractional derivative of the wave function of the delta function potential is given with the help of the jump condition we study some delta function potential fields for the delta function potential well an alternate expression of the wave function the h function form of it was given by dong and xu j math phy

linearized crank nicolson scheme for the nonlinear time space fractional schrodinger equations abstract in this paper a crank nicolson difference scheme is first derived for solving the nonlinear time space fractional schrodinger equations the truncation error and stability of the scheme are discussed in detail the existence of the numerical solution is shown by the brouwer fixed point theorem for improving the calculating efficiency a three level linearized difference scheme is also proposed and analyzed both schemes are subsequently extended to the nonlinear coupled equations and some similar results are given and proved several numerical experiments are included to verify the accuracy and efficiency of the two types of schemes and comparison with the related work is presented

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

green s function for the time dependent scattering problem in the fractional quantum mechanics we study the green s function of the time dependent 3d space fractional schrodinger equation for the scattering problem in the fractional quantum mechanics the green s function is expressed in terms of fox s h function and in a computable series form we get the asymptotic formula of the green s function and apply it to obtain the approximate wave function for the fractional quantum scattering problem

asymptotic behavior of stochastic fractional power dissipative equations on rn abstract we study a reaction fractional diffusion equation with additive noise on the entire space r n with particular interest in the asymptotic behavior of solutions we first transform the equation into a random equation whose solutions generate a random dynamical system a priori estimates for solutions are derived when the nonlinearity satisfies certain growth conditions using estimates for far field values of solutions and a cut off technique asymptotic compactness is proved thus the existence of a random attractor in l 2 r n is established

fox function representation of non debye relaxation processes applying the liouville riemann fractional calculus we derive and solve a fractional operator relaxation equation we demonstrate how the exponent of the asymptotic power law decay t relates to the order of the fractional operatordv dtv 0 1 continuous time random walk ctrw models offer a physical interpretation of fractional order equations and thus we point out a connection between a special type of ctrw and our fractional relaxation model exact analytical solutions of the fractional relaxation equation are obtained in terms of fox functions by using laplace and mellin transforms apart from fractional relaxation fox functions are further used to calculate fourier integrals of kohlrausch williams watts type relaxation functions because of its close connection to integral transforms the rich class of fox functions forms a suitable framework for discussing slow relaxation phenomena

time fractional schrodinger equation the schrodinger equation is considered with the first order time derivative changed to a caputo fractional derivative the time fractional schrodinger equation the resulting hamiltonian is found to be non hermitian and nonlocal in time the resulting wave functions are thus not invariant under time reversal the time fractional schrodinger equation is solved for a free particle and for a potential well probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative new identities for the mittag leffler function are also found and presented in an appendix

dynamics of the 3 d fractional complex ginzburg landau equation abstract we study the initial boundary value problem of the fractional complex ginzburg landau equation in three spatial dimensions with the dissipative effect given by a fractional laplacian a priori estimates are derived when the nonlinearity satisfies certain growth conditions using galerkin s method the existence of a global smooth solution is established uniqueness is also proved furthermore the existence of a global attractor is proved and estimates of the hausdorff and fractal dimensions for the global attractor are obtained

techniques for assessing polygonal approximations of curves given the enormous number of available methods for finding polygonal approximations to curves techniques are required to assess different algorithms some of the standard approaches are shown to be unsuitable if the approximations contain varying numbers of lines instead we suggest assessing an algorithm s results relative to an optimal polygon and describe a measure which combines the relative fidelity and efficiency of a curve segmentation we use this measure to compare the application of 23 algorithms to a curve first used by teh and chin 1989 their integral square errors ises are assessed relative to the optimal ise in addition using an example of pose estimation it is shown how goal directed evaluation can be used to select an appropriate assessment criterion

