superintegrability and higher order integrals for quantum systems we refine a method for finding a canonical form of symmetry operators of arbitrary order for the schrodinger eigenvalue equation h 2 v e on any 2d riemannian manifold real or complex that admits a separation of variables in some orthogonal coordinate system the flat space equations with potentials v x iy k 1 x iy k 1 in cartesian coordinates and v r2 r2cos 2k r2sin 2k the tremblay turbiner and winternitz system in polar coordinates have each been shown to be classically superintegrable for all rational numbers k we apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k we develop the classical analog of the quantum canonical form for a symmetry it is clear that our methods will generalize to other hamiltonian systems

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

on euler superintegrable systems we discuss the euler approach to the construction and investigation of the superintegrable systems with additional quadratic and cubic integrals of motions

second order superintegrable systems in conformally flat spaces ii the classical two dimensional stackel transform this paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in conformally flat spaces here we study the stackel transform or coupling constant metamorphosis as an invertible mapping between classical superintegrable systems on different spaces through the use of this tool we derive and classify for the first time all two dimensional 2d superintegrable systems the underlying spaces are exactly those derived by koenigs in his remarkable paper giving all 2d manifolds with zero potential that admit at least three second order symmetries our derivation is very simple and quite distinct we also show that every superintegrable system is the stackel transform of a superintegrable system on a constant curvature space

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

superintegrability and higher order polynomial algebras we present a method to obtain higher order integrals and polynomial algebras for two dimensional quantum superintegrable systems separable in cartesian coordinates from ladder operators all systems with a second and a third order integral of motion separable in cartesian coordinates were studied the integrals of motion of two of them do not generate a cubic algebra we construct for these hamiltonians a higher order polynomial algebra from their ladder operators we obtain quintic and seventh order polynomial algebras we also give for the polynomial algebras of order 7 realizations in terms of deformed oscillator algebras these realizations and finite dimensional unitary representations allow us to obtain the energy spectrum we also apply the construction to the caged anisotropic harmonic oscillator and a system involving the fourth painleve transcendent

riemann functions and the group e 1 1 historically lie algebras of first order symmetry operators have proven to be a useful method for finding equivalence classes of riemann functions here this idea is extended to higher order symmetries the approach is to seek self adjoint linear hyperbolic partial differential equations that separate variables in more than one coordinate system under the action of the group e 1 1 the equations derived admit no nontrivial first order operators and can only be obtained from second order symmetry operators using this symmetry structure a new equivalence class of riemann functions can then be found

perturbed harmonic oscillator ladder operators eigenenergies and eigenfunctions for the x2 λx2 1 gx2 interaction the perturbed ladder operators method is applied to the resolution of the perturbed harmonic oscillator wave equation for the case where the perturbation is expandable in a convergent series of hermite polynomials hk when v x b2x2 sigma kckhk b12 x it is found that the use of a hermite polynomials basis together with the use of binomial coefficient functions in the quantum number greatly simplifies the determination of the perturbed ladder and factorisation functions thus one obtains analytical expressions of the eigenenergies and eigenfunctions up to any order of the perturbation without increasing intricacy thorough calculation has been given for a perturbing potential v x function even in x as an illustrative application of the procedure the resolution of the schrodinger equation with a potential function v x x2 lambda x2 1 gx2 g 0 is reinvestigated

hill s determinant approach to single mode spin boson model an efficient and stable numerical scheme based on hill s determinant is proposed for computing the whole spectrum of the single mode spin boson model its shown that for the low lying levels a simple version of hill s determinant method works well while for high levels a normalization of the determinant is necessary and this normalization results in an entire function whose zeros correspond to the energy levels it is very convenient to use this entire function to compute all the energy levels numerically some high levels of the model are computed and compared with irish s general rotating wave approximation a scaling law in the large photon number limit is numerically tested and several statistical properties of the spectrum are investigated

