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

poincare group and relativistic wave equations in dimensions using the generalized regular representation an explicit construction of the unitary irreducible representations of the 2 1 poincare group is presented a detailed description of the angular momentum and spin in 2 1 dimensions is given on this base the relativistic wave equations for all spins including fractional are constructed

exact quantization of a pt symmetric reversible lienard type nonlinear oscillator we carry out an exact quantization of a pt symmetric reversible lienard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically the associated time independent classical hamiltonian is of nonstandard type and is invariant under a combined coordinate reflection and time reversal transformation we use the von roos symmetric ordering procedure to write down the appropriate quantum hamiltonian while the quantum problem cannot be tackled in coordinate space we show how the problem can be successfully solved in momentum space by solving the underlying schrodinger equation therein we explicitly obtain the eigenvalues and eigenfunctions in momentum space and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator which is also confirmed by a semiclassical modified bohr sommerfeld quantization rule while the eigenfunctions are completely different

a new effective mass hamiltonian and associated lame equation bound states a new quantum model with rational functions for the potential and effective mass is proposed in a stretchable region outside which both are constant starting from a generalized effective mass kinetic energy operator the matching and boundary conditions for the envelope wavefunctions are derived it is shown that in a mapping to an auxiliary constant mass schrodinger picture one obtains one period associated lame well bounded by two wells or barriers depending on the values of the ordering parameter the results for bound states of this new solvable model are provided for a wide variation of the parameters

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

kernel methods in machine learning we review machine learning methods employing positive definite kernels these methods formulate learning and estimation problems in a reproducing kernel hilbert space rkhs of functions defined on the data domain expanded in terms of a kernel working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions the latter include nonlinear functions as well as functions defined on nonvectorial data we cover a wide range of methods ranging from binary classifiers to sophisticated methods for estimation with structured data

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

exact time dependence of solutions to the time dependent schrodinger equation solutions of the schrodinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time independent operators using time dependent unitary transformations exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time dependent hamiltonians in d dimensions including non hermitean symmetric hamiltonians hamiltonians are constructed using time dependent unitary spatial transformations comprising dilatations translations and rotations and solutions are found in several forms as eigenfunctions of a quadratic invariant as coherent state eigenfunctions of boson operators as plane wave solutions from which the general solution is obtained as an integral transform by means of the fourier transform and as distributional solutions for which the initial wavefunction is the dirac function for the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d 1 including barut girardello and perelomov coherent states i e vector coherent states which are shown to be related to eigenfunctions of the quadratic invariant by the transformation this transformation which leaves the ermakov equation invariant implements su 1 1 transformations on linear dynamical invariants coherent states are derived also for the time dependent linear potential exact solutions are found for hamiltonians with electromagnetic interactions in which the time dependent magnetic and electric fields are not necessarily spatially uniform as an example it is shown how to find exact solutions of the time dependent schrodinger equation for the dirac magnetic monopole in the presence of time dependent magnetic and electric fields of a specified form

jacobi polynomials bernstein type inequalities and dispersion estimates for the discrete laguerre operator abstract the present paper is about bernstein type estimates for jacobi polynomials and their applications to various branches in mathematics this is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of schrodinger equations whose hamiltonian is given by the generalized laguerre operator more precisely we show that dispersive estimates for the schrodinger equation associated with the generalized laguerre operator are connected with bernstein type inequalities for jacobi polynomials we use known uniform estimates for jacobi polynomials to establish some new dispersive estimates in turn the optimal dispersive decay estimates lead to new bernstein type inequalities

an exactly solvable schrodinger equation with finite positive position dependent effective mass the solution of the one dimensional schrodinger equation is discussed in the case of position dependent mass the general formalism is specified for potentials that are solvable in terms of generalized laguerre polynomials and mass functions that are positive and bounded on the whole real x axis the resulting four parameter potential is shown to belong to the class of implicit potentials closed expressions are obtained for the bound state energies and the corresponding wave functions including their normalization constants the constant mass case is obtained by a specific choice of the parameters it is shown that this potential contains both the harmonic oscillator and the morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials possible generalizations of the method are outlined

analytic solutions darboux transformation operators and supersymmetry for a generalized one dimensional time dependent schrodinger equation abstract in this paper analytically investigated is a generalized one dimensional time dependent schrodinger equation using darboux transformation operator technique we construct the first order darboux transformation and the real valued condition of transformed potential for the generalized schrodinger equation to prove the equivalence of the supersymmetry formalism and the darboux transformation we investigate the relationship among first order darboux transformation supersymmetry and factorization of the corresponding effective mass hamiltonian furthermore the nth order darboux transformations are constructed by means of different method finally by using darboux transformation some analytical solutions are generated in a recursive manner for some examples of the schrodinger equation

