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

qualitative analysis of certain generalized classes of quadratic oscillator systems we carry out a systematic qualitative analysis of the two quadratic schemes of generalized oscillators recently proposed by quesne j math phys 56 012903 2015 by performing a local analysis of the governing potentials we demonstrate that while the first potential admits a pair of equilibrium points one of which is typically a center for both signs of the coupling strength the other points to a centre for 0 on the other hand the second potential reveals only a center for both the signs of from a linear stability analysis we carry out our study by extending quesne s scheme to include the effects of a linear dissipative term an important outcome is that we run into a remarkable transition to chaos in the presence of a periodic force term fcos t

superintegrable systems on the two dimensional sphere s2 and the hyperbolic plane h2 the existence of superintegrable systems with n 2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two dimensional sphere s2 and on the hyperbolic plane h2 the approach used is based on enforcing the conditions for the existence of two independent integrals further than the energy this is done in a way which allows us to discuss at once the cases of the sphere s2 and the hyperbolical plane h2 by considering the curvature as a parameter different superintegrable potentials are obtained as the solutions of certain systems of two dependent second order partial differential equations the euclidean results are directly recovered for 0 and the superintegrable potentials on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent superintegrable potentials for the values 1 and 1 some new superintegrable potentials are f

reconstruction of a three dimensional object from a limited range of views abstract we define the radon transform r and the back projection r adjoint of r on the space of c rapidly decreasing functions j and on the space of distributions of compact support e we show that r r is a convolution operator the result is true in the case of full views as well as in the case of a limited range of views an application to the medical image reconstruction is given

a quantum quasi harmonic nonlinear oscillator with an isotonic term the properties of a nonlinear oscillator with an additional term kg x2 characterizing the isotonic oscillator are studied the nonlinearity affects to both the kinetic term and the potential and combines two nonlinearities associated to two parameters and kg in such a way that for 0 all the characteristics of the standard isotonic system are recovered the first part is devoted to the classical system and the second part to the quantum system this is a problem of quantization of a system with position dependent mass of the form m x 1 1 x2 with a dependent non polynomial rational potential and with an additional isotonic term the schrodinger equation is exactly solved and the kg dependent wave functions and bound state energies are explicitly obtained for both 0

high precision calculation of the eigenvalues for the x2 λx2 1 gx2 potential an analytic continuation procedure using a taylor series is used to produce very accurate wavefunctions and eigenvalues for the one dimensional anharmonic oscillator governed by the potential v x x2 x2 1 gx2

generalized quantum nonlinear oscillators exact solutions and rational extensions we construct exact solutions and rational extensions to quantum systems of generalized nonlinear oscillators our method is based on a connection between nonlinear oscillator systems and schrodinger models for certain hyperbolic potentials the rationally extended models admit discrete spectrums and corresponding closed form solutions are expressed through jacobi type xm exceptional orthogonal polynomials

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

on some mixed finite element methods with numerical integration in this paper a new family of mixed finite volume methods is analyzed for the approximation of a reaction diffusion problem all the methods are obtained starting from the dual mixed formulation of the problem and then employing the lowest order raviart thomas finite element spaces plus a suitable quadrature formula for the mass matrix this allows for the use of different averages of the inverse diffusion coefficient to enforce the constitutive law for the fluxes at the interelement boundary in a finite volume fashion the soundness of the methods is supported by an error analysis which shows optimal cal o h convergence rate with respect to the standard mixed finite element norm in the case of both smooth and piecewise smooth coefficients numerical results on test problems with both smooth and nonsmooth coefficients support the theoretical estimates

optimization for limited angle tomography in medical image processing this paper aims to reduce the problems of incomplete data in computed tomography which happens frequently in medical image process and analysis e g when the high density region of objects can only be penetrated by x rays at a limited angular range as the projection data are available only in an angular range the incomplete data problem can be attributed to the limited angle problem which is an ill posed inverse problem image reconstruction based on total variation tv reduces the problem and gives better performance on edge preserving reconstruction however the artificial parameter can only be determined through considerable experimentation in this paper an effective tv objective function is proposed to reduce the inverse problem in the limited angle tomography this novel objective function provides a robust and effective reconstruction without any artificial parameter in the iterative processes using the tv as a multiplicative constraint the results demonstrate that this reconstruction strategy outperforms some previous ones