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

invariant classification of second order conformally flat superintegrable systems in this paper we continue the work of kalnins et al in classifying all second order conformally superintegrable laplace type systems over conformally flat spaces using tools from algebraic geometry and classical invariant theory the results obtained show through stackel equivalence that the list of known nondegenerate superintegrable systems over three dimensional conformally flat spaces is complete in particular a seven dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems this manifold is foliated by the nonlinear action of the conformal group in three dimensions two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation this foliation is explicitly described using algebraic varieties formed from representations of the conformal group the proof of these results rely heavily on grobner basis calculations using the computer algebra software packages maple and singular

double exponential sinc collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function we show that the double exponential sinc collocation method provides an efficient uniformly accurate solution to the one dimensional time independent schrodinger equation for a general class of rational potentials of the form v x p x q x the derived algorithm is based on the discretization of the hamiltonian of the schrodinger equation using sinc expansions this discretization results in a generalized eigenvalue problem the eigenvalues of which correspond to approximations of the energy values of the starting hamiltonian a systematic numerical study is conducted beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree

composition diamond lemma for associative conformal algebras abstract we prove the composition diamond lemma for associative conformal algebras as some corollaries we prove that the word problem for some homogeneous associative conformal algebras is solvable while it is unsolvable in general

hamiltonians separable in cartesian coordinates and third order integrals of motion we present in this article all hamiltonian systems in e 2 that are separable in cartesian coordinates and that admit a third order integral both in quantum and in classical mechanics many of these superintegrable systems are new and it is seen that there exists a relation between quantum superintegrable potentials invariant solutions of the korteweg de vries equation and the painleve transcendents

wilson polynomials and the generic superintegrable system on the 2 sphere for complex two dimensional riemannian spaces every classical or quantum second order superintegrable system can be obtained from a single generic 3 parameter potential on the complex 2 sphere by delicate limit operations and through stackel transforms between manifolds here we derive families of finite and infinite dimensional irreducible representations of the corresponding quadratic quantum algebra for the 2 sphere and point out their role in explaining the degeneracy of the energy eigenspaces corresponding to bound state and continuous spectra of quantum and wave equation analogs of this system the algebra is exactly the one that describes the wilson and racah polynomials in their full generality

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

superintegrability and associated polynomial solutions euclidean space and the sphere in two dimensions in this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems we do this for the corresponding systems defined in euclidean space and on the two dimensional sphere we present all of these cases from a unified point of view in particular all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros the principal new results are the details of the polynomial bases for each of the nonsubgroup bases not just the subgroup cartesian and polar coordinate cases and the details of the structure of the quadratic algebras we also study the polynomial eigenfunctions in elliptic coordinates of the n dimensional isotropic quantum oscillator

elliptic basis for the zernike system heun function solutions the differential equation that defines the zernike system originally proposed to classify wavefront aberrations of the wavefields in the disk of a circular pupil had been shown to separate in three distinct coordinate systems obtained from polar coordinates on a half sphere here we find and examine the separation in the generic elliptic coordinate system on the half sphere and its projected disk where the solutions separated in jacobi coordinates contain heun polynomials

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

definite integrals as solutions for the x2 λx2 1 gx2 potential presents exact solutions and eigenvalues for the schrodinger equation with the x2 x2 1 gx2 interaction on the half axis x or 0 the solutions are given in the form of definite integrals and the eigenvalues by means of a well defined limiting procedure

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

on the hill determinant method the conditions of applicability of the hill determinant method to the potentials v r v1r v2r2 and v x x2 lambda x2 1 gx2 are investigated it is shown that the procedure leads to the actual eigenvalues in the former case provided the ansatz is properly chosen but it completely fails to yield the right answer in the latter one

supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion the main result of this article is that we show that from supersymmetry we can generate new superintegrable hamiltonians we consider a particular case with a third order integral and apply mielnik s construction in supersymmetric quantum mechanics we obtain a new superintegrable potential separable in cartesian coordinates with a quadratic and quintic integrals and also one with a quadratic integral and an integral of order of 7 we also construct a superintegrable system written in terms of the fourth painleve transcendent with a quadratic integral and an integral of order of 7

extended numerov method for computing eigenvalues of specific schrodinger equations numerov s method and an extended version of it are introduced for computing eigenvalues of schrodinger equations with potentials v x which are even functions with respect to x furthermore it is assumed that the wavefunctions tend to zero for x to or infinity the derived results are compared with previously derived numerical data and with available exact values

