analytic smoothing effect for a system of schrodinger equations with three wave interaction we consider the global cauchy problem for a system of schrodinger equations with quadratic interaction two types of analytic smoothing effect for the solutions are formulated in the small data setting under the mass resonance condition one is the usual analytic smoothing effect in space variables in terms of the generator of galilei transforms we prove the existence and uniqueness of global solutions which are analytic with respect to galilei generators for sufficiently small data with exponential decay at infinity in space n with n 3 the other is analytic smoothing effect in space time variables in terms of generator of pseudo conformal and galilei transforms we prove the existence and uniqueness of global solutions which are analytic with respect to pseudo conformal and galilei generators for sufficiently small data with exponential decay in 4 we also discuss the associated lagrange structure

a two level atom coupled to a two dimensional supersymmetric and shape invariant system models a class of bound state problems which represents the coupling of a two level atom with a two dimensional supersymmetric system involving two shape invariant potentials is introduced we study two models with different coupling hamiltonians

on unscented kalman filtering for state estimation of continuous time nonlinear systems this paper considers the application of the unscented kalman filter ukf to continuous time filtering problems where both the state and measurement processes are modeled as stochastic differential equations the mean and covariance differential equations which result in the continuous time limit of the ukf are derived the continuous discrete ukf is derived as a special case of the continuous time filter when the continuous time prediction equations are combined with the update step of the discrete time ukf the filter equations are also transformed into sigma point differential equations which can be interpreted as matrix square root versions of the filter equations

unscented kalman filter aspects and adaptive setting of scaling parameter this technical note deals with the unscented kalman filter for state estimation of nonlinear stochastic dynamic systems with a special focus on the scaling parameter of the filter its standard choice is analyzed and its impact on the estimation quality is discussed on the basis of the analysis a novel method for adaptive setting of the parameter in the unscented kalman filter is proposed the results are illustrated in a numerical example

three level coupled systems and parasupersymmetric shape invariance a class of bound state problems which represents the coupling of a three level atom with a two dimensional system involving two shape invariant potentials is introduced we consider second order parasupersymmetric quantum mechanical models and using an algebraic formulation for shape invariant potential systems resolve the eigenvalue problem for these coupled systems considering two possible kinds for the coupling hamiltonian linear and nonlinear in the potential ladder operators an application is given for a couple of shape invariant potentials harmonic oscillator morse potentials

gaussian filters for nonlinear filtering problems we develop and analyze real time and accurate filters for nonlinear filtering problems based on the gaussian distributions we present the systematic formulation of gaussian filters and develop efficient and accurate numerical integration of the optimal filter we also discuss the mixed gaussian filters in which the conditional probability density is approximated by the sum of gaussian distributions a new update rule of weights for gaussian sum filters is proposed our numerical tests demonstrate that new filters significantly improve the extended kalman filter with no additional cost and the new gaussian sum filter has a nearly optimal performance

quantum fluid dynamics description of a multiterm potential nonlinear schrodinger langevin equation via quantum fluid dynamics the author presents a general method to obtain a solution of the nonlinear schrodinger langevin equation with a multiterm potential for the description of interactions in non conservative systems the nonlinear complete set of equations forming the basis of the model are derived the corresponding solutions are exhibited in terms of auxiliary ordinary differential equations which provide relations between the potential coefficients and other coefficients prescribing the fluid particle behaviour

isospectral problem interplay between liouville equations darboux transforms and mckean trubowitz flows systems of liouville equations are generated by isospectral transforms of the schrodinger operator and describe the isospectral evolution of both a potential and eigenfunctions relations of these equations with the mckean trubowitz isospectral flows are studied the role of these flows among other isospectral flows is discussed

exact solutions for anharmonic oscillators completes the author s investigation see phys lett a vol 81 p 17 8 1981 and presents exact solutions and eigenvalues for anharmonic oscillators the solutions being given in the form of definite integrals the conditions for the validity of these results are investigated

potentials of the heun class we review different methods of generating potentials such that the one dimensional schrodinger equation odse can be transformed into the hypergeometric equation we compare our results with previous studies and complement the subject with new findings our main result is to derive new classes of potentials such that the odse can be transformed into the heun equation and its confluent cases the generalized heun equation is also considered