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

highly accurate numerical schemes for multi dimensional space variable order fractional schrdinger equations as a natural generalization of the fractional schrdinger equation the variable order fractional schrdinger equation has been exploited to study fractional quantum phenomena in this paper we develop an exponentially accurate jacobigausslobatto collocation jgl c method to solve the variable order fractional schrdinger equations in one dimension 1d and two dimensions 2d in this method the aforementioned problem is reduced to a system of ordinary differential equations odes in the time variable as a result we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations one based on the implicit rungekutta irk method of fourth order and the other based on jacobigaussradau collocation jgr c method the validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1d and 2d the convergence of the methods is graphically analyzed the results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable order nonlinear partial differential equations

on the cost of fast controls for some families of dispersive or parabolic equations in one space dimension in this paper we consider the cost of fast controls for a large class of linear equations of parabolic or dispersive type in one space dimension in small time by extending the work of tenenbaum and tucsnak j differential equations 243 2007 pp 70 100 we are able to give precise upper bounds on the time dependance of the cost of fast controls when the time of control t tends to 0 we also give a lower bound of the cost of fast controls for the same class of equations which proves the optimality of the power of t involved in the cost of the control these general results are then applied to treat notably the case of linear kdv equations and fractional heat or schrodinger equations

levy path integrals of particle on circle and some applications fractional quantum mechanics in the non relativistic case is described by the space fractional schrodinger equation containing the fractional riesz operator which is a non local operator the non locality makes it difficult to solve the fractional schrodinger equation in the local potentials in this paper we study the solution of the fractional schrodinger equation of a free particle moving on a circle by use of the levy path integrals approach we present the levy path integrals propagator of a particle on circle and then use it to get the wave functions and energy eigenvalues of the free particle case in addition the laplace transform energy time transform and momentum representation of the free particle kernel are also obtained the results of this paper contain the ones in standard quantum mechanics as special cases the problem of a particle on circle is a fundamental problem of path integrals with topological constraints in quantum systems and the method of levy path integrals can be generalized to study more complex local potentials

the fractional schrodinger equation with hardy type potentials and sign changing nonlinearities abstract we look for solutions to a fractional schrodinger equation of the following form 2 u v x x u f x u k x u q 2 u o n r n 0 where v is bounded and close to periodic potential and x is a hardy type potential we assume that v is positive and f has the subcritical growth but not higher than u q 2 u if is positive and small enough we find a ground state solution i e a critical point of the energy being minimizer on the nehari manifold if is negative we show that there is no ground state solutions we are also interested in an asymptotic behaviour of solutions as 0 and k 0

the fractional schrodinger equation for delta potentials the fractional schrodinger equation is solved for the delta potential and the double delta potential for all energies the solutions are given in terms of fox s h function

the avalanche principle and negative curvature we use the geometric structure of the hyperbolic upper half plane to provide a new proof of the avalanche principle introduced by m goldstein and w schlag in the context of mathrm sl 2 mathbb r matrices this approach allows to interpret and extend this result to arbitrary text cat 1 metric spaces through the proof we deduce a polygonal schur theorem for these spaces

a fast linearized conservative finite element method for the strongly coupled nonlinear fractional schrodinger equations abstract in this paper a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional schrodinger equations we prove that the scheme preserves both the mass and energy which are defined by virtue of some recursion relationships using the sobolev inequalities and then employing the mathematical induction the discrete scheme is proved to be unconditionally convergent in the sense of l 2 norm and h 2 norm which means that there are no any constraints on the grid ratios then the prior bound of the discrete solution in l 2 norm and l norm are also obtained moreover we propose an iterative algorithm by which the coefficient matrix is independent of the time level and thus it leads to toeplitz like linear systems that can be efficiently solved by krylov subspace solvers with circulant preconditioners this method can reduce the memory requirement of the proposed linearized finite element scheme from o m 2 to o m and the computational complexity from o m 3 to o m log m in each iterative step where m is the number of grid nodes finally numerical results are carried out to verify the correction of the theoretical analysis simulate the collision of two solitary waves and show the utility of the fast numerical solution techniques