symmetries of the hydrogen atom and algebraic families we show how the schrodinger equation for the hydrogen atom in two dimensions gives rise to an algebraic family of harish chandra pairs that codifies hidden symmetries the hidden symmetries vary continuously between so 3 so 2 1 and the euclidean group o 2 r2 we show that the solutions of the schrodinger equation may be organized into an algebraic family of harish chandra modules furthermore we use jantzen filtration techniques to algebraically recover the spectrum of the schrodinger operator this is a first application to physics of the algebraic families of harish chandra pairs and modules developed in the work of bernstein et al int math res notices rny147 2018 rny146 2018 we show how the schrodinger equation for the hydrogen atom in two dimensions gives rise to an algebraic family of harish chandra pairs that codifies hidden symmetries the hidden symmetries vary continuously between so 3 so 2 1 and the euclidean group o 2 r2 we show that the solutions of the schrodinger equation may be organized into an algebraic family of harish chandra modules furthermore we use jantzen filtration techniques to algebraically recover the spectrum of the schrodinger operator this is a first application to physics of the algebraic families of harish chandra pairs and modules developed in the work of bernstein et al int math res notices rny147 2018 rny146 2018

maximal superintegrability on n dimensional curved spaces a unified algebraic construction of the classical smorodinsky winternitz systems on the nd sphere euclidean and hyperbolic spaces through the lie groups so n 1 iso n and so n 1 is presented firstly general expressions for the hamiltonian and its integrals of motion are given in a linear ambient space n 1 and secondly they are expressed in terms of two geodesic coordinate systems on the nd spaces themselves with an explicit dependence on the curvature as a parameter on the sphere the potential is interpreted as a superposition of n 1 oscillators furthermore each lie algebra generator provides an integral of motion and a set of 2n 1 functionally independent ones are explicitly given in this way the maximal superintegrability of the nd euclidean smorodinsky winternitz system is shown for any value of the curvature

applications of the differentiability of eigenvectors and eigenvalues to a perturbed harmonic oscillator the potential interaction x2 1 gx2 g 0 of the harmonic oscillator h0 d2 dx2 x2 considered as an operator in the space l2 is bounded this together with the nondegeneracy of the eigenvalues implies that the eigenvectors of the perturbed harmonic oscillator as functions of the parameters and g are strongly differentiable the eigenvalues are therefore differentiable functions for every real and every real g 0 in particular the first eigenvalue e1 as a function of is strictly concave e 1 0 emphasis is given to the inequality that follows from the strict concavity of the function e1

spherical geometry zernike s separability and interbasis expansion coefficients free motion on a 3 sphere properly projected on the 2 dimensional manifold of a disk yields the zernike system which exhibits the fundamental properties of superintegrability these include separability in a variety of coordinate systems polynomial solutions and a particular subset of clebsch gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the askey scheme deriving these results from the initial formulation in spherical geometry provides the zernike system with interest beyond its optical applications

integrable and superintegrable quantum systems in a magnetic field integrable quantum mechanical systems with magnetic fields are constructed in two dimensional euclidean space the integral of motion is assumed to be a first or second order hermitian operator contrary to the case of purely scalar potentials quadratic integrability does not imply the separation of variables in the schrodinger equation moreover quantum and classical integrable systems do not necessarily coincide the hamiltonian can depend on the planck constant in a nontrivial manner

interaction λx2 1 gx2 revisited abstract it is shown that the refined non perturbative hill s method solves in a very neat way the schrodinger equation e x 2 x 2 1 gx 2 0 the refinement of the standard hill s method consists in finding a renormalized frequency to optimize the calculation this method appears to be preferable to the variational one recently used by a k mitra in the same context

grobner shirshov bases for irreducible sln 1 modules abstract we determine the grobner shirshov bases for finite dimensional irreducible representations of the special linear lie algebra sl n 1 and construct explicit monomial bases for these representations we also show that each of these monomial bases is in 1 1 correspondence with the set of semistandard young tableaux of a given shape