families of quasi exactly solvable extensions of the quantum oscillator in curved spaces we introduce two new families of quasi exactly solvable qes extensions of the oscillator in a d dimensional constant curvature space for the first three members of each family we obtain closed form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the bethe ansatz method we prove that the first member of each family has a hidden sl 2 symmetry and is connected with a qes equation of the first or second type respectively one dimensional results are also derived from the d dimensional ones with d 2 thereby getting qes extensions of the mathews lakshmanan nonlinear oscillator

the patch test as a validation of a new finite element for the solution of convection diffusion equations abstract a first goal of this work is to show how the patch test a well known tool established in the field of continuum mechanics can be fruitfully employed to demonstrate the convergence and robustness of a numerical method for the solution of convection dominated flow problems a new finite element passing the patch test is then proposed for the discrestization of current continuity equations in semiconductors numerical results relative to the study of two dimensional devices are included to validate the effectiveness of the method

spectral properties of a confined nonlinear quantum oscillator in one and three dimensions we analyze the spectral behaviour of a nonlinear quantum oscillator model under confinement the underlying potential is given by a harmonic oscillator interaction plus a nonlinear term that can be weakened or strengthened through a parameter numerical eigenvalues of the model in one and three dimensions are presented the asymptotic behaviour of the eigenvalues for confinement relaxation and for vanishing nonlinear term in the potential is investigated our findings are compared with existing results

exponentially fitted shape functions for advection dominated flow problems in two dimensions we sketch out the basic properties of three novel exponentially fitted shape functions that generalize to two dimensional triangular elements the one dimensional functions used in the well known scharfetter gummel method this scheme is widely employed in the finite element approximation of the drift diffusion equations arising in semiconductor device modelling

an accurate method for numerical calculations in quantum mechanics an accurate method for numerical calculations of matrix elements and for solving the eigenvalue problem in quantum mechanics is presented methods for numerical interpolation differentiation and integration provide 15 digit accuracy with double precision arithmetic operations a method for solution of the eigenvalue problem of an ordinary differential equation by using discretization and matrix eigenvalue methods provides 13 to 15 digit accuracy the efficiency of the proposed methods is demonstrated by the applications to bound states for the linear harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential

limited angle reconstruction using stabilized algorithms the stability of solutions to the limited angle tomography reconstruction problem obtained by using the projections onto convex sets pocs technique are examined although pocs techniques provide a feasible solution to the reconstruction problem the solution is only one sample from the intersection of the closed convex sets that define the solution space a method for evaluating the ensemble of possible solution waveforms that are in the neighborhood of a solution is presented the ensemble characteristics are used to construct an inverse filter which is then applied to the computed solution the results obtained using this method are less sensitive to noise amplification and are less dependent on both starting data and the number of iterations an estimate of the object dependent extrapolation that is possible using either linear or nonlinear constraints is provided

finite element methods for convection diffusion problems using exponential splines on triangles abstract a new family of petrov galerkin finite element methods on triangular grids is constructed for singularly perturbed elliptic problems in two dimensions it uses divergence free trial functions that form a natural generalization of one dimensional exponential trial functions this family includes an improved version of the divergence free finite element method used in the pltmg code numerical results show that the new method is able to compute strikingly accurate solutions on coarse meshes an analysis of the use of slotboom variables shows that they are theoretically unsatisfactory and explains why certain petrov galerkin methods lose stability when generalized from one to two dimensions