meixner pollaczek polynomials and the heisenberg algebra an alternative proof is given for the connection between a system of continuous hahn polynomials and identities for symmetric elements in the heisenberg algebra which was first observed by bender mead and pinsky phys rev lett 56 2445 1986 j math phys 28 509 1987 the continuous hahn polynomials turn out to be meixner pollaczek polynomials use is made of the connection between laguerre polynomials and meixner pollaczek polynomials the rodrigues formula for laguerre polynomials an operational formula involving meixner pollaczek polynomials and the schrodinger model for the irreducible unitary representations of the three dimensional heisenberg group

remarks on the solution of the position dependent mass schrodinger equation an approximate method is proposed to solve the position dependent mass pdm schrodinger equation the procedure suggested here leads to the solution of the pdm schrodinger equation without transforming the potential function to the mass space or vice versa the method based on the asymptotic taylor expansion of the function produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the pdm schrodinger equation the results show that the pdm and constant mass schrodinger equations are not isospectral the calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm

a new pt symmetric complex hamiltonian with a real spectrum we construct an isospectrum system in terms of a real and a complex potential to show that the underlying pt symmetric complex hamiltonian possesses a real spectrum which is shared by its real partner

generalized laguerre polynomials with position dependent effective mass visualized via wigner s distribution functions we construct analytically and numerically the wigner distribution functions for the exact solutions of the position dependent effective mass schrodinger equation for two cases belonging to the generalized laguerre polynomials using a suitable quantum canonical transformation expectation values of position and momentum operators are obtained analytically in order to verify the universality of heisenberg s uncertainty principle

ordering ambiguity versus representation in this work we show that the ordering ambiguity on quantization depends on the representation choice this property is then used to solve unambiguously some particular systems finally we speculate on the consequences for more involved cases

exactly solvable position dependent mass hamiltonians related to non compact semi simple lie groups we suggest a generalized procedure to obtain exactly solvable position dependent mass hamiltonians in one dimension the second order casimir invariant of the regular representation of a non compact semi simple lie group g the spectral properties of which are well known is used to introduce exactly solvable hamiltonians a brief description of the procedure is presented and its application to quantum systems associated with sl 2 r is detailed

discrete repulsive oscillator wavefunctions for the study of infinite discrete systems on phase space the three dimensional lorentz algebra and group so 2 1 and so 2 1 provide a discrete model of the repulsive oscillator its eigenfunctions are found in the principal irreducible representation series where the compact generator that we identify with the position operator has the infinite discrete spectrum of the integers z while the spectrum of energies is a double continuum the right and left moving wavefunctions are given by hypergeometric functions that form a dirac basis for 2 z under contraction the discrete system limits to the well known quantum repulsive oscillator numerical computations of finite approximations raise further questions on the use of dirac bases for infinite discrete systems

the unitary irreducible representations of sl 2 r in all subgroup reductions we use the canonical transform realization of sl 2 r in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group explicit results are given for all mixed bases and subgroup reductions these provide the full multiparameter set of integral transforms and series expansions associated to sl 2 r

the metaplectic group within the heisenberg weyl ring the heisenberg weyl ring contains the metaplectic group of canonical transforms acting unitarily on l 2 r these ring elements are characterized through i the integral transform kernels ii coset distributions and iii classical functions under any quantization scheme the isomorphism under group composition leads to several new relations involving twisted products and quantization of gaussian classical functions the wigner inversion operator is a special central group element it is shown that the only quantization scheme invariant under metaplectic transformations is the weyl scheme the structure studied here appears to be relevant to the study of wave optics with aberration

an exactly soluble schrodinger equation with smooth position dependent mass the one dimensional generalized schrodinger equation for a system with smooth potential and mass step is resolved exactly the wave function depends on the heun s function which is a solution of a second order fuchsian equation with four singularities the behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step two limiting cases are also studied when the width of the mass step is vanishing and when the smooth potential and mass step tend to an abrupt potential and mass step

numerical solution of a thermoviscoelastic contact problem by a penalty method we consider the numerical approximation of a one dimensional quasi static contact problem in linear thermoviscoelasticity a finite element approximation based on a penalized problem is proposed and analyzed we furnish an a priori estimate of the difference between the true and numerical solutions the results of some computations are also presented