gain of analyticity for semilinear schrodinger equations abstract we discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear schrodinger equations with gauge invariant nonlinearity we prove that if the initial data decays exponentially then the solution becomes real analytic in the space variable and a gevrey function of order 2 in the time variable except in the initial plane our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm

comment on a new method for the nonlinear transformation of means and covariances in filters and estimators with authors reply the above paper julier et al ieee trans automat contr vol 45 pp 477 82 2000 generalizes the kalman filter to nonlinear systems by transforming approximations of the probability distributions through the nonlinear process and measurement functions this comment derives exactly the same estimator by linearizing the process and measurement functions by a statistical linear regression through some regression points in contrast with the extended kalman filter which uses an analytic linearization in one point this insight allows one 1 to understand predict the performance of the estimator for specific applications and 2 to make adaptations to the estimator i e the choice of the regression points and their weights in those cases where the original formulation does not assure good results in reply the authors state that the commenters conclusion is unnecessarily narrow interpretation of results

ladder operators for subtle hidden shape invariant potentials ladder operators can be constructed for all potentials that present the integrability condition known as shape invariance satisfied by most of the exactly solvable potentials using the superalgebra of supersymmetric quantum mechanics we construct the ladder operators for two exactly solvable potentials that present a subtle hidden shape invariance

coherent states for isospectral oscillator hamiltonians coherent states for a family of isospectral oscillator hamiltonians are derived from a suitable choice of annihilation and creation operators the fock bargmann representation is also considered

extended supersymmetry for the bound states of the generalized hulthen potential hierarchy using the associated hypergeometric differential equation we analytically solve the bound states corresponding to a hierarchy of the radial potential v0 e r 1 e r c e r 1 e r 2 as a generalization of the hulth n potential then an analytic solution corresponding to a special case for which the parameter c is expected to be in terms of l l 1 is also derived meanwhile without introducing a superpotential and in the framework of supersymmetric quantum mechanics it is shown that these bound states can be calculated by two different algebraic methods based on these two approaches it is noted that the bound states realize an extended supersymmetry structure

on the αx2 βx4 interaction by applying the laplace transform method the authors obtain a class of exact solutions for the schrodinger equation with the potential v x alpha x2 beta x4 x 0 alpha or 0 beta 0 the corresponding energy spectrum being continuous a particular feature of these eigen solutions is that they may not be accessible to perturbation theory the properties of the solutions obtained are investigated and their relation to the case where x or 0 is included is discussed

improved cubature kalman filter for gnss ins based on transformation of posterior sigma points error tightly coupled gnss ins has been widely approved as a promising substitute for standalone gnss in urban areas navigation however due to the frequent gnss signal outages the filter used in gnss ins should be insensitive to the less informative observations in this paper a novel sigma points update method is proposed to enhance the robustness of cubature kalman filter ckf under the circumstance of unavailable observations first the problems of existing sampling based filters are analyzed then by transforming the posterior sigma points error matrix from prediction phase of filtering to the posterior domain of update the updated sigma points are expected to capture the covariance more precisely than traditional sigma points finally an improved ckf ickf is developed by embedding these points into the bayesian estimation framework and the upper bounds of error covariance matrices are analyzed theoretically signal outages with different durations are simulated and results demonstrate that ickf outperforms state of the art methods

algebraic shape invariant potentials as the generalized deformed oscillator within the framework of supersymmetric quantum mechanics we study the simplified version of the potential algebra of the shape invariance condition in k steps where k is an arbitrary positive integer the associated potential algebra is found to be equivalent to the generalized deformed oscillator algebra that has a built in zk grading structure the algebraic realization of the shape invariance condition in k steps is therefore formulated by the method of the zk graded deformed oscillator based on this formulation we explicitly construct the general algebraic properties for shape invariant potentials in k steps in which the parameters of partner potentials are related to each other by the translation a1 a0 the obtained results include the cyclic shape invariant potentials of period k as a special case

remarks on analytic smoothing effect for the schrodinger equation we study analytic smoothing effect of solutions to the schrodinger equation with cauchy data decaying exponentially at infinity the domain of analyticity in the space variables of solutions is described under weight conditions on the data in terms of the corresponding supporting function the domain of analyticity in the time variable is characterized by means of weight conditions of gaussian type on the data a generalization of various isometrical identities related to the analytic smoothing effect is introduced

