unitary rotation of square pixellated images the unitary rotation of square pixellated images is based on the finite su 2 oscillator model which describes systems whose values for position momentum and energy are discrete and finite in a two dimensional position space this allows the construction of angular momentum states orthonormal and complete for which rotations are defined as multiplication by phases that carry the rotation angle the decomposition of a digital square images in terms of these angular momentum states determines a unitary hence invertible rotation of the image whose kernel can be computed as a four dimensional array of real numbers

finite dimensional schwinger basis deformed symmetries wigner function and an algebraic approach to quantum phase schwinger s finite d dimensional periodic hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the wigner function these subalgebras are shown to be admissible endowed with the non negative norm of hilbert space vectors hence they provide the desired canonical basis for the algebraic formulation of the quantum phase problem certain equivalence classes in the space of labels are identified within each subalgebra and connections with area preserving canonical transformations are examined the generalized representations of the wigner function are examined in the finite dimensional cyclic schwinger basis these representations are shown to conform to all fundamental conditions of the generalized phase space wigner distribution as a specific application of the schwinger basis the number phase unitary operator pair in is studied and based on the admissibility of the underlying q oscillator subalgebra an algebraic approach to the unitary quantum phase operator is established this being the focus of this work connections with the susskind glogower carruthers nieto phase operator formalism as well as standard action angle wigner function formalisms are examined in the infinite period limit the concept of continuously shifted fock basis is introduced to facilitate the fock space representations of the wigner function

finite two dimensional oscillator ii the radial model a finite two dimensional radial oscillator of n 1 2 points is proposed with the dynamical lie algebra so 4 su 2 x su 2 y examined in part i of this work but reduced by a subalgebra chain so 4 so 3 so 2 as before there are a finite number of energies and angular momenta the casimir spectrum of the new chain provides the integer radii 0 n and the 2 1 discrete angles on each circle are obtained from the finite fourier transform of angular momenta the wavefunctions of the finite radial oscillator are so 3 clebsch gordan coefficients we define here the hankel hahn transforms with dual hahn polynomials as finite n unitary approximations to hankel integral transforms with bessel functions obtained in the contraction limit n

a contribution to the unaliased discrete time wigner distribution this paper presents a new closed form analytical definition of the unaliased discrete time wigner distribution dtwd which arises naturally from its continuous time counterpart and is straightforwardly implementable for digital signal processing this dtwd possesses properties that resemble those of the continuous time case it is shown that this dtwd is the sum of two terms designated as the even and odd terms that has interesting symmetric properties similar to those of the discrete fourier transform moreover this dtwd captures important features of the continuous time case and it collapses to the power spectral density function when smoothed with the proper functions these smoothing functions are limited to be separable the unaliasedness is validated by computer simulations

the variational iteration method for a reliable treatment of the linear and the nonlinear goursat problem abstract the linear and the nonlinear goursat problems are investigated the he s variational iteration method vim is used to construct exact solutions to these problems the linear and the nonlinear structures are handled in a like manner without any need to restrictive assumptions the accuracy level of the obtained results reveal the power of this method over existing numerical methods

the fractional fourier transform and time frequency representations the functional fourier transform frft which is a generalization of the classical fourier transform was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community to which it may however be potentially useful the frft depends on a parameter spl alpha and can be interpreted as a rotation by an angle spl alpha in the time frequency plane an frft with spl alpha spl pi 2 corresponds to the classical fourier transform and an frft with spl alpha 0 corresponds to the identity operator on the other hand the angles of successively performed frfts simply add up as do the angles of successive rotations the frft of a signal can also be interpreted as a decomposition of the signal in terms of chirps the authors briefly introduce the frft and a number of its properties and then present some new results the interpretation as a rotation in the time frequency plane and the frft s relationships with time frequency representations such as the wigner distribution the ambiguity function the short time fourier transform and the spectrogram these relationships have a very simple and natural form and support the frft s interpretation as a rotation operator examples of frfts of some simple signals are given an example of the application of the frft is also given