the exact solution of the schrodinger equation with a polynomially spatially varying mass the schrodinger equation with a position dependent mass sepdm is employed in many areas of quantum physics exact solutions for the sepdm lie at the center of interest of the professional public because it helps us to understand the behavior of quantum particles in the cases in which their mass varies spatially for this purpose we used the mass function represented by a quartic polynomial and a quadratic potential function which extends the current class of exact solutions of the sepdm the exact analytical solution of the problem is expressed as a linear combination of local heun functions heun s equation contains many parameters resulting in its general nature we studied how limit changes in some of these parameters will affect the solution of the sepdm the obtained solutions are particularly suitable for the transfer matrix method and solutions of scattering problems this is demonstrated by the calculation of bound states the schrodinger equation with a position dependent mass sepdm is employed in many areas of quantum physics exact solutions for the sepdm lie at the center of interest of the professional public because it helps us to understand the behavior of quantum particles in the cases in which their mass varies spatially for this purpose we used the mass function represented by a quartic polynomial and a quadratic potential function which extends the current class of exact solutions of the sepdm the exact analytical solution of the problem is expressed as a linear combination of local heun functions heun s equation contains many parameters resulting in its general nature we studied how limit changes in some of these parameters will affect the solution of the sepdm the obtained solutions are particularly suitable for the transfer matrix method and solutions of scattering problems this is demonstrated by the calculation of bound states

algebraic solution for the natanzon confluent potentials an algebraic method is used to solve the bound state problem for the most general natanzon potential for which the schrodinger equation can be reduced to the confluent hypergeometric form hence confluent potentials the solution is obtained straightforwardly a simple argument is given to sustain that the natanzon potential is the most general confluent potential

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

simple bound state energy spectra solutions of dirac equation for hulthen and eckart potentials within the framework of simple similarity transformation method we present simple schrodinger like equation for dirac equation with hulthen and eckart lorentz vectors potentials using the asymptotic iteration method we have managed to obtain the explicit analytical solution of the bound state energy spectra as well as the eigenfunctions for both potentials discussion related to the eigenfunction for both cases is also considered we have shown that the wave function is sensitive for both potentials as well as the parameter

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

the schrodinger equation with position dependent mass it is shown that the traditional hartree second order shooting method for eigenvalues can be modified to give accurate energy levels for a schrodinger equation involving a position dependent mass some numerical applications are given to demonstrate the effectiveness of the method

laplace and segal bargmann transforms on hermitian symmetric spaces and orthogonal polynomials let d g k be a complex bounded symmetric domain of tube type in a complex jordan algebra v and let dr j d d be its real form in a formally real euclidean jordan algebra j v dr h l is a bounded realization of the symmetric cone in j we consider representations of h that are gotten by the generalized segal bargmann transform from a unitary g space of holomorphic functions on d to an l2 space on dr we prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the laplace transform we find the extension to d of the spherical functions on dr and find their expansion in terms of the l spherical polynomials on d which are jack symmetric polynomials we prove that the coefficients are orthogonal polynomials in an l2 space the measure being the harish chandra plancherel measure multiplied by the symbol of the berezin transform we prove the difference equation and recurrence relation for those polynomials by considering the action of the lie algebra and the cayley transform on the polynomials on d finally we use the laplace transform to study generalized laguerre functions on symmetric cones

solvable potentials generated by sl 2 r canonical transformations are used to build realisations of sl 2 r in terms of the basic quantum mechanical operators q and p the results are used to construct solvable potentials in the framework of one dimensional quantum mechanics

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

remarks on a dynamical higher order theory of laminated plates and its application in random vibration response abstract this paper presents an analysis of the equations governing the dynamics of sheardeformable composite plates without recourse to a variational procedure it is noted that the operator associated with the governing equations is nonsymmetric using a first order perturbation technique it is shown to be positive definite in addition using the biorthogonality condition the dynamic response of the plate is formulated

isospectral hamiltonian for position dependent mass for an arbitrary quantum system and coherent states by means of the unitary transformation a new way for discussing the ordering prescription of the schrodinger equation with a position dependent mass pdm for isospectral hamiltonian operators is presented we show that the ambiguity parameter choices in the kinetic part of the hamiltonian can be explained through an exact susy qm symmetry as well as a consequence of an accidental symmetry under the z2 action by making use of the unitary transformation we construct coherent states for a family of pdm isospectral hamiltonians from a suitable choice of ladder operators we show that these states preserve the usual structure of klauder perelomov s states and thus saturate and minimize the position momentum uncertainty relation pmur under some special restrictions we show that pmur properties can be used to determine the sign of the superpotential

a unified treatment of exactly solvable and quasi exactly solvable quantum potentials by exploiting the hidden algebraic structure of the schrodinger hamiltonian namely the sl 2 we propose a unified approach of generating both exactly solvable and quasi exactly solvable potentials we obtain in this way two new classes of quasi exactly solvable systems one of which is of periodic type while the other hyperbolic

a squeeze like operator approach to position dependent mass in quantum mechanics we provide a squeeze like transformation that allows one to remove a position dependent mass from the hamiltonian methods to solve the schrodinger equation may then be applied to find the respective eigenvalues and eigenfunctions as an example we consider a position dependent mass that leads to the integrable morse potential and therefore to well known solutions