superintegrable n 2 systems quadratic constants of motion and potentials of drach the properties of superintegrable systems in two degrees of freedom possessing three independent globally defined constants of motion are studied using as an approach the existence of hidden symmetries and the generalized noether s theorem the potentials are obtained as solution of a system of two partial differential equations first the case of standard lagrangians is studied and then the method is applied to the case of lagrangians with a pseudo euclidean kinetic term finally the results are related with other approaches and with a family of potentials admitting a second integral of motion cubic in the velocities obtained by drach

quantum superintegrable systems with quadratic integrals on a two dimensional manifold there are two classes of quantum integrable systems on a manifold with quadratic integrals the liouville and the lie integrable systems as it happens in the classical case the quantum liouville quadratic integrable systems are defined on a liouville manifold and the schrodinger equation can be solved by separation of variables in one coordinate system the lie integrable systems are defined on a lie manifold and are not generally separable ones but can be solved therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems the quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion there are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones analytic formulas for the involved integrals are calculated in all the cases all the known quan

the hill determinant method in application to the sextic oscillator limitations and improvement using the sextic anharmonic oscillator a chi 2 b chi 4 c chi 6 as a testing ground it is shown why the hill determinant method has a limited range of applicability a possible improvement is suggested and discussed

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

linear algebraic approach to grobner shirshov basis theory we construct a new efficient algorithm for finding grobner shirshov bases for noncommutative algebras and their representations this algorithm uses the macaulay matrix f s macaulay on some formula in elimination proc london math soc 33 1 1902 3 27 and can be viewed as a representation theoretic analogue of the f4 algorithm developed by j c faugere we work out some examples of universal enveloping algebras of lie algebras and of their representations to illustrate the algorithm

on the schrodinger equation for the interaction x2 λx2 1 gx2 by forming the 6 6 pade approximants to the energy perturbation series for the interaction x2 lambda x2 1 gx2 as obtained from the hypervirial relations the ground and the first three excited state energy levels are calculated for g or 2 the results of the present calculation are in good agreement with those of the variational calculations the authors also present another set of exact solutions to the schrodinger equation for the interaction x2 lambda x2 1 gx2

fine structure for 3d second order superintegrable systems three parameter potentials a classical or quantum superintegrable system of second order is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion quadratic in the momenta the maximum possible for n 3 on conformally flat spaces with nondegenerate i e four parameter potentials the extreme case we have worked out the structure and classified most of the possible spaces and potentials here we extend the analysis to a more degenerate class of functionally linearly independent superintegrable systems the three parameter potential case we show that for true three parameter potentials the algebra of constants of the motion no longer closes at order 6 but still all such systems are stackel transforms of systems on complex euclidean space or the complex 3 sphere this is a significant step towards the complete structure analysis of all types of second order superintegrable systems

on the schrodinger equation for the interaction x2 λx2 1 gx2 the method of obtaining the explicit expressions of all the even odd parity exact solutions of the schrodinger equation for the interaction x2 lambda x2 1 gx2 is discussed when the couplings lambda and g satisfy some specific relations in the general case a simple equation for approximating the energy eigenvalues has been developed

completeness of multiseparable superintegrability on the complex 2 sphere the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability here we demonstrate for nondegenerate potentials how to establish a complete list of such potentials that are superintegrable on the complex 2 sphere using essentially algebraic means we classify all such potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters the method of proof generalizes to other spaces and dimensions we show for the 2 sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

exact solvability of superintegrable benenti systems we establish quantum and classical exact solvability for two large classes of maximally superintegrable benenti systems in n dimensions with arbitrarily large n namely we solve the hamilton jacobi and schrodinger equations for the systems in question the results obtained are illustrated for a model with the cubic potential