linear canonical transformations and quantum phase a unified canonical and algebraic approach the algebra of generalized linear quantum canonical transformations is examined in the perspective of schwinger s unitary canonical operator basis formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space the representations of the wigner function in the generalized action angle unitary operator pair for certain hamiltonian systems with dynamical symmetry is examined this generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action angle wigner function

quantum time of arrival goursat problem the construction of quantum time of arrival operator conjugate to the system hamiltonian leads to a particular linear homogeneous goursat problem in this work we demonstrate how to approximate the solution of the mentioned differential equation both semi analytically and numerically with the goal of calculating the largest eigenvalue of the associated confined time of arrival operator in the analytical approximation we used the partial sum expansion of the solution and showed that the resulting largest eigenvalue converges as the number of terms increases the result shows that for the parameters considered in this paper the approximation is sufficient up to the fourth order correction term in the numerical approximation we develop a non iterative formula to obtain the numerical solution of the goursat problem the performance of the non iterative method is compared with the known numerical techniques in literature numerical results show that the non iterative algorithm is more accurate and faster

the action angle wigner function a discrete finite and algebraic phase space formalism the action angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space in this work a discrete and finite dimensional phase space formalism in which the phase space variables are discrete and the time is continuous is developed and the fundamental properties of the discrete weyl wigner moyal quantization are derived the action angle wigner function is shown to exist in the semi discrete limit of this quantization scheme a comparison with other formalisms which are not explicitly based on canonical discretization is made fundamental properties that an action angle phase space distribution respects are derived the dynamical properties of the action angle wigner function are analysed for discrete and finite dimensional model hamiltonians the limit of the discrete and finite dimensional formalism including a discrete analogue of the gaussian wavefunction spread viz the binomial wavepacket is examined and shown by examples that standard continuum quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity

on the numerical solution of the goursat problem abstract in this paper we present a nonlinear trapezoidal formula for the solution of the goursat problem the new scheme implements the harmonic mean hm averaging of the functional values rather than the arithmetic mean am or the geometric mean gm averaging a comparison is made with the existing techniques and the results obtained show better approximations related to the accuracy level in favor of the hm strategy

integrable models from twisted half loop algebras this paper is devoted to the construction of new integrable quantum mechanical models based on certain subalgebras of the half loop algebra of various results about these subalgebras are proven by presenting them in the notation of the st petersburg school these results are then used to demonstrate the integrability and find the symmetries of two types of physical system twisted gaudin magnets and calogero type models of particles on several half lines meeting at a point

an extended weyl wigner transformation for special finite spaces abstract we extend the weyl wigner transformation to those particular degrees of freedom described by a finite number of states using a technique of constructing operator bases developed by schwinger discrete transformation kernels are presented instead of continuous coordinate momentum pair system and systems such as the one dimensional canonical continuous coordinate momentum pair system and the two dimensional rotation system are described by special limits expressions are explicitly given for the spin one half case

quantum twisted yangians symmetry baxterisation and centralizers based on the quantum twisted yangians integrable systems with special boundary conditions called soliton nonpreserving snp may be constructed in the present article we focus on the study of subalgebras of the quantum twisted yangians and we show that such a subalgebra provides an exact symmetry of the rational transfer matrix we discuss how the spectrum of a generic transfer matrix may be obtained by focusing only on two types of special boundaries it is also shown that the subalgebras emerging from the asymptotics of tensor product representations of the quantum twisted yangian turn out to be dual to the quantum brauer algebra to deal with general boundaries in the trigonometric case we propose a new algebra which also provides the appropriate framework for the baxterisation procedure in the snp case

contracted and expanded integrable structures we propose a generic framework to obtain certain types of contracted and centrally extended algebras this is based on the existence of quadratic algebras reflection algebras and twisted yangians naturally arising in the context of boundary integrable models a quite old misconception regarding the expansion of the e2 algebra into sl2 is resolved using the representation theory of the aforementioned quadratic algebras we also obtain centrally extended algebras associated with rational and trigonometric q deformed r matrices that are solutions of the yang baxter equation

