superintegrable classical zernike system we consider the differential equation that zernike proposed to classify aberrations of wavefronts in a circular pupil as if it were a classical hamiltonian with a non standard potential the trajectories turn out to be closed ellipses we show that this is due to the existence of higher order invariants that close into a cubic higgs algebra the zernike classical system thus belongs to the class of superintegrable systems its hamilton jacobi action separates into three vertical projections of polar coordinates of sphere polar and equidistant coordinates on half hyperboloids and also in elliptic coordinates on the sphere

symmetric versus hermitian formulations of quantum mechanics a non hermitian hamiltonian that has an unbroken symmetry can be converted by means of a similarity transformation to a physically equivalent hermitian hamiltonian this raises the following question in which form of the quantum theory the non hermitian or the hermitian one is it easier to perform calculations this paper compares both forms of a non hermitian ix3 quantum mechanical hamiltonian and demonstrates that it is much harder to perform calculations in the hermitian theory because the perturbation series for the hermitian hamiltonian is constructed from divergent feynman graphs for the hermitian version of the theory dimensional continuation is used to regulate the divergent graphs that contribute to the ground state energy and the one point green s function the results that are obtained are identical to those found much more simply and without divergences in the non hermitian symmetric hamiltonian the contribution to the ground state energy of the hermitian version of the theory involves graphs with overlapping divergences and these graphs are extremely difficult to regulate in contrast the graphs for the non hermitian version of the theory are finite to all orders and they are very easy to evaluate

pseudo hermitian versus hermitian position dependent mass hamiltonians in a perturbative framework we formulate a systematic algorithm for constructing a whole class of hermitian position dependent mass hamiltonians which to lowest order of perturbation theory allow a description in terms of symmetric hamiltonians the method is applied to the hermitian analogue of the symmetric cubic anharmonic oscillator a new example is provided by a hamiltonian approximately equivalent to a symmetric extension of the one parameter trigonometric poschl teller potential

applications of rigged hilbert spaces in quantum mechanics and signal processing simultaneous use of discrete and continuous bases in quantum systems is not possible in the context of hilbert spaces but only in the more general structure of rigged hilbert spaces rhs in addition the relevant operators in rhs but not in hilbert space are a realization of elements of a lie enveloping algebra and support representations of semigroups we explicitly construct here basis dependent rhs of the line and half line and relate them to the universal enveloping algebras of the weyl heisenberg algebra and su 1 1 respectively the complete sub structure of both rhs and of the operators acting on them is obtained from their algebraic structures or from the related fractional fourier transforms this allows us to describe both quantum and signal processing states and their dynamics two relevant improvements are introduced i new kinds of filters related to restrictions to subspaces and or the elimination of high frequency fluctuations and ii an operatorial structure that starting from fix objects describes their time evolution

zernike functions rigged hilbert spaces and potential applications we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing

generalized zernike or disc polynomials an application in quantum optics an application of the generalized zernike or disc polynomials recently introduced in the literature is shown resorting to the lie algebra based investigation of the dynamics of quantum systems driven by two mode interaction hamiltonians further properties of the associated disc functions are deduced also a generalization of the disc polynomials towards the hahn polynomials is suggested

pt symmetric cubic anharmonic oscillator as a physical model there is a factor of 2 error in equation 61 of this paper correcting this error leads to minor changes in equations 62 and 63 please see pdf for details

a systematic study on the exact solution of the position dependent mass schrodinger equation an algebraic method of constructing potentials for which the schrodinger equation with position dependent mass can be solved exactly is presented a general form of the generators of su 1 1 algebra has been employed with a unified approach to the problem our systematic approach reproduces a number of earlier results and also leads to some novelties we show that the solutions of the schrodinger equation with position dependent mass are free from the choice of parameters for position dependent mass two classes of potentials are constructed that include almost all exactly solvable potentials

recursive computation of generalised zernike polynomials abstract an algorithmic approach for generating generalised zernike polynomials by differential operators and connection matrices is proposed this is done by introducing a new order of generalised zernike polynomials such that it collects all the polynomials of the same total degree in a column vector the connection matrices between these column vectors composed by the generalised zernike polynomials and a family of polynomials generated by a rodrigues formula are given explicitly this yields a rodrigues type formula for the generalised zernike polynomials themselves with properly defined differential operators another consequence of our approach is the fact that the generalised zernike polynomials obey a rather simple partial differential equation we recall also how to define hermite zernike polynomials

an efficient pruning method for decision alternatives of owa operators in this paper we present an efficient method for pruning decision alternatives in the case of using ordered weighted averaging owa operators for decision making the proposed method helps to identify inferior alternatives that are less likely to be selected out of competing alternatives as the owa aggregation proceeds it thus enables us to diminish the number of alternatives before applying the owa operators the reordering process unique to the owa aggregation plays an important role in identifying inferior alternatives the efficacy of the proposed method is demonstrated by simulation analysis in which artificial decision problems of diverse sizes are generated and then examined with four scenarios pruning alternatives pruning alternatives with rank order owa weights pruning alternatives with a normalized decision problem and pruning alternatives with both a normalized decision problem and rank order owa weights the proposed method is easy to use and the simulation results show that the number of alternatives can be reduced drastically by applying this method