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

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

quasi exact solvability on the sphere sn an exactly solvable es potential on the sphere sn is reviewed and a related quasi exactly solvable qes potential is found and studied after mapping the sphere to a simplex it is found that the metric of constant curvature is in polynomial form and both the es and the qes potentials are rational functions their hidden algebra is gln realized in a finite dimensional representation by first order differential operators acting on rpn it is shown that variables in the schrodinger eigenvalue equation can be separated in polyspherical coordinates and there is always complete integrability the qes system is completely integrable for n 2 and non maximally superintegrable for n 3 there is no separable coordinate system in which it is exactly solvable we point out that by taking contractions of superintegrable systems such as induced by inonu wigner lie algebra contractions we can find other qes superintegrable systems and we illustrate this by contracting our sn system to a qes non maximal sup

combined state adding and state deleting approaches to type iii multi step rationally extended potentials applications to ladder operators and superintegrability type iii multi step rationally extended harmonic oscillator and radial harmonic oscillator potentials characterized by a set of k integers m1 m2 mk such that m1 m2 mk with mi even resp odd for i odd resp even are considered the state adding and state deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators the eigenstates of the hamiltonians are shown to separate into mk 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebras these ladder operators are then used to build a higher order integral of motion for seven new infinite families of superintegrable two dimensional systems separable in cartesian coordinates the finite dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one dimensional hamiltonian eigenstates and provide an algebr

higher order superintegrability of separable potentials with a new approach to the post winternitz system the higher order superintegrability of separable potentials is studied it is proved that these potentials possess in addition to the two quadratic integrals a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions some systems related to the harmonic oscillator such as the generalized sw system and the ttw system were studied in previous papers now a similar analysis is presented for superintegrable systems related to the kepler problem in this way a new proof of the superintegrability of the post winternitz system is presented and the explicit expression for the integral is obtained finally the relations between the superintegrable systems with quadratic constants of motion separable in several different coordinate systems and the superintegrable systems with higher order constants of motion are analyzed

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

quantum integrals from coalgebra structure quantum versions of the hydrogen atom and the harmonic oscillator are studied on non euclidean spaces of dimension n integrals of arbitrary order are constructed via a multi dimensional version of the factorization method thus confirming the conjecture of riglioni 2013 j phys a math theor 46 265207 the systems are extended via coalgebra extension of representations although not all integrals are expressible in these generators as an example dimensional reduction is applied to four dimensional systems to obtain extension and new proofs of the superintegrability of known families of hamiltonians

shifted 1 n expansion approach to the interaction v r r2 λr2 1 gr2 the authors use the shifted 1 n expansion method to calculate the energy eigenvalues of the non polynomial oscillator v r r2 r2 1 gr2 the results thus obtained are in excellent agreement with exact numerical results it is also shown that the non polynomial interaction can be made supersymmetric and in this case also the results obtained by the shifted 1 n method matches extremely well with the exact analytic results given by supersymmetry

accurate calculation of the eigenvalues of the x2 λx2 1 gx2 potential pour calculer les valeurs propres de h on calcule trois quantites qui convergent vers ces valeurs propres la difference entre 2 de ces quantites est une mesure de l erreur faite dans le calcul

some observations on the nature of solutions for the interaction v x x2 λx2 1 gx2 the authors study the nature of exact solutions for the nonpolynomial interaction in particular it is shown that all the solutions are supersymmetric it is also shown that this problem admits a dynamical su 2 symmetry so that an arbitrarily large part of the spectrum but not the whole spectrum can be determined exactly

superintegrable deformations of superintegrable systems quadratic superintegrability and higher order superintegrability the superintegrability of four hamiltonians hr hr r a b c d where hr are known hamiltonians and is a certain function defined on the configuration space and depended on a parameter is studied the new hamiltonians and the associated constants of motion jri i 1 2 3 are continous functions of the parameter the first part is concerned with separability and quadratic superintegrability the integrals of motion are quadratic in the momenta and the second part is devoted to the existence of higher order superintegrability the results obtained in the second part are related with the tremblay turbiner winternitz and the post winternitz systems

nondegenerate three dimensional complex euclidean superintegrable systems and algebraic varieties a classical or quantum second order superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent second order constants of the motion polynomial in the momenta the maximum possible such systems have remarkable properties multi integrability and multiseparability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with quasiexactly solvable systems here we announce a complete classification of nondegenerate i e four parameter potentials for complex euclidean 3 space we characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints the euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation there are exactly ten nondegenerat