commuting quantum traces the case of reflection algebras we formulate a systematic construction of commuting quantum traces for reflection algebras this is achieved by introducing two dual sets of generalized reflection equations with associated consistent fusion procedures products of their respective solutions yield commuting quantum traces

quantum symmetries and the weyl wigner product of group representations in the usual formulation of quantum mechanics groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex hilbert space in accordance with wigner s theorem in the phase space formulation they have real true unitary representations in the space of square integrable functions on phase space each such phase space representation is a weyl wigner product of the corresponding hilbert space representation with its contragredient and these can be recovered by factorizing the weyl wigner product however not every real unitary representation on phase space corresponds to a group of automorphisms so not every such representation is in the form of a weyl wigner product and can be factorized the conditions under which this is possible are examined examples are presented

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

schwinger and pegg barnett approaches and a relationship between angular and cartesian quantum descriptions ii phase spaces following the discussion in state space language presented in a preceding paper we work on the passage from the phase space description of a degree of freedom described by a finite number of states without classical counterpart to one described by an infinite and continuously labelled number of states with this it is possible to relate an original schwinger idea to the pegg barnett approach to the phase problem in phase space language this discussion shows that one can obtain the weyl wigner formalism for both cartesian and angular coordinates as limiting elements of the discrete phase space formalism

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

an algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group an algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group is formulated for the first time explicitly in detail for both the finite dimensional and infinite dimensional hilbert spaces for the finite dimensional hilbert space the su 2 algebraic representation is used while for the infinite dimensional hilbert space the heisenberg weyl algebraic representation is employed the fourier expansion technique is generalized to the operator space which is suitable for analysis of irregular spectra the polynomial operator basis is also used for the complement which is appropriate for analysis of some simple hamiltonians the proposed new approach is applied to solve the classical inverse sturm liouville problem and to study the problems of quantum regular and irregular spectra

fusion and analytical bethe ansatz for thea 1 1 open spin chain we generalize the fusion procedure for the a 1 1 open spin chain 2 and we show that the transfer matrix satisfies a crossing property we use these results to solve the a 1 1 open spin chain with uq su symmetry by means of the analytical bethe ansatz method our results coincide with the known ones obtained via the nested bethe ansatz

extended cahill glauber formalism for finite dimensional spaces i fundamentals the cahill glauber approach for quantum mechanics on phase space is extended to the finite dimensional case through the use of discrete coherent states all properties and features of the continuous formalism are appropriately generalized the continuum results are promptly recovered as a limiting case the jacobi theta functions are shown to have a prominent role in the context

continuum quantum systems as limits of discrete quantum systems i state vectors abstract dynamical systems on continuum hilbert spaces may be realized as limits of dynamical systems on discrete possibly finite dimensional hilbert spaces in this first of four papers on the topic the continuum and discrete spaces are interfaced to one another algebraically convergence of vectors is defined in such a way as to preserve inner products and a necessary and sufficient coordinate wise criterion for convergence is proved

the discrete harmonic oscillator harper s equation and the discrete fractional fourier transform certain solutions to harper s equation are discrete analogues of and approximations to the hermite gaussian functions they are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator and they lead to a definition of a discrete fractional fourier transform ft the discrete fractional ft is essentially the time evolution operator of the discrete harmonic oscillator

on the structure of quantum phase space the space of labels characterizing the elements of schwinger s basis for unitary quantum operators is endowed with a structure of symplectic type this structure is embodied in a certain algebraic cocycle whose main features are inherited by the symplectic form of classical phase space in consequence the label space may be taken as the quantum phase space it plays in the quantum case the same role played by phase space in classical mechanics some differences coming inevitably from its nonlinear character

a unified definition for the discrete time discrete frequency and discrete time frequency wigner distributions various discrete definitions of the wigner distribution wd for discrete time signals have been proposed in previous works the formulation developed in this paper leads to natural and unified definitions of discrete versions of the wd they are directly related to the continuous and preserve most of its properties the discretization is first considered in the time domain dtwd in the frequency domain dfwd and then in both domains simultaneously dtfwd in each case the aliasing problem is studied and generalized interpolation formulas allowing the reconstruction of the continuous wd are derived the dtfwd is particulary relevant for computer implementation of the wd