solvable potentials of shape invariance in two steps within the framework of supersymmetric quantum mechanics we obtain a class of solvable potentials of shape invariance in two steps where the parameters a1 and a2 of partner potentials are related to each other by translation a2 a1 it is found that discontinuity at some x points is a characteristic of the two step superpotentials therefore giving rise to dirac delta function singularities to the corresponding potentials

an algebraic q deformed form for shape invariant systems a quantum deformed theory applicable to all shape invariant bound state systems is introduced by defining q deformed ladder operators we show that these new ladder operators satisfy new q deformed commutation relations in this context we construct an alternative q deformed model that preserves the shape invariance property presented by the primary system q deformed generalizations of morse scarf and coulomb potentials are given as examples

analyticity in time and smoothing effect of solutions to nonlinear schrodinger equations in this paper we consider analyticity in time and smoothing effect of solutions to nonlinear schrodinger equations where we prove that if satisfies then there exists a unique solution of 1 and positive constants t c 0 c 1 such that is analytic in time and space variables for and and has an analytic continuation on and in the case the condition 2 can be relaxed as follows where m 0 if n 1 p 1 m 1 if n 2 and m 1 if n 3 p 1

supersymmetry factorisation of the schrodinger equation and a hamiltonian hierarchy the author presents a systematic procedure for constructing a hierarchy of non relativistic hamiltonians with the property that the adjacent members of the hierarchy are supersymmetric partners i e they share the same eigenvalue spectrum except for the missing ground state and the eigenvectors are simply related

analytic smoothing effect for nonlinear schrodinger equation with quintic nonlinearity abstract we prove the global existence of analytic solutions to nonlinear schrodinger equation with quintic nonlinearity in n space dimensions for sufficiently small cauchy data with exponential decay the smallness assumption on the data is imposed in terms of the critical sobolev space h n 2 1 2 moreover a characterization of some class of analytic functions is given

a note on parasupersymmetric quantum mechanics of arbitrary order we revisit parasupersymmetric quantum mechanics of arbitrary order and present a set of nontrivial relations which characterizes the most general multilinear part of the associated parasupersymmetric algebra we then show that the formulation of multilinear relations leads immediately to a polynomial of parasupersymmetric hamiltonian in terms of the corresponding parasupercharges the deduction of higher derivative supersymmetric quantum mechanics directly via this parasupersymmetric formulation is discussed the complete degenerate structure of the energy spectrum for parasupersymmetric quantum mechanics of order p is systematically analyzed finally the notion of cyclic symmetry is introduced and the algebra of cyclic charge operators of arbitrary order is developed based on the parasupersymmetric formalism

kalman filtering for a generalized class of nonlinear systems and a new gaussian quadrature technique the class of nonlinear systems treated in this technical note consists of the discrete time nonlinear systems that are formed by the interconnection of linear systems through static nonlinearities with few inputs this special structure is exploited to reduce the dimension of the integrals involved in the propagation of mean values and covariances thus permitting accurate calculations furthermore a new quadrature scheme suitable for nonlinear kalman filtering is introduced the proposed techniques are applied to a seven dimensional numerical example the results show that they can increase the performance significantly

computationally efficient kalman filtering for a class of nonlinear systems this paper deals with recursive state estimation for the class of discrete time nonlinear systems whose nonlinearity consists of one or more static nonlinear one variable functions this class contains several important subclasses the special structure is exploited to permit accurate computations without an increase in computational cost the proposed method is compared with standard extended kalman filter unscented kalman filter and gauss hermite kalman filter in three illustrative examples the results show that it yields good results with small computational cost

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

analyticity of solutions to a quadratic system of schrodinger equations without mass resonance condition abstract we consider the cauchy problem for a quadratic system of schrodinger equations in space dimensions n 4 without mass resonance condition analytic smoothing effect for the quadratic system of schrodinger equations under the mass resonance condition for data which satisfy exponentially decaying condition has been studied in 14 in this study we construct global solutions in the analytic hardy space via the phase modulation operator based on the scale critical sobolev space h n 2 2 with data like the test functions of fourier hyperfunctions under the condition m 1 m 2 1 4 n m 1 between two masses m 1 and m 2