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

remarks on the inhomogeneous fractional nonlinear schrodinger equation using a sharp gagliardo nirenberg type inequality well posedness issues of the initial value problem for a fractional inhomogeneous schrodinger equation are investigated

remarks on damped fractional schrodinger equation with pure power nonlinearity we investigate the initial value problem for a semi linear fractional damped schrodinger equation global existence and scattering are proved depending on the size of the damping coefficient

bound states for multiple dirac δ wells in space fractional quantum mechanics using the momentum space approach we obtain bound states for multiple dirac wells in the framework of space fractional quantum mechanics introducing first an attractive dirac comb potential i e dirac comb with strength g g 0 in the space fractional schrodinger equation we show that the problem of obtaining eigenenergies of a system with n dirac wells can be reduced to a problem of obtaining the eigenvalues of an n n matrix as an illustration we use the present matrix formulation to derive expressions satisfied by the bound state energies of n 1 2 3 delta wells we also obtain the corresponding wave functions and express them in terms of fox s h function

an analysis of implicit conservative difference solver for fractional klein gordon zakharov system abstract in this paper we propose an efficient linearly implicit conservative difference solver for the fractional klein gordon zakharov system first of all we present a detailed derivation of the energy conservation property of the system in the discrete setting then by using the mathematical induction it is proved that the proposed scheme is uniquely solvable subsequently by virtue of the discrete energy method and a cut off function technique it is shown that the proposed solver possesses the convergence rates of o t 2 h 2 in the sense of l and l2 norms respectively and is unconditionally stable finally numerical results testify the effectiveness of the proposed scheme and exhibit the correctness of theoretical results

existence and stability of standing waves for nonlinear fractional schrodinger equations with hartree type nonlinearity abstract in this paper we consider the nonlinear fractional schrodinger equations with hartree type nonlinearity we obtain the existence of standing waves by studying the related constrained minimization problems via applying the concentration compactness principle by symmetric decreasing rearrangements we also show that the standing waves up to translations and phases are positive symmetric nonincreasing functions moreover we prove that the set of minimizers is a stable set for the initial value problem of the equations that is a solution whose initial data is near the set will remain near it for all time

maximum norm error analysis of a difference scheme for the space fractional cnls the difference method for the space fractional coupled nonlinear schrodinger equations cnls is studied the fractional centered difference is used to approximate the space fractional laplacian this scheme conserves the discrete mass and energy due to the nonlocal nature of fractional laplacian in the classic sobolev space it is hard to obtain the error estimation in l to overcome this difficulty the fractional sobolev space h 2 and a fractional norm equivalence in h 2 are introduced then the convergence of order o h 2 2 in l is proved by fractional sobolev inequality where h is the mesh size and is the time step numerical examples are given to illustrate the theoretical results at last

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

the global solution for a class of systems of fractional nonlinear schrodinger equations with periodic boundary condition in this paper we consider a class of systems of fractional nonlinear schrodinger equations we prove the existence and uniqueness of the global solution to the periodic boundary value problem by using the faedo galerkin method

on hss like iteration method for the space fractional coupled nonlinear schrodinger equations the implicit conservative difference scheme with the fractional centered difference formula which is unconditionally stable is employed to discretize the space fractional coupled nonlinear schrodinger equations the coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric diagonal plus toeplitz matrix in this paper the hss like iteration method is proposed to solve the discretized linear system theoretical analyses show that the hss like iteration method is unconditionally convergent numerical examples are presented to illustrate the effectiveness of the hss like iteration method

existence and stability of standing waves for nonlinear fractional schrodinger equation with logarithmic nonlinearity abstract in this paper we consider the nonlinear fractional logarithmic schrodinger equation by using a compactness method we construct a unique global solution of the associated cauchy problem in a suitable functional framework we also prove the existence of ground states as minimizers of the action on the nehari manifold finally we prove that the set of minimizers is a stable set for the initial value problem that is a solution whose initial data is near the set will remain near it for all time