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

complete sets of invariants for dynamical systems that admit a separation of variables consider a classical hamiltonian h in n dimensions consisting of a kinetic energy term plus a potential if the associated hamilton jacobi equation admits an orthogonal separation of variables then it is possible to generate algorithmically a canonical basis q p where p1 h p2 pn are the other second order constants of the motion associated with the separable coordinates and qi qj pi pj 0 qi pj ij the 2n 1 functions q2 qn p1 pn form a basis for the invariants we show how to determine for exactly which spaces and potentials the invariant qj is a polynomial in the original momenta we shed light on the general question of exactly when the hamiltonian admits a constant of the motion that is polynomial in the momenta for n 2 we go further and consider all cases where the hamilton jacobi equation admits a second order constant of the motion not necessarily associated with orthogonal separable coordinates or even separable coordinates at all in each of these cases we construct an addition

isochronic potentials and new family of superintegrable systems a general scheme for constructing superintegrable systems for separated hamiltonians in an arbitrary number of degrees of freedom is presented the resulting family contains previously known superintgrable systems with separated hamiltonians in cartesian coordinates at least however in general the models belonging to the family admit additional integrals which are nonpolynomial functions of momenta an application of the method for the construction of superintegrable models of liouville type is described

grobner shirshov bases for lie superalgebras and their universal enveloping algebras abstract we show that a set of monic polynomials in a free lie superalgebra is a grobner shirshov basis for a lie superalgebra if and only if it is a grobner shirshov basis for its universal enveloping algebra we investigate the structure of grobner shirshov bases for kac moody superalgebras and give explicit constructions of grobner shirshov bases for classical lie superalgebras

nondegenerate 2d complex euclidean superintegrable systems and algebraic varieties a classical or quantum superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion polynomial in the momenta the maximum possible if the constants are all quadratic the system is second order superintegrable the kepler coulomb system is the best known example such systems have remarkable properties multi integrability and multi separability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with qes systems for complex riemannian spaces with n 2 the structure and classification of second order superintegrable systems is complete here however we present a new and conceptually simpler approach to the classification for complex euclidean 2 space in which the possible superintegrable systems with nondegenerate potentials correspond to points on an algebraic variety specifically we determine a variety in six variables subject to two cubic and one quartic polynomial constraints each point on the variety corresponds to a superintegrable system the euclidean group e 2 acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

superintegrability in a two dimensional space of nonconstant curvature a hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion this property has been extensively studied in the case of two dimensional spaces of constant possibly zero curvature when all the independent integrals are either quadratic or linear in the canonical momenta in this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions this is done by examining in detail one of the spaces of revolution found by g koenigs we determine that there are essentially three distinct potentials which when added to the free hamiltonian of this space have this type of superintegrability separation of variables for the associated hamilton jacobi and schrodinger equations is discussed the classical and quantum quadratic algebras associated with each of these potentials are determined

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

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

conformal laplace superintegrable systems in 2d polynomial invariant subspaces 2nd order conformal superintegrable systems in n dimensions are laplace equations on a manifold with an added scalar potential and independent 2nd order conformal symmetry operators they encode all the information about helmholtz eigenvalue superintegrable systems in an efficient manner there is a 1 1 correspondence between laplace superintegrable systems and stackel equivalence classes of helmholtz superintegrable systems in this paper we focus on superintegrable systems in two dimensions n 2 where there are 44 helmholtz systems corresponding to 12 laplace systems for each laplace equation we determine the possible two variate polynomial subspaces that are invariant under the action of the laplace operator thus leading to families of polynomial eigenfunctions we also study the behavior of the polynomial invariant subspaces under a stackel transform the principal new results are the details of the polynomial variables and the conditions on parameters of the potential corresponding to polynomial solutions the hidden gl 3 algebraic structure is exhibited for the exact and quasi exact systems for physically meaningful solutions the orthogonality properties and normalizability of the polynomials are presented as well finally for all helmholtz superintegrable solvable systems we give a unified construction of one dimensional 1d and two dimensional 2d quasi exactly solvable potentials possessing polynomial solutions and a construction of new 2d pt symmetric potentials is established