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

an algebraic construction of generalized coherent states associated with q deformed models for primary shape invariant systems generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics we show that this generalized formalism is able a to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system b to reproduce as particular cases results already known for shape invariant systems such as standard harmonic oscillator and p schl teller potentials as well as quantum deformed harmonic oscillator models and c point to a formalism that provides a unified description of the different kind of coherent states for quantum systems deformed or not deformed

analytic wave front set for solutions to schrodinger equations this paper is a continuation of a martinez s nakamura v sordoni analytic smoothing effect for the schrodinger equation with long range perturbation comm pure appl math lix 2006 1330 1351 where an analytic smoothing effect was proved for long range type perturbations of the laplacian h0 on rn in this paper we consider short range type perturbations h of the laplacian on rn and we characterize the analytic wave front set of the solution to the schrodinger equation e ithf in terms of that of the free solution e ith0f for t 0 in the backward non trapping region this result is an analytic analogue of results by hassel and wunsch a hassel j wunsch the schrodinger propagator for scattering metrics ann of math 162 2005 487 523 and nakamura s nakamura wave front set for solutions to schrodinger equations j funct anal 256 2009 1299 1309

the generalized moment based filter can we solve the filtering problem from the only knowledge of few moments of the noise terms in this technical note by exploiting set of distributions based filtering we solve this problem without introducing additional assumptions on the distributions of the noises e g gaussianity or on the final form of the estimator e g linear estimator given the moments e g mean and variance of random variable x it is possible to define the set of all distributions that are compatible with the moments information this set can be equivalently characterized by its extreme distributions a family of mixtures of dirac s deltas the lower and upper expectation of any function g of x are obtained in correspondence of these extremes and can be computed by solving a linear programming problem the filtering problem can then be solved by running iteratively this linear programming problem in this technical note we discuss theoretical properties of this filter we show the connection with set membership estimation and its practical applications

non hermitian matrix description of the calpcalt symmetric anharmonic oscillators the symmetric differential schrodinger equation h e with the operator h h x p2 ax4 i x3 cx2 i x h x on l2 is studied at a 0 it is rearranged as a linear algebraic diagonalization with rigorous proof our non variational construction of bound states offers an infinite dimensional analogue to the recent finite dimensional quasi exact solution available at the less common a 0

cubature kalman filtering for continuous discrete systems theory and simulations in this paper we extend the cubature kalman filter ckf to deal with nonlinear state space models of the continuous discrete kind to be consistent with the literature the resulting nonlinear filter is referred to as the continuous discrete cubature kalman filter cd ckf we use the ito taylor expansion of order 1 5 to transform the process equation modeled in the form of stochastic ordinary differential equations into a set of stochastic difference equations building on this transformation and assuming that all conditional densities are gaussian distributed the solution to the bayesian filter reduces to the problem of how to compute gaussian weighted integrals to numerically compute the integrals we use the third degree cubature rule for a reliable implementation of the cd ckf in a finite word length machine it is structurally modified to propagate the square roots of the covariance matrices the reliability and accuracy of the square root version of the cd ckf are tested in a case study that involves the use of a radar problem of practical significance the problem considered herein is challenging in the context of radar in two respects high dimensionality of the state and increasing degree of nonlinearity the results presented herein indicate that the cd ckf markedly outperforms existing continuous discrete filters

square root quadrature kalman filtering the quadrature kalman filter qkf is a recursive nonlinear filtering algorithm developed in the kalman filtering framework it computes the mean and covariance of all conditional densities using the gauss hermite quadrature rule in this correspondence we develop a square root extension of the quadrature kalman filter using matrix triangularizations the square root quadrature kalman filter sqkf propagates the mean and the square root of the covariance although equivalent to the qkf algebraically the sqkf exhibits excellent numerical characteristics but at the expense of increased computational complexity we also present possible refinements of the generic sqkf

new bases of representation for the unitary parasupersymmetry algebra representation bases of unitary parasupersymmetry algebra of arbitrary order p is constructed by some one dimensional models which are shape invariant with respect to the main quantum number n consequently the isospectral hamiltonians and their exact solutions are obtained as labelled by the main quantum number n