explicit lower bounds for the cost of fast controls for some 1 d parabolic or dispersive equations and a new lower bound concerning the uniform controllability of the 1 d transport diffusion equation in this paper we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional laplace operator we notably deduce the following striking result in the case of the heat equation controlled on the boundary the miller s conjecture formulated in geometric bounds on the growth rate of null controllability cost for the heat equation in small time j differential equations 204 2004 pp 202 226 is not verified moreover we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one dimensional convection diffusion equation with negative speed controlled on the left boundary proving that the conjecture formulated in j m coron and s guerrero singular optimal control a linear 1 d parabolic hyperbolic example asymptot anal 44 2005 pp 237 257 concerning this problem is also not verified at least for negative speeds the proof is based on complex analysis and more precisely on a representation formula for entire functions of exponential type and is quite related to the moment method of fattorini and russell

fourier pseudospectral method on generalized sparse grids for the space fractional schrodinger equation abstract in this paper the time splitting fourier pseudospectral method on the generalized sparse grids is applied to solve the space fractional schrodinger equation we give a containment relation between different level index sets of the generalized sparse grids and it can be used in designing the reference generalized sparse grids which are finer than other considered grids thus the numerical solution on the reference generalized sparse grids can be used as the reference true solution of the equation then the fully discrete algorithm is obtained in the numerical experiments we compare the numerical results on the generalized sparse grids with those on the full grids for the interpolation of the gaussian multiplied by a factor and for the computation of the schrodinger equation with two kinds of non smooth potentials the advantages of the fourier pseudospectral method on the generalized sparse grids with the level index set of parameter k 1 2 3 are manifest in the approximation with high resolution here the sparsity of the generalized sparse grids will become weak when the parameter k becomes large moreover the advantage of the generalized sparse grids is more pronounced in solving the schrodinger equation with the higher dimension the square well potential or the fractional laplacian

split step alternating direction implicit difference scheme for the fractional schrodinger equation in two dimensions in this paper we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space fractional schrodinger equation with the riesz fractional derivative the scheme is constructed by combining the split step method for handling the nonlinearity with the alternating direction implicit adi method for resolving the multi dimensions difficulty the riesz space fractional derivative is approximated by the second order accurate fractional centered difference based on matrix analysis we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems the unconditional convergence is proved rigorously in the linear case numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme

existence and stability of standing waves for nonlinear fractional schrodinger equations in this paper we consider the nonlinear fractional schrodinger equations by studying constrained minimization problems and applying the method of concentration compactness and commutator estimates we obtain the existence of standing waves for nonlinear fractional schrodinger equations under some assumptions moreover we prove that the set of minimizers is a stable set for the initial value problem of the equations in the sense that a solution which starts near the set will remain near it for all time

a circulant preconditioner for fractional diffusion equations the implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable is employed to discretize fractional diffusion equations the resulting systems are toeplitz like and then the fast fourier transform can be used to reduce the computational cost of the matrix vector multiplication the preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant hence the convergence rate of the proposed iterative algorithm is superlinear numerical experiments are carried out to demonstrate that our circulant preconditioner works very well even though for cases of variable diffusion coefficients

existence of the global smooth solution to the period boundary value problem of fractional nonlinear schrodinger equation in this paper we prove the existence and uniqueness of the global smooth solution to the period boundary value problem of fractional nonlinear schrodinger equation by using energy method

curvature and tangential deflection of discrete arcs a theory based on the commutator of scatter matrix pairs and its application to vertex detection in planar shape data this paper introduces a new theory for the tangential deflection and curvature of plane discrete curves our theory applies to discrete data in either rectangular boundary coordinate or chain coded formats its rationale is drawn from the statistical and geometric properties associated with the eigenvalue eigenvector structure of sample covariance matrices specifically we prove that the nonzero entry of the commutator of a piar of scatter matrices constructed from discrete arcs is related to the angle between their eigenspaces and further we show that this entry is in certain limiting cases also proportional to the analytical curvature of the plane curve from which the discrete data are drawn these results lend a sound theoretical basis to the notions of discrete curvature and tangential deflection and moreover they provide a means for computationally efficient implementation of algorithms which use these ideas in various image processing contexts as a concrete example we develop the commutator vertex detection cvd algorithm which identifies the location of vertices in shape data based on excessive cummulative tangential deflection and we compare its performance to several well established corner detectors that utilize the alternative strategy of finding approximate curvature extrema