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

an analysis of a conforming exponentially fitted finite element method for a convection diffusion problem in this paper we present a convergence analysis for a conforming exponentially fitted galerkin finite element method with triangular elements for a linear singularly perturbed convection diffusion problem with a singular perturbation parameter e it is shown that the error for the finite element solution in the energy norm is bounded by o h e1 2 u 2 e 1 2 u 1 if a regular family of triangular meshes is used in the case that a problem contains only exponential boundary layers the method is shown to be convergent at a rate of h1 2 h ln e on anisotropic layer fitted meshes

superintegrable systems on 3 dimensional curved spaces eisenhart formalism and separability the eisenhart geometric formalism which transforms an euclidean natural hamiltonian h t v into a geodesic hamiltonian t with one additional degree of freedom is applied to the four families of quadratically superintegrable systems with multiple separabilities in the euclidean plane first the separability and superintegrability of such four geodesic hamiltonians t r r a b c d in a three dimensional curved space are studied and then these four systems are modified with the addition of a potential u r leading to h r t r u r second we study the superintegrability of the four hamiltonians h r h r r where r is a certain position dependent mass that enjoys the same separability as the original system h r all the hamiltonians here studied describe superintegrable systems on non euclidean three dimensional manifolds with a broken spherical symmetry

a geometric approach to a generalized virial theorem the virial theorem introduced by clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics is studied by making use of symplectic formalism as an approach in the case of both the hamiltonian and lagrangian systems the possibility of establishing virial like theorems from one parameter groups of non strictly canonical transformations is analysed the case of systems with a position dependent mass is also discussed using the modern symplectic approach to quantum mechanics we arrive at the quantum virial theorem in full analogy with the classical case

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

potential r2 λr2 1 gr2 and the analytic continued fractions the author constructs the exact wavefunctions and an analytic continued fractional green function for the one and three dimensional schrodinger equation with the potential v r h r2 r2 lambda r2 1 gr2 h 1 4 lambda g 0 in the numerical application of the formulae the energy and norm of the boundary state psi may be approximated by both their lower and upper estimates

generalized nonlinear oscillators with quasi harmonic behaviour classical solutions the classical nonlinear oscillator proposed by mathews and lakshmanan q appl math 32 215 1974 and including a position dependent mass in the kinetic energy term is generalized in two different ways by adding an extra term to the potential the solutions of the euler lagrange equation are shown to exhibit richer behaviour patterns than those of the original nonlinear oscillator

incomplete data problems in x ray computerized tomography i singular value decomposition of the limited angle transform etude des problemes intrinseques relatifs aux donnees incompletes dans la tomographie aux rayons x assistee par ordinateur decomposition de la transformation de radon

exact and quasiexact solvability of second order superintegrable quantum systems ii relation to separation of variables we make explicit the intimate relationship between quasiexact solvability as expounded for example by ushveridze quasi exactly solvable models in quantum mechanics iop bristol 1993 and the technique of separation of variables as it applies to specific superintegrable quantum hamiltonians it is the multiseparability of superintegrable systems that forces the existence of interesting families of polynomial solutions characteristic of quasiexact solvability that enables us to solve these systems in distinct ways and that gives us the basis of a classification theory this connection is generalized in terms of the understanding of the role of finite solutions of quantum hamiltonians

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

eigenenergies of the r2 lambda r2 1 gr2 and 1 2 r2 or gr4 1 g alpha r2 potentials obtained by the hypervirial method the energy levels of the schrodinger equation involving the potentials v r r2 lambda r2 1 gr2 and v or r 1 2 r2 or gr4 1 g alpha r2 are calculated by using hypervirial and pade approximant methods

non bayesian detection and detectability of anomalies from a few noisy tomographic projections the detection of an anomaly from a few noisy tomographic projections is addressed from the statistical point of view an unknown scene is composed of a background considered as a deterministic nuisance parameter with a possibly hidden anomaly because the full pixel by pixel reconstruction is impossible a parametric non bayesian approach is proposed to fill up the gap in the missing data an optimal statistical test which eliminates the background and detects the anomaly is designed the potential advantage of such an approach is its capacity to detect an anomaly target hidden in background designed by an adversary to mask the anomaly a key issue in the non bayesian anomaly detection i e the problem of anomaly detectability is stated and solved in this paper in the case of a bivariate polynomial background defined on an unknown rectangular support the size of detectable anomaly reaches its maximum defined by the number of elementary cells of x ray detector and degree of the polynomial function

special function solutions of a spectral problem for a nonlinear quantum oscillator we construct exact solutions of a spectral problem involving the schrodinger equation for a nonlinear one parameter oscillator potential in contrast to a previous analysis of the problem carinena et al 2007 ann phys 322 434 59 where solutions were given through a rodrigues type formula our approach leads to closed form representations of the solutions in terms of special functions not containing any derivative operators we show normalizability and orthogonality of our solutions as well as correct reduction of the problem to the harmonic oscillator model if the parameter in the potential gets close to zero