cubature kalman filters in this paper we present a new nonlinear filter for high dimensional state estimation which we have named the cubature kalman filter ckf the heart of the ckf is a spherical radial cubature rule which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear bayesian filter specifically we derive a third degree spherical radial cubature rule that provides a set of cubature points scaling linearly with the state vector dimension the ckf may therefore provide a systematic solution for high dimensional nonlinear filtering problems the paper also includes the derivation of a square root version of the ckf for improved numerical stability the ckf is tested experimentally in two nonlinear state estimation problems in the first problem the proposed cubature rule is used to compute the second order statistics of a nonlinearly transformed gaussian random variable the second problem addresses the use of the ckf for tracking a maneuvering aircraft the results of both experiments demonstrate the improved performance of the ckf over conventional nonlinear filters

algebraic nature of shape invariant and self similar potentials self similar potentials generalize the concept of shape invariance which was originally introduced to explore exactly solvable potentials in quantum mechanics in this paper it is shown that the previously introduced algebraic approach to the latter can be generalized to the former the infinite lie algebras introduced in this context are shown to be closely related to the q algebras the associated coherent states are investigated

finite horizon suboptimal control of markov jump linear parameter varying systems this paper presents a strategy to compute a suboptimal solution to the finite horizon linear quadratic control problem for markov jump linear parameter varying systems the system parameters depend

script pscript t symmetric models in more dimensions and solvable square well versions of their angular schrodinger equations from the partial differential calogero s three body and smorodinsky winternitz superintegrable hamiltonians in two variables we separate the respective angular schrodinger equations and study the possibilities of their minimal symmetric complexification the simultaneous loss of the hermiticity and solvability of the respective angular potentials v is compensated by their replacement by solvable purely imaginary and piece wise constant multiple wells v0 we demonstrate that the spectrum remains real and that it exhibits a rich four series structure in the double well case

exactly solvable potentials and the concept of shape invariance the author analyses the relationship between shape invariance and exactly solvable potentials using earlier results he shows that all actually known shape invariant potentials with a single parameter can be recovered quite simply extension to the case of shape invariant potentials with two parameters as well as to non shape invariant but exactly solvable potentials are also possible

algebraic shape invariant potentials in two steps the simplified algebraic structure of the shape invariance condition in two steps is developed by imposing an extra relation on the two superpotentials of the corresponding two step potentials this simplified version of potential algebra is found to be similar to that of the shape invariance condition in one step the solvable potentials of shape invariance in two steps with a translation change of parameters a1 a0 are shown to possess such a simplified version of potential algebra the condition under which the potential algebra of shape invariance in more than two steps becomes simplified is also discussed

a numerical integration perspective on gaussian filters this paper proposes a numerical integration perspective on the gaussian filters a gaussian filter is approximation of the bayesian inference with the gaussian posterior probability density assumption being valid there exists a variation of gaussian filters in the literature that derived themselves from very different backgrounds from the numerical integration viewpoint various versions of gaussian filters are only distinctive from each other in their specific treatments of approximating the multiple statistical integrations a common base is provided for the first time to analyze and compare gaussian filters with respect to accuracy efficiency and stability factor this study is expected to facilitate the selection of appropriate gaussian filters in practice and to help design more efficient filters by employing better numerical integration methods

supersymmetry operator transformations and exactly solvable potentials a large class of potentials can be solved algebraically by using supersymmetry and shape invariance the authors apply operator transformations f transformations to these algebraically solvable problems to obtain a larger class of solvable potentials the natanzon class of potentials which are not shape invariant the important condition which is independent of supersymmetry for finding new potentials from old ones using operator transformations is that the resulting schrodinger equation has a potential which does not depend on the state as a special case of the f transformation they rederive the previously known connection between the 3d harmonic oscillator the hydrogen atom and the morse potential they also discuss the lack of commutivity of susy and the f transformations

an exact solution of the schrodinger equation for a multiterm potential the authors show that the schrodinger equation with the potential bx cx2 dx3 ex4 is exactly solvable on the half line x or 0 provided two simple relations between b c d and e hold some remarks concerning the b d 0 case are made

a class of exactly solvable potentials related to the jacobi polynomials the author investigates a family of solvable potentials related to the jacobi polynomials this one dimensional potential family depends on three parameters and is restricted to the domain xo so it can be interpreted as the radial part of a central potential in three dimensions with l 0 closed expressions are obtained for the bound state energy spectrum and the wavefunctions the supersymmetric partner of this potential is also determined and it is found not to belong to the same potential family it is shown that this potential family is a special subclass of the general six parameter natanzon potential class and similarities with another subclass the ginocchio potentials are pointed out some aspects of supersymmetric quantum mechanics and shape invariance are also discussed in connection with the potential family under study