delta function potential with a complex coupling we explore the hamiltonian operator where is the dirac delta function and z is an arbitrary complex coupling constant for a purely imaginary z h has a spectral singularity at for re z 0 h has a real positive continuous spectrum that is free from spectral singularities for this latter case we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive definite inner product that renders h self adjoint this allows us to address the intriguing question of the nonlocal aspects of the equivalent hermitian hamiltonian for the system in particular we compute the energy expectation values for various gaussian wave packets to show that the non hermiticity effect diminishes rapidly outside an effective interaction region

on the reality of the eigenvalues for a class of symmetric oscillators we study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays where is a real polynomial and we prove that if for some we have for all then the eigenvalues are all positive real we then sharpen this to a larger class of polynomial potentials

a novel algorithm for fast computation of zernike moments abstract zernike moments zms have been successfully used in pattern recognition and image analysis due to their good properties of orthogonality and rotation invariance however their computation by a direct method is too expensive which limits the application of zms in this paper we present a novel algorithm for fast computation of zernike moments by using the recursive property of zernike polynomials the inter relationship of the zernike moments can be established as a result the zernike moment of order n with repetition m znm can be expressed as a combination of zn 2 m and zn 4 m based on this relationship the zernike moment znm for n m can be deduced from zmm to reduce the computational complexity we adopt an algorithm known as systolic array for computing these latter moments using such a strategy the multiplication number required in the moment calculation of zmm can be decreased significantly comparison with known methods shows that our algorithm is as accurate as the existing methods but is more efficient

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

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

eigenstates of paraparticle creation operators eigenstates of the parabose and parafermi creation operators are constructed in the dirac contour representation the parabose eigenstates correspond to the dual vectors of the parabose coherent states in order p 2 conserved charge parabose creation operator eigenstates are also constructed the contour forms of the associated resolutions of unity are obtained

symmetric effective mass schrodinger equations we outline a general method of obtaining exact solutions of symmetric schrodinger equations with a position dependent effective mass using this method exact solutions of some symmetric potentials have been obtained we have also discussed the construction of isospectral symmetric potentials

generalized zernike or disc polynomials we investigate generalized zernike or disc polynomials p m n z z which are orthogonal 2d polynomials in the unit disc 0 zz 1 is a free parameter these polynomials can be expressed by jacobi polynomials of transformed arguments in connection with a simple angle dependence a limiting procedure leads to laguerre 2d polynomials lm n z z furthermore we introduce the corresponding orthonormalized disc functions the disc polynomials and disc functions obey two differential equations a first order and a second order one with a certain degree of freedom and the operators of lowering and raising of the indices are found these operators can be closed to a lie algebra su 1 1 su 1 1 new generating functions are derived from an operational representation which is alternative to the rodrigues type representation the one dimensional analogue of the disc polynomials which are orthogonal polynomials in the interval 0 r 1 with weight factors 1 r2 x are ultraspherical or gegenbauer polynomials in a new standardization the lowering and raising operators to the corresponding orthonormalized functions form a simple su 1 1 lie algebra this is given in the appendix in sketched form

application of nonlinear deformation algebra to a physical system with poschl teller potential nonlinear deformation algebra are realized in a physical system with p schl teller potential the raising and lowering operators satisfying this algebra are constructed from which the eigenproblem of the system can be exactly solved by the operator method the physical meaning of two deforming functions involved in this algebra is also found in addition algebra is obtained naturally and discussions on the coherent state are also made

pseudo hermiticity and some consequences of a generalized quantum condition we exploit the hidden symmetry structure of a recently proposed non hermitian hamiltonian and of its hermitian equivalent one this sheds new light on the pseudo hermitian character of the former and allows access to a generalized quantum condition special cases lead to hyperbolic and morse like potentials in the framework of a coordinate dependent mass model

a new class of zernike moments for computer vision applications a modified direct method for the computation of the zernike moments is presented in this paper the presence of many factorial terms in the direct method for computing the zernike moments makes their computation process a very time consuming task although the computational power of the modern computers is impressively increasing the calculation of the factorial of a big number is still an inaccurate numerical procedure the main concept of the present paper is that by using stirling s approximation formula for the factorial and by applying some suitable mathematical properties a novel factorial free direct method can be developed the resulted moments are not equal to those computed by the original direct method but they are a sufficiently accurate approximation of them besides their variability does not affect their ability to describe uniquely and distinguish the objects they represent this is verified by pattern recognition simulation examples

calculation of the hidden symmetry operator in symmetric quantum mechanics in a recent paper it was shown that if a hamiltonian h has an unbroken symmetry then it also possesses a hidden symmetry represented by the linear operator the operator commutes with both h and the inner product with respect to is associated with a positive norm and the quantum theory built on the associated hilbert space is unitary in this paper it is shown how to construct the operator for the non hermitian symmetric hamiltonian h 1 2p2 1 2x2 ix3 using perturbative techniques it is also shown how to construct the operator for h 1 2p2 1 2x2 x4 using nonperturbative methods