finite two dimensional oscillator i the cartesian model a finite two dimensional oscillator is built as the direct product of two finite one dimensional oscillators using the dynamical lie algebra su 2 x su 2 y the position space in this model is a square grid of points while the ordinary continuous two dimensional quantum oscillator has a symmetry algebra u 2 the symmetry algebra of the finite model is only u 1 x u 1 y because it lacks rotations in the position and momentum plane we show how to import an so 2 group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid we thus propose a finite analogue for fractional u 2 fourier transforms

numerical solution of the goursat problem on a triangular domain with mixed boundary conditions abstract for the goursat problem we consider a triangular domain with mixed dirichlet and impedance boundary conditions imposed on it we develop an algorithm for its numerical solution mainly based on runge kutta method and trapezoidal formula iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition error estimates are derived examples are presented to illustrate the effectiveness of the method

algebraic bethe ansatz for 19 vertex models with reflection conditions in this work we solve the 19 vertex models with the use of algebraic bethe ansatz for diagonal reflection matrices sklyanin k matrices the eigenvectors eigenvalues and bethe equations are given in a general form quantum spin chains of spin one derived from the 19 vertex models were also discussed

quantum spin chain with soliton non preserving boundary conditions we consider the case of an integrable quantum spin chain with soliton non preserving boundary conditions this is the first time that such boundary conditions have been considered in the spin chain framework we construct the transfer matrix of the model we study its symmetry and we find explicit expressions for its eigenvalues moreover we derive a new set of bethe ansatz equations by means of the analytical bethe ansatz method

nonlinear integral equations for complex affine toda models associated with simply laced lie algebras a set of coupled nonlinear integral equations nlie is derived for a class of models connected to the quantum group simply laced lie algebra which are solvable using the bethe ansatz these equations describe arbitrary excited states of a system with finite spatial length l they generalize the simpler nlie of the sine gordon massive thirring model to affine toda field theory with imaginary coupling constant as an application the central charge and all the conformal weights of the uv conformal field theory are extracted in a straightforward manner the quantum group truncation for q at a root of unity is discussed in detail in the uv limit we recover through this procedure the rcfts with extended conformal symmetry

construction of positive time frequency distributions a general method for constructing nonnegative definite joint time frequency distributions tfds satisfying the marginals of time s t sup 2 and frequency s f sup 2 is presented as nonnegative definite distributions with the correct marginals these tfds are members of the cohen posch class several examples illustrating properties of these tfds are presented for both synthetic and real signals

the canonical kravchuk basis for discrete quantum mechanics the well known kravchuk formalism of the harmonic oscillator obtained from the direct discretization method is shown to be a new way of formulating discrete quantum phase space it is shown that the kravchuk oscillator hamiltonian has a well defined unitary canonical partner which we identify with the quantum phase of the kravchuk oscillator the generalized discrete wigner function formalism based on the action and angle variables is applied to the kravchuk oscillator and its continuous limit is examined

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

discrete time discrete frequency time frequency analysis a formulation of a discrete time discrete frequency wigner distribution for analysis of discrete time periodic signals is given using an approach involving group representation theory this approach is motivated by a well known connection between group theory and the continuous wigner distribution the advantage of this approach is that the resulting discrete distribution satisfies mathematical properties analogous to those satisfied by the continuous distribution after outlining the relationship between group representation theory and time frequency analysis we derive the discrete distribution and exhibit many of its mathematical properties these include time and frequency marginals the weyl correspondence and covariance in particular the interpretation of covariance for the discrete distribution is shown to be different than that for the continuous distribution finally we note some unusual features of this discrete distribution which are a consequence of the group theoretic derivation

alias free generalized discrete time time frequency distributions a definition of generalized discrete time time frequency distribution that utilizes all of the outer product terms from a data sequence so that one can avoid aliasing is introduced the new approach provides 1 proper implementation of the discrete time spectrogram 2 correct evaluation of the instantaneous frequency of the underlying continuous time signal and 3 correct frequency marginal the formulation provides a unified framework for implementing members of cohen s class which was formulated in the continuous time domain some requirements for the discrete time kernel in the new approach are discussed in association with desirable distribution properties some experimental results are provided to illustrate the features of the proposed method