further polynomial type eigenfunctions the method developed previously by the author for obtaining eigenfunctions in the form polynomial exponential polynomial for a linear second order differential equation in normal form is extended to embrace the general case when the interaction potential or square of the refractive index profile has the form polynomial polynomial various situations arise in the development of the theory some of which can be concluded algebraically but the majority require numerical calculations regarding the rank of quite a general matrix containing many unknown parameters

explicit wavefunctions for shape invariant potentials by operator techniques the authors obtain explicit expressions for the wavefunctions of all the known shape invariant potentials by using the operator method

analytic smoothing effect for global solutions to nonlinear schrodinger equations abstract we prove the global existence of analytic solutions to the cauchy problem for the cubic schrodinger equation in space dimension n 3 for sufficiently small data with exponential decay at infinity minimal regularity assumption regarding scaling invariance is imposed on the cauchy data

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

advanced point mass method for nonlinear state estimation state estimation of discrete time nonlinear non gaussian stochastic systems by point mass approach which is based on discretization of state space by a regular grid and numerical solution of bayesian recursive relations is treated the stress is laid to grid design which is crucial for estimator quality and significantly affects the computational demands of the estimator boundary based grid design thrifty convolution and multigrid design with grid splitting and merging are proposed the main advantages of these techniques are nonnegligible support delimitation time saving computation of convolution and effective processing of multimodal probability density functions respectively the techniques are involved into the basic point mass approach and a new general purpose more sophisticated point mass algorithm is designed computational demands and estimation quality of the designed algorithm are presented and compared with the particle filter in a numerical example

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

quantum dynamics of a two level system coupled to a shape invariant potential the quantum dynamics of a two level system coupled to a shape invariant potential is investigated shape invariant potentials share an integrability condition called shape invariance which identifies an underlying algebraic structure in general infinite dimensional that transforms the potential parameters such as strength range and diffuseness we determine the time evolution operator the density operator of the system and obtain the temporal behaviour of various dynamical variables by considering either pure or mixed initial states of the system constructed with the generalized coherent state of the shape invariant coupling potential we consider specific examples of shape invariant coupling potentials harmonic oscillator poschl teller and self similar potentials the results obtained for all dynamical variables exhibit rapid oscillations which periodically collapse and regenerate in different ways depending on the coupling potential nature

dirac equation for a spin charged particle on the 2d sphere s2 and the hyperbolic plane h2 using the idea of shape invariance with respect to the main quantum number n we represent lie algebras u 2 and u 1 1 the induced metric by the casimir operator of lie algebras u 2 and u 1 1 leads us to obtain new solutions of the dirac equation corresponding to a spin charged particle on the 2d sphere s2 and the hyperbolic plane h2 in the presence of a magnetic monopole it is shown that the related new spinors represent the supersymmetry algebra and that they satisfy shape invariance equations with respect to n

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

small analytic solutions to the hartree equation we consider the cauchy problem for the hartree equation in space dimension d 3 we assume that the interaction potential v belongs to the weak ld 2 space we prove that if the initial data is sufficiently small in the l2 sense and either or its fourier transform f satisfies a real analytic condition then the solution u t is also real analytic for any t 0 we also prove that if and v satisfy some strong condition then u t can be extended to an entire function on cd for any t 0 a part of our method is applicable to the final value problem we remark that no l2 smallness condition is imposed on first and higher order partial derivatives of and f

a systematization of the unscented kalman filter theory in this paper we propose a systematization of the discrete time unscented kalman filter ukf theory we gather all available ukf variants in the literature present corrections to theoretical inconsistencies and provide a tool for the construction of new ukf s in a consistent way this systematization is done mainly by revisiting the concepts of sigma representation unscented transformation ut scaled unscented transformation sut ukf and square root unscented kalman filter srukf inconsistencies are related to 1 matching the order of the transformed covariance and cross covariance matrices of both the ut and the sut 2 multiple ukf definitions 3 issue with some reduced sets of sigma points described in the literature 4 the conservativeness of the sut 5 the scaling effect of the sut on both its transformed covariance and cross covariance matrices and 6 possibly ill conditioned results in srukf s with the proposed systematization the symmetric sets of sigma points in the literature are formally justified and we are able to provide new consistent variations for ukf s such as the scaled srukf s and the ukf s composed by the minimum number of sigma points furthermore our proposed srukf has improved computational properties when compared to state of the art methods