a geometric projection space reconstruction algorithm abstract we present a method to reconstruct images from finite sets of noisy projections that may be available only over limited or sparse angles the algorithm calculates the maximum a posteriori map estimate of the full sinogram which is an image of the 2 d radon transform of the object from the available data it is implemented using a primal dual constrained optimization procedure that solves a partial differential equation in the primal phase with an efficient local relaxation algorithm and uses a simple lagrange multiplier update in the dual phase the sinogram prior probability is given by a markov random field mrf that includes information about the mass center of mass and convex hull of the object and about the smoothness fundamental constraints and periodicity of the 2 d radon transform the object is reconstructed using convolution backprojection applied to the estimated sinogram we show several reconstructed objects which are obtained from simulated limited and sparse angle data using the described algorithm and compare these results with images obtained using convolution backprojection directly

the ill conditioned nature of the limited angle tomography problem the instability of inverting the limited angle radon transform is studied by constructing the singular value decomposition of this operator connections are established with previously known series expansions and with the discrete prolate spheroidal wave functions mollification methods are shown to delay but not prevent the onset of instability as the angular range of projections is decreased

spectrum generating algebras for position dependent mass oscillator schrodinger equations the interest of quadratic algebras for position dependent mass schrodinger equations is highlighted by constructing spectrum generating algebras for a class of d dimensional radial harmonic oscillators with d 2 and a specific mass choice depending on some positive parameter via some minor changes the one dimensional oscillator on the line with the same kind of mass is included in this class the existence of a single unitary irreducible representation belonging to the positive discrete series type for d 2 and of two of them for d 1 is proved the transition to the constant mass limit 0 is studied and deformed su 1 1 generators are constructed these operators are finally used to generate all the bound state wavefunctions by an algebraic procedure

biorthogonal decompositions of the radon transform the concept of singular value decompositions is a valuable tool in the examination of ill posed inverse problems af g such as the inversion of the radon transform a singular value decomposition depends on the determination of suitable orthogonal systems of eigenfunctions of the operators a a a a in this paper we consider a new approach which generalizes this concept by application of biorthogonal instead of orthogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces although it is preferable to use eigenfunctions it is still possible to consider biorthogonal function systems which are not eigenfunctions of the operator with respect to the radon transform for functions with support in the unit ball we apply the system of appell polynomials which is a natural generalization of the univariate system of gegenbauer ultraspherical polynomials to the multivariate case the corresponding biorthogonal decompositions show some advantages in comparison with the known singular value decompositions vice versa by application of our decompositions we are able to prove new properties of the appell polynomials

integrable potentials on spaces with curvature from quantum groups a family of classical integrable systems defined on a deformation of the two dimensional sphere hyperbolic and anti de sitter spaces is constructed through hamiltonians defined on the non standard quantum deformation of a sl 2 poisson coalgebra all these spaces have a non constant curvature that depends on the deformation parameter z as particular cases the analogues of the harmonic oscillator and kepler coulomb potentials on such spaces are proposed another deformed hamiltonian is also shown to provide superintegrable systems on the usual sphere hyperbolic and anti de sitter spaces with a constant curvature that exactly coincides with z according to each specific space the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers the non deformed limit z 0 of all these hamiltonians can then be regarded as the zero curvature limit contraction which leads to the corresponding super integrable systems on the flat euclidean and minkowskian spaces

adart an adaptive algebraic reconstruction algorithm for discrete tomography in this paper we suggest an algorithm based on the discrete algebraic reconstruction technique dart which is capable of computing high quality reconstructions from substantially fewer projections than required for conventional continuous tomography adaptive dart adart goes a step further than dart on the reduction of the number of unknowns of the associated linear system achieving a significant reduction in the pixel error rate of reconstructed objects the proposed methodology automatically adapts the border definition criterion at each iteration resulting in a reduction of the number of pixels belonging to the border and consequently of the number of unknowns in the general algebraic reconstruction linear system to be solved being this reduction specially important at the final stage of the iterative process experimental results show that reconstruction errors are considerably reduced using adart when compared to original dart both in clean and noisy environments