generalized fractional schrodinger equation with space time fractional derivatives in this paper the generalized fractional schrodinger equation with space and time fractional derivatives is constructed the equation is solved for free particle and for a square potential well by the method of integral transforms fourier transform and laplace transform and the solution can be expressed in terms of mittag leffler function the green function for free particle is also presented in this paper finally we discuss the relationship between the cases of the generalized fractional schrodinger equation and the ones in standard quantum

some physical applications of fractional schrodinger equation the fractional schrodinger equation is solved for a free particle and for an infinite square potential well the fundamental solution of the cauchy problem for a free particle the energy levels and the normalized wave functions of a particle in a potential well are obtained in the barrier penetration problem the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined in the quantum scattering problem according to the fractional schrodinger equation the green s function of the lippmann schwinger integral equation is given

existence and upper semicontinuity of random attractors for non autonomous fractional stochastic ginzburg landau equations this paper deals with the dynamical behavior of solutions for nonautonomous stochastic fractional ginzburg landau equations driven by multiplicative noise with 0 1 we first transform the stochastic equation into a random equation the solutions of which generate a random dynamical system consequently we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain finally we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero

bound state for the fractional schrodinger equation with unbounded potential this paper considers the fractional schrodinger equation with unbounded potential we prove the existence of ground state by lagrange multiplier method the standing wave with prescribed frequency is also obtained by nehari s manifold approach in the last part we obtain that the standing wave is bound state

analysis of the damped nonlinear space fractional schrodinger equation abstract in this paper we verify the unique existence of the global smooth solution of the damped nonlinear space fractional schrodinger dnfs equation and show it follows a conformal mass conservation law we propose a conformal mass preserving linearized scheme it is rigorously proved that this scheme preserves the discrete conformal mass furthermore we prove that the proposed scheme admits a unique solution and is of second order convergence in space and first order convergence in time some numerical experiments are carried out to validate the theoretical analysis

continuity of the lyapunov exponent for quasiperiodic operators with analytic potential we study regularity properties of the lyapunov exponent l of one frequency quasiperiodic operators with analytic potential under no assumptions on the diophantine class of the frequency we prove joint continuity of l in frequency and energy at every irrational frequency

dynamical behavior of non autonomous fractional stochastic reaction diffusion equations abstract in this paper we investigate the asymptotic behavior of solutions for the non autonomous fractional stochastic reaction diffusion equations driven by multiplicative noise with s 0 1 we prove the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain u by the compactness of sobolev embedding h s u l 2 u which is different from the previous work 46

on the blow up solutions for the nonlinear fractional schrodinger equation abstract this paper is dedicated to the blow up solutions for the nonlinear fractional schrodinger equation arising from pseudorelativistic boson stars first we compute the best constant of a gg n inequality by the profile decomposition theory and variational arguments then we find the sharp threshold mass of the existence of finite time blow up solutions finally we study the dynamical properties of finite time blow up solutions around the sharp threshold mass by giving a refined compactness lemma

preconditioned modified hermitian and skew hermitian splitting iteration methods for fractional nonlinear schrodinger equations abstract a variant of preconditioned modified hermitian and skew hermitian splitting pmhss iteration method is proposed for a class of toeplitz like complex linear equations arising from the space fractional coupled nonlinear schrodinger equations theoretical analysis show that the new method is convergent unconditionally moreover the choice of the theoretical optimal parameter is further studied in detail which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the toeplitz matrix and the extreme absolute values of the diagonal matrix in addition the splitting can be used as a preconditioner for krylov subspaces method numerical experiments show that the proposed method is efficient and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem

a new algorithm for dominant points detection and polygonization of digital curves a new algorithm for detecting dominant points and polygonal approximation of digitized closed curves is presented it uses an optimal criterion for determining the region of support of each boundary point and a new mechanism for selecting the dominant points the algorithm does not require an input parameter and can handle shapes that contain features of multiple sizes efficiently in addition the approximating polygon preserves the symmetry of the shape