an algebraic construction of generalized coherent states for shape invariant potentials generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics we show that this generalized formalism is able to a supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system continuity of labelling resolution of unity temporal stability and action identity b reproduce results already known for shape invariant systems such as harmonic oscillator double anharmonic poschl teller and self similar potentials and c point to a formalism that provides a unified description of the different kind of coherent states for quantum systems

deformed shape invariance and exactly solvable hamiltonians with position dependent effective mass known shape invariant potentials for the constant mass schrodinger equation are taken as effective potentials in a position dependent effective mass pdem one the corresponding shape invariance condition turns out to be deformed its solvability imposes the form of both the deformed superpotential and the pdem a lot of new exactly solvable potentials associated with a pdem background are generated in this way a novel and important condition restricting the existence of bound states whenever the pdem vanishes at an end point of the interval is identified in some cases the bound state spectrum results from a smooth deformation of that of the conventional shape invariant potential used in the construction in others one observes a generation or suppression of bound states depending on the mass parameter values the corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

real time updating of structural mechanics models using kalman filtering modified constitutive relation error and proper generalized decomposition the paper aims at proposing a new strategy for real time identification or updating of structural mechanics models defined as dynamical systems the main idea is to introduce the modified constitutive relation error concept which is a practical tool that enables to efficiently solve identification problems with highly corrupted data into the kalman filtering which is a classical framework for data assimilation furthermore a pgd based model reduction method is performed in order to optimize capabilities of the online updating strategy performances of the proposed approach in terms of robustness gain and computational cost reduction are illustrated on several unsteady thermal applications

a generalized jaynes cummings hamiltonian and supersymmetric shape invariance a class of shape invariant bound state problems which represent two level systems are introduced it is shown that the coupled channel hamiltonians obtained correspond to the generalization of the jaynes cummings hamiltonian

analytic smoothing effect for the schrodinger equation with long range perturbation we study microlocal analytic singularity of solutions to schr odinger equation with analytic coefficients using microlocal weight estimate developped for estimating the phase space tunneling we prove microlocal smoothing estimates that generalize results by l robbiano and c zuily we suppose the schr odinger operator is a long range type perturbation of the laplacian and we employ positive commutator type estimates to prove the smoothing property

a master function approach for describing shape invariance parameters with respect to which quantum number they are shape invariant most of the one dimensional shape invariance potentials fall into two classes for each of which one of the quantum numbers describing the shape invariance is assigned on this basis the shape invariance parameters of every class are separately described in terms of a master function its weight function and an appropriate quantum number

some relations between extended and unscented kalman filters the unscented kalman filter ukf has become a popular alternative to the extended kalman filter ekf during the last decade ukf propagates the so called sigma points by function evaluations using the unscented transformation ut and this is at first glance very different from the standard ekf algorithm which is based on a linearized model the claimed advantages with ukf are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model we point out several less known links between ekf and ukf in terms of two conceptually different implementations of the kalman filter the standard one based on the discrete riccati equation and one based on a formula on conditional expectations that does not involve an explicit riccati equation first it is shown that the sigma point function evaluations can be used in the classical ekf rather than an explicitly linearized model second a less cited version of the ekf based on a second order taylor expansion is shown to be quite closely related to ukf the different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications

analyticity and global existence of small solutions to some nonlinear schrodinger equations in this paper we will study the nonlinear schrodinger equations begin gathered i partial t u tfrac 1 2 delta u left u right 2 u t x in mathbb r times mathbb r x n hfill u 0 x phi x x in mathbb r x n hfill end gathered it is shown that the solutions of exist and are analytic in space variables fort 0 if x h2n 1 2 xn decay exponentially as x andn 2

quantum dynamics of parasupersymmetric and shape invariant coupled systems a class of bound state problems which represents the coupling of a three level atom with a two dimensional system involving two shape invariant potentials was introduced in a previous paper in this paper we considered second order parasupersymmetric quantum mechanical models and two possible kinds for the coupling hamiltonian linear and nonlinear in the potential ladder operators in the present paper using an algebraic formulation for shape invariant potential systems we study the quantum dynamics of these coupled systems and obtain the temporal behaviour of some dynamical variables related with the atom and the coupling potentials an application is given for a couple of shape invariant potentials poschl teller self similar potentials