some solutions of a supersymmetric nonpolynomial oscillator a comparison between the swkb and wkb methods it is shown that the non polynomial oscillator represented by v x x2 lambda x2 1 gx2 can be given a supersymmetric form if the parameter lambda satisfies a certain constraint in this case one finds an exact analytic expression for the ground state excited states corresponding to this potential are computed for several values of g and lambda using the supersymmetric wkb method the ordinary wkb method and also numerical integration the results are compared

second order superintegrable systems in conformally flat spaces i two dimensional classical structure theory this paper is part of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in real or complex conformally flat spaces here we consider classical superintegrable systems with nondegenerate potentials in three dimensions we show that there exists a standard structure for such systems based on the algebra of 3 3 symmetric matrices and that the quadratic algebra always closes at order 6 we show that the spaces of truly second third fourth and sixth order constants of the motion are of dimension 6 4 21 and 56 respectively and we construct explicit bases for the fourth and sixth order constants in terms of products of the second order constants

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

unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold in this paper we prove that the two dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the poisson algebra of the integrals of motion there are six general fundamental classes of superintegrable systems analytic formulas for the involved integrals are calculated in all the cases all the known superintegrable systems are classified as special cases of these six general classes

the schrodinger equation for the interaction potential x2 λx2 1 gx2 and the first heun confluent equation the authors present systematically the solutions and the quasi polynomial solutions for the schrodinger equation with the x2 lambda x2 1 gx2 interaction potential with the help of the first heun confluent equation or spheroidal heun equation

completeness of multiseparable superintegrability in e2 c the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability examples of this type of system are well known in this paper we demonstrate how to establish a complete list of such potentials using essentially algebraic means our approach is to classify all nondegenerate potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters this is carried out for two dimensional complex euclidean space though the method generalizes to other spaces and dimensions we show that all these superintegrable systems correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

quantum superintegrable zernike system we consider the differential equation that zernike proposed to classify aberrations of wavefronts in a circular pupil whose value at the boundary can be nonzero on this account the quantum zernike system where that differential equation is seen as a schrodinger equation with a potential is special in that it has a potential and a boundary condition that are not standard in quantum mechanics we project the disk on a half sphere and there we find that in addition to polar coordinates this system separates into two additional coordinate systems non orthogonal on the pupil disk which lead to schrodinger type equations with poschl teller potentials whose eigen solutions involve legendre gegenbauer and jacobi polynomials this provides new expressions for separated polynomial solutions of the original zernike system that are real the operators which provide the separation constants are found to participate in a superintegrable cubic higgs algebra

on the simultaneous eigenproblem for the x2 λ x2 1 gx2 1 interaction extension of gallas results an ansatz for the eigenfunctions of the x2 x2 1 gx2 1 interaction has been recently introduced by blecher and leach using this ansatz gallas has obtained five solution pairs for 2g 3g 2 gallas results are generalised and extended here

maximal superintegrability of benenti systems for a class of hamiltonian systems naturally arising in the modern theory of separation of variables we establish their maximal superintegrability by explicitly constructing the additional integrals of motion

superintegrable systems in darboux spaces almost all research on superintegrable potentials concerns spaces of constant curvature in this paper we find by exhaustive calculation all superintegrable potentials in the four darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta in addition to the hamiltonian these are two dimensional spaces of nonconstant curvature it turns out that all of these potentials are equivalent to superintegrable potentials in complex euclidean 2 space or on the complex 2 sphere via coupling constant metamorphosis or equivalently via stackel multiplier transformations we present a table of the results

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

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

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

universal integrals for superintegrable systems on n dimensional spaces of constant curvature an infinite family of classical superintegrable hamiltonians defined on the n dimensional spherical euclidean and hyperbolic spaces are shown to have a common set of 2n 3 functionally independent constants of the motion among them two different subsets of n integrals in involution including the hamiltonian can always be explicitly identified as particular cases we recover in a straightforward way most of the superintegrability properties of the smorodinsky winternitz and generalized kepler coulomb systems on spaces of constant curvature and we introduce as well new classes of quasi maximally superintegrable potentials on these spaces results presented here are a consequence of the poisson coalgebra symmetry of all the hamiltonians together with an appropriate use of the phase spaces associated with poincare and beltrami coordinates