a numerical method for solving a goursat cauchy boundary value problem in this paper we propose a method to construct numerically the solution pair of a second order hyperbolic partial differential equation when two types of boundary conditions on a triangular domain are imposed specifically the pde is w x x x t w t t x t q x w x t email protected x l x t where the unknown functions are w x t and dq x one set of boundary conditions refer to one set of the triangle s boundaries we call them goursat type and the other set of boundary conditions both refer to the third boundary of the triangle we call them cauchy type to solve numerically this boundary value problem we make a change of dependent variables and place a mesh on the triangular domain with lines that are the characteristic curves of the hyperbolic pde then we shall integrate along the characteristic curves of the hyperbolic pde in such a way that numerical values of w x t and dq x will be produced at mesh nodes located on vertical lines the calculations will proceed from the vertical boundary of the domain going backwards to the origin o of the oxt coordinate system from one vertical line to another the trapezoidal formula will be used for numerical integration numerical examples will be provided

wigner distribution function for finite systems we construct a wigner distribution function for finite data sets it is based on a finite optical system a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry the dynamical group for this model is su 2 and the wave functions are sets of n 2l 1 data points the wigner distribution function assigns classical c numbers to the operators of position momentum and wave guide mode

fusion procedure for open chains the authors have generalized sklyanin s approach of constructing open integrable quantum spin chains to the case of pt invariant r matrices they formulate a fusion procedure for such chains in particular they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated applications of these results include constructing open integrable higher spin chains as well as obtaining functional equations for transfer matrix eigenvalues which may be solved by an analytical bethe ansatz

the angle angular momentum quantum phase space the angle angular momentum quantum phase space is considered and the corresponding heisenberg weyl group is studied when 2j 1 is a power of a prime z 2j 1 is a galois field and stronger results can be proved sl 2 z 2j 1 transformations are explicitly constructed and various properties of the displacement operators are studied central extensions of the abelian group by z 2j 1 are studied and they provide all the ways of constructing the heisenberg weyl group from g and z 2j 1

continuum quantum systems as limits of discrete quantum systems ii state functions in this second of four papers on the eponymous topic pointwise convergence of a discrete state function to a continuum state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel as examples and as a prerequisite for the sequels the normal approximation theorem and the convergence of the kravchuk functions to the hermite gaussians are expressed in terms of the algebraic notion of convergence

continuum quantum systems as limits of discrete quantum systems iii operators convergence of a discrete operator to a continuum operator is defined as examples the circular rotor the one dimensional box the harmonic oscillator and the fractional fourier transform are realized as limits of finite dimensional quantum systems limits thus defined preserve algebraic structure the results prepare for a sequel in which some affine canonical transforms will be discretized

commuting quantum traces for quadratic algebras consistent tensor products on auxiliary spaces hereafter denoted fusion procedures and commuting transfer matrices are defined for general quadratic algebras nondynamical and dynamical inspired by results on reflection algebras applications of these procedures then yield integer indexed families of commuting hamiltonians

integrable su 3 spin chain combining different representations the general expression for the local matrix of a quantum chain with the site space in any representation of su 3 is obtained this is made by generalizing from the fundamental representation and imposing the fulfilment of the yang baxter equation then a non homogeneous spin chain combining different representations of su 3 is solved by developing a method inspired in the nested bethe ansatz the solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled bethe equations a conjecture about the solution of a chain with the site states in different representations of su n is presented the thermodynamic limit of the ground state is calculated

new integrable quantum chains combining different kinds of spins a general procedure to generate new integrable hamiltonians combining to any kind of spins distributed arbitrarily on the line is given as a concrete application anisotropic chains formed by spin 1 2 and spin 1 operators at alternating sites are presented and solved exactly by the bethe ansatz ba the authors compute the ground state and excitation energies and momentum the higher order ba equations are derived depending on the choice these new hamiltonians exhibit or not conformal invariance in their low energy spectrum