the theory of coupled differential equations in supersymmetric quantum mechanics there is a close relationship between the theory of coupled differential equations and supersymmetric quantum mechanics the authors set up the bridge connecting these fields with two theorems concerning both the coupling and noncoupling cases

on the analytic smoothing effect for the hartree equation with a short range interaction potential abstract we consider the cauchy problem for the hartree equation in space dimension d 2 we assume that the interaction potential v x is short range more precisely we consider the case where v belongs to the weak l d space with 1 d we prove that if 2 d resp 1 2 the initial data is small in the sense of the homogeneous sobolev space h 2 1 resp the homogeneous weighted sobolev space f h 1 2 and the fourier transform f satisfies a real analytic condition then the corresponding solution u t is also real analytic for any t 0 we remark that no h 2 1 resp f h 1 2 smallness condition is imposed on first and higher order partial derivatives of f when 2 d resp 1 2

gauss quadrature rules for a generalized hermite weight function abstract in this paper we give an algorithm for finding the coefficients in the three term recurrence relation for the polynomials orthogonal with respect to the weight function w x x b 2 exp x c 2 on r where b c r 2 1 the algorithm is based on the discretization technique proposed by gautschi with the lanczos algorithm using the coefficients obtained by this algorithm we construct the gauss quadrature rules relative to the weight function w x to show the performance of the new algorithm we give some numerical examples

analytic well posedness of periodic gkdv abstract in the periodic case it is proved that the cauchy problem for the generalized korteweg de vries equation gkdv is locally well posed in a class of analytic functions that can be extended holomorphically in a symmetric strip of the complex plane around the x axis thus the uniform analyticity radius of the solution does not change as time progresses also information about the regularity of the solution in the time variable is provided

isospectral susy potentials the authors introduce a new linear transformation in order to construct isospectral potentials within the framework of supersymmetric quantum mechanics under this transformation the adjoint character of the ladder operator is destroyed but with preservation of the energy spectrum leading to some interesting consequences such as the concept of pseudo partnership of the two components connection with the darboux theorem as well as with some other recent methods of construction is analysed

new developments in state estimation for nonlinear systems based on an interpolation formula accurate state estimators for nonlinear systems can be derived the estimators do not require derivative information which makes them simple to implement

single and coupled channel radial inverse scattering with supersymmetric transformations the present status of the three dimensional inverse scattering method with supersymmetric transformations is reviewed for the coupled channel case we first revisit in a pedagogical way the single channel case where the supersymmetric approach is shown to provide a complete efficient and elegant solution to the inverse scattering problem for the radial schrodinger equation with short range interactions a special emphasis is put on the differences between conservative and non conservative transformations i e transformations that do or do not conserve the behaviour of solutions of the radial schrodinger equation at the origin in particular we show that for the zero initial potential a non conservative transformation is always equivalent to a pair of conservative transformations these single channel results are illustrated on the inversion of the neutron proton triplet eigenphase shifts for the s and d waves we then summarize and extend our previous works on the coupled channel case i e on systems of coupled radial schrodinger equations and stress remaining difficulties and open questions of this problem by putting it in perspective with the single channel case we mostly concentrate on two channel examples to illustrate general principles while keeping mathematics as simple as possible in particular we discuss the important difference between the equal threshold and different threshold problems for equal thresholds conservative transformations can provide non diagonal jost and scattering matrices iterations of such transformations in the two channel case are studied and shown to lead to practical algorithms for inversion a convenient particular technique where the mixing parameter can be fitted without modifying the eigenphases is developed with iterations of pairs of conjugate transformations this technique is applied to the neutron proton triplet s d scattering matrix for which exactly solvable matrix potential models are constructed for different thresholds conservative transformations do not seem to be able to provide a non trivial coupling between channels in contrast a single non conservative transformation can generate coupled channel potentials starting from the zero potential and is a promising first step towards a full solution to the coupled channel inverse problem with threshold differences

adomian method for solving some coupled systems of two equations coupled systems of two linear and nonlinear differential equations for second and first orders respectively can be solved with some techniques and adomian decomposition method a few simple examples are also studied to show with analytical results how the adm works efficiently