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

globally convergent edge preserving regularized reconstruction an application to limited angle tomography we introduce a generalization of a deterministic relaxation algorithm for edge preserving regularization in linear inverse problems this algorithm transforms the original possibly nonconvex optimization problem into a sequence of quadratic optimization problems and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex we prove that our more general algorithm is globally convergent i e converges to a local minimum from any initialization under less restrictive conditions even when the original cost functional is nonconvex we apply this algorithm to tomographic reconstruction from limited angle data by formulating the problem as one of regularized least squares optimization the results demonstrate that the constraint of piecewise smoothness applied through the use of edge preserving regularization can provide excellent limited angle tomographic reconstructions two edge preserving regularizers one convex the other nonconvex are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited angle scenarios and to explore how factors such as the choice of error norm angular sampling rate and amount of noise affect the reconstruction quality and algorithm performance these simulation results show that for this application the nonconvex regularizer produces consistently superior results

on the x2 λx2 1 gx2 interaction presents an infinite set of exact solutions of odd parity for the one dimensional schrodinger equation with the interaction x2 lambda x2 1 gx2 these complement the exact solutions of even parity that have been obtained recently by flessas phys lett a vol 84 p 121 1981

exact analytical eigenfunctions for the x2 λ x2 1 gx2 interaction the author discusses an ansatz for the eigenfunctions of the x2 x2 1 gx2 interaction introduced by blecher and leach 1987 the author shows that the ansatz provides pairs of solutions in five cases having and g connected by 6g2 4g and conjectures the possible existence of an infinite number of such solution pairs

deformed shape invariance symmetry and potentials in curved space with two known eigenstates we consider two families of extensions of the oscillator in a d dimensional constant curvature space and analyze them in a deformed supersymmetric framework wherein the starting oscillator is known to exhibit a deformed shape invariance property we show that the first two members of each extension family are also endowed with such a property provided some constraint conditions relating the potential parameters are satisfied in other words they are conditionally deformed shape invariant since in the second step of the construction of a partner potential hierarchy the constraint conditions change we impose compatibility conditions between the two sets to build potentials with known ground and first excited states to extend such results to any members of the two families we devise a general method wherein the first two superpotentials the first two partner potentials and the first two eigenstates of the starting potential are built from some generating function w r and its accompanying function w r we consider two families of extensions of the oscillator in a d dimensional constant curvature space and analyze them in a deformed supersymmetric framework wherein the starting oscillator is known to exhibit a deformed shape invariance property we show that the first two members of each extension family are also endowed with such a property provided some constraint conditions relating the potential parameters are satisfied in other words they are conditionally deformed shape invariant since in the second step of the construction of a partner potential hierarchy the constraint conditions change we impose compatibility conditions between the two sets to build potentials with known ground and first excited states to extend such results to any members of the two families we devise a general method wherein the first two superpotentials the first two partner potentials and the first two eigenstates of the starting potential are built from some generating function w r and its accompanying functio

speed of estimation in positron emission tomography and related inverse problems we study a continuous idealization of the image reconstruction problem in positron emission tomography considered as an example of bivariate density estimation based on indirect observations given a large sample of indirect observations we consider the size of the equivalent sample of observations whose original exact positions would allow equally accurate estimation of the image of interest both for indirect and for direct observations we establish exact minimax rates of convergence of estimation for all possible estimators over suitable smoothness classes of functions the approach of the paper is applicable to a wide class of linear inverse problems

varepsilon optimal non bayesian anomaly detection for parametric tomography the non bayesian detection of an anomaly from a single or a few noisy tomographic projections is considered as a statistical hypotheses testing problem it is supposed that a radiography is composed of an imaged nonanomalous background medium considered as a deterministic nuisance parameter with a possibly hidden anomaly because the full voxel by voxel reconstruction is impossible an original tomographic method based on the parametric models of the nonanomalous background medium and radiographic process is proposed to fill up the gap in the missing data exploiting this ldquoparametric tomography rdquo a new detection scheme with a limited loss of optimality is proposed as an alternative to the nonlinear generalized likelihood ratio test which is untractable in the context of nondestructive testing for the objects with uncertainties in their physical geometrical properties the theoretical results are illustrated by the processing of real radiographies for the nuclear fuel rod inspection