approximation of smooth functions on compact two point homogeneous spaces estimates of kolmogorov n widths dn bpr lq and linear n widths n bpr lq 1 q of sobolev s classes bpr r 0 1 p on compact two point homogeneous spaces ctphs are established for part of p q 1 1 sharp orders of dn bpr lq or n bpr lq were obtained by bordin et al j funct anal 202 2 2003 307 in this paper we obtain the sharp orders of dn bpr lq and n bpr lq for all the remaining p q our proof is based on positive cubature formulas and marcinkiewicz zygmund type inequalities on ctphs

super exponential decay rates for eigenvalues and singular values of integral operators on the sphere abstract this paper brings results about the behavior of sequences of eigenvalues or singular values of integral operators generated by square integrable kernels on the real m dimensional unit sphere m 2 under smoothness assumptions on the generating kernels given via laplace beltrami differentiability we obtain super exponential decay rates for the eigenvalues of the generated positive integral operators and for singular values of those integral operators which are non positive we show an optimal type result and provide a list of parametric families of kernels which are of interest for numerical analysis and geostatistical communities and satisfy the smoothness assumptions for the positive case

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

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

orthogonal stochastic duality functions from lie algebra representations we obtain stochastic duality functions for specific markov processes using representation theory of lie algebras the duality functions come from the kernel of a unitary intertwiner between representations which provides generalized orthogonality relations for the duality functions in particular we consider representations of the heisenberg algebra and mathfrak su 1 1 both cases lead to orthogonal self duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes

the clebsch gordan coefficients of the three dimensional lorentz algebra in the parabolic basis starting from the oscillator representation of the three dimensional lorentz algebra so 2 1 we build a lie algebra of second order differential operators which realizes all series of self adjoint irreducible representations the choice of the common self adjoint extention over a two chart function space determines whether they lead to single or multivalued representations over the corresponding lie group the diagonal operator defining the basis is the parabolic subgroup generator the direct product of two such algebras allows for the calculation of all clebsch gordan coefficients explicitly as solutions of schrodinger equations for poschl teller potentials over one d d two d c or three c c charts all coefficients are given in terms of up to two 2f1 hypergeometric functions

single variable realisation of the su 1 1 spectrum generating algebra and discrete eigenvalue spectra of a class of potentials a wide range of operators can be expressed in terms of the generators of the su 1 1 algebra the spectral properties of which are well known we attempt to find a realisation of the algebra in its most general form and thus evolve a unified approach to the problem of finding the spectra of hamiltonians amenable to the technique rather than taking each case separately we extend the analysis to obtain the spectra of potentials whose coordinate dependence is implicit

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

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

a lie algebraic approach to effective mass schrodinger equations we use lie algebraic techniques to obtain exact solutions of the effective mass schrodinger equation in particular we use the su 1 1 algebra both as a spectrum generating algebra and as a potential algebra to obtain exact solutions of effective mass schrodinger equations corresponding to a number of potentials we also discuss the construction of isospectral hamiltonians for which both the mass and the potential are different

a numerical method for a nonlinear system of integro differential equations arising in thermoviscoelasticity this paper deals with the numerical solution of nonlinear integro differential equations modelling a one dimensional quasi static contact problem in thermoviscoelasticity a finite element approximation is proposed and analysed and some numerical results are given

the gel fand realization and the exceptional representations of sl 2 r it is shown that the canonical representation space of gel fand and co workers is particularly appropriate for problems requiring explicit reduction under the noncompact so 1 1 and e 1 bases for both the principal and exceptional series of representations of sl 2 r we use this realization to set up complete orthonormal sets of eigendistributions corresponding to the three subgroup reductions namely sl 2 r so 1 1 sl 2 r e 1 and sl 2 r so 2 and evaluate the unitary transformations connecting these reductions these overlap matrix elements appear as the applications of these distributions to a set of well defined test functions using the rigorous theory of analytic continuation we show that the results for the exceptional representations have the same analytic forms as the corresponding results for the principal series some of these results are essential prerequisites for the solution of the clebsch gordan problem series and coefficients of sl 2 r in the so 1 1 basis

error analysis for a finite element approximation of a thermoviscoelastic contact problem in this article a finite element approximation based on a variational inequality to the solution of a one dimensional quasi static signorini contact problem in linear thermoviscoelasticity is proposed stability and error estimates are obtained

on the solutions of the position dependent effective mass schrodinger equation of a nonlinear oscillator related with the isotonic oscillator we study exact solutions of the position dependent effective mass schrodinger equation by considering the new solvable nonlinear oscillator that relates to the isotonic oscillator through the method of point canonical transformations we construct exactly solvable potentials by choosing physically important position dependent mass distributions we also provide the energy spectrum of the bound states and the wavefunctions of the solvable potentials

potentials generated by su 1 1 a systematic procedure for deriving a class of potentials with the underlying symmetry group su 1 1 starting the commutation relations for the generators of su 1 1 is presented