on harmonic oscillators on the two dimensional sphere s2 and the hyperbolic plane h2 ii two harmonic oscillators isotropic and nonisotropic 2 1 are studied on the two dimensional sphere s2 and the hyperbolic plane h2 both systems are integrable and super integrable with constants of motion quadratic in the momenta these properties are shown to derive from a complex factorization for the constants of motion which holds for arbitrary values of the curvature and the dynamics of the euclidean harmonic 1 1 and 2 1 oscillators is directly recovered for 0 the harmonic oscillators on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent potentials for the values 1 and 1 finally a particular potential is proposed for representing the general spherical hyperbolic n 1 anisotropic harmonic oscillator on a two dimensional manifold of constant curvature

the quantum harmonic oscillator on the sphere and the hyperbolic plane κ dependent formalism polar coordinates and hypergeometric functions a nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates r by making use of a curvature dependent formalism the curvature is considered as a parameter and then the radial schrodinger equation becomes a dependent gauss hypergeometric equation the energy spectrum and the wave functions are exactly obtained in both the sphere s2 0 and the hyperbolic plane h2 0 a comparative study between the spherical and the hyperbolic quantum results is presented

superintegrable quantum u 3 systems and higher rank factorizations a class of two dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one dimensional factorized systems by using standard methods to find the shape invariant intertwining operators we arrive at a so 6 dynamical algebra and its hamiltonian hierarchies we pay attention to those associated to certain unitary irreducible representations that can be displayed by means of three dimensional polyhedral lattices we also discuss the role of superpotentials in this new context

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

two dimensional exponential fitting and applications to drift diffusion models a class of methods is introduced that generalizes the so called exponential fitting method to two dimensional problems and more specifically to equations of the type operatorname div underset raise0 3em hbox smash scriptscriptstyle nabla u u underset raise0 3em hbox smash scriptscriptstyle nabla psi f with psi given and underset raise0 3em hbox smash scriptscriptstyle nabla psi big in a part of the domain the basic idea is the following i write the equation as operatorname div e psi underset raise0 3em hbox smash scriptscriptstyle nabla rho f through the change of variables u rho e psi ii discretize the symmetric equation by means of a method that uses some kind of harmonic average for the coefficient e psi mixed hybrid etc iii write the discrete scheme in terms of the nodal values of u e psi rho this produces an m matrix

spectrum generating algebra for the continuous spectrum of a free particle in lobachevski space in this paper we construct a spectrum generating algebra sga for a quantum system with purely continuous spectrum the quantum free particle in a lobachevski space with constant negative curvature the sga contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the hamiltonian among themselves in our case the geometrical symmetry algebra is so 3 1 and the sga is so 4 2 we start with a representation of so 4 2 by functions on a realization of the lobachevski space given by a two sheeted hyperboloid where the lie algebra commutators are the usual poisson dirac brackets then we introduce a quantized version of the representation in which functions are replaced by operators on a hilbert space and poisson dirac brackets by commutators eigenfunctions of the hamiltonian are given and naive ladder operators are identified the previously defined naive ladder operators shift the eigenvalues by a

polynomial based algorithms for computed tomography ii this paper is concerned with two dimensional parallel beam tomography we investigate the computational effectiveness of the general approach outlined in a previous paper by one of us that approach was based on reconstructions by means of convolution with a polynomial kernel here we consider only the case of uniformly spaced views to take advantage of the symmetry in this case we give a class of algorithms and calculate several numerical examples we also obtain estimates on the expected mean square error when the data is perturbed by white noise these estimates lead to several surprising applications

rational extension and jacobi type xm solutions of a quantum nonlinear oscillator we construct a rational extension of a recently studied nonlinear quantum oscillator model our extended model is shown to retain exact solvability admitting a discrete spectrum and corresponding closed form solutions that are expressed through jacobi type xm exceptional orthogonal polynomials

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators