noncommutative fiber products and lattice models abstract we establish a connection between the representation theory of certain noncommutative singular varieties and two dimensional lattice models specifically we consider noncommutative biparametric deformations of the fiber product of two kleinian singularities of type a special examples are closely related to the schrodinger algebra s 1 the affine lie algebra a 1 1 and a finite w algebra associated to sl 4 the algebras depend on two scalars and two polynomials that must satisfy the mazorchuk turowska equation mte which we re interpret as a quantization of the ice rule local current conservation in statistical mechanics solutions to the mte previously classified by the author and d rosso can accordingly be expressed in terms of multisets of higher spin vertex configurations on a twisted cylinder we first reduce the problem of describing the category of weight modules to the case of a single configuration l secondly we classify all simple weight modules over the corresponding algebras a l in terms of the connected components of the cylinder minus l lastly we prove that a l are crystalline graded rings as defined by nauwelaerts and van oystaeyen and describe the center of a l explicitly in terms of l along the way we prove several new results about twisted generalized weyl algebras and their simple weight modules

parametric symmetries in exactly solvable real and pt symmetric complex potentials in this paper we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex pt symmetric potentials we focus our attention on the conventional potentials such as the generalized poschl teller gpt scarf i and pt symmetric scarf ii which are invariant under certain parametric transformations the resulting set of potentials is shown to yield a completely different behavior of the bound state solutions further the supersymmetric partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and pt symmetric complex potentials these potentials are also observed to be shape invariant si in nature we subsequently take up a study of the newly discovered rationally extended si potentials corresponding to the above mentioned conventional potentials whose bound state solutions are associated with the exceptional orthogonal polynomials eops we discuss the transformations of the corresponding

supersymmetric quantum mechanics engineered hierarchies of integrable potentials and related orthogonal polynomials within the context of supersymmetric quantum mechanics and its related hierarchies of integrable quantum hamiltonians and potentials a general programme is outlined and applied to its first two simplest illustrations going beyond the usual restriction of shape invariance for intertwined potentials it is suggested to require a similar relation for hamiltonians in the hierarchy separated by an arbitrary number of levels n by requiring further that these two hamiltonians be in fact identical up to an overall shift in energy a periodic structure is installed in the hierarchy which should allow for its resolution specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated while the methods of supersymmetric quantum mechanics then lead to generalised rodrigues formulae and recursion relations for such polynomials the approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra in this paper these ideas are presented and solved explicitly for the cases n 1 and n 2 the latter case is related to the generalised laguerre polynomials for which indeed new results are thereby obtained in the context of dressing chains and deformed polynomial heisenberg algebras some partial results for n 3 also exist in the literature which should be relevant to a complete study of the n 3 general periodic hierarchies

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

construction of classical superintegrable systems with higher order integrals of motion from ladder operators we construct integrals of motion for multidimensional classical systems from ladder operators of one dimensional systems this method can be used to obtain new systems with higher order integrals we show how these integrals generate a polynomial poisson algebra we consider a one dimensional system with third order ladder operators and found a family of superintegrable systems with higher order integrals of motion we obtain also the polynomial algebra generated by these integrals we calculate numerically the trajectories and show that all bounded trajectories are closed

new families of isospectral hydrogen like potentials by applying algebraic techniques we construct a two parametric family of strictly isospectral hydrogen like potentials as well as some of its one parametric limits an additional one parametric almost isospectral family of hydrogen like potentials is also investigated it is argued that the construction of a susy partner hamiltonian using a factorization energy less than the ground state energy of the departure hamiltonian is unnecessarily restrictive

solutions to the painleve v equation through supersymmetric quantum mechanics in this paper we shall use the algebraic method known as supersymmetric quantum mechanics susy qm to obtain solutions to the painleve v pv equation a second order nonlinear ordinary differential equation for this purpose we will apply first the susy qm treatment to the radial oscillator in addition we will revisit the polynomial heisenberg algebras phas and we will study the general systems ruled by them for first order phas we obtain the radial oscillator while for third order phas the potential will be determined by solutions to the pv equation this connection allows us to introduce a simple technique for generating solutions of the pv equation expressed in terms of confluent hypergeometric functions finally we will classify them into several solution hierarchies

classical ladder operators polynomial poisson algebras and classification of superintegrable systems we recall results concerning one dimensional classical and quantum systems with ladder operators we obtain the most general one dimensional classical systems respectively with a third and a fourth order ladder operators satisfying polynomial heisenberg algebras these systems are written in terms of the solutions of quartic and quintic equations they are the classical equivalent of quantum systems involving the fourth and fifth painleve transcendents we use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of lissajous s figures

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

darboux transformations and the symmetric fourth painleve equation this paper is concerned with the group symmetries of the fourth painlev equation piv a second order nonlinear ordinary differential equation it is well known that the parameter space of piv admits the action of the extended affine weyl group as shown by noumi and yamada the action of as b cklund transformations of piv provides a derivation of its symmetric form sp4 the dynamical system sp4 is also equivalent to the isomonodromic deformation of an associated three by three matrix linear system lax pair the action of the generators of on this lax pair is derived using the darboux transformation for an associated third order operator

factorization of operators ii we extend the methods of a previous paper j math phys 21 2508 1980 factorizing the general scalar third order differential operator and obtain a miura transformation for the boussinesq equation we give a general factorized eigenvalue problem we also give a hamiltonian structure associated with the factorized eigenvalue problem we derive several isospectral flows some of klein gordon type

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

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

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

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

new ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two dimensional systems new ladder operators are constructed for a rational extension of the harmonic oscillator associated with type iii hermite exceptional orthogonal polynomials and characterized by an even integer m the eigenstates of the hamiltonian separate into m 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebra these ladder operators are used to construct a higher order integral of motion for two superintegrable two dimensional systems separable in cartesian coordinates the polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite dimensional unitary irreducible representations

new families of superintegrable systems from hermite and laguerre exceptional orthogonal polynomials in recent years many exceptional orthogonal polynomials eop were introduced and used to construct new families of 1d exactly solvable quantum potentials some of which are shape invariant in this paper we construct from hermite and laguerre eop and their related quantum systems new 2d superintegrable hamiltonians with higher order integrals of motion and the polynomial algebras generated by their integrals of motion we obtain the finite dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum we also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of eop

distorted heisenberg algebra and coherent states for isospectral oscillator hamiltonians the dynamical algebra associated with a family of isospectral oscillator hamiltonians is studied through the analysis of its representation in the basis of energy eigenstates it is shown that this representation becomes similar to that of the standard heisenberg algebra and it is dependent on a parameter omega or 0 we call it the distorted heisenberg algebra where omega is the distortion parameter the corresponding coherent states for an arbitrary omega are derived and some particular examples are discussed in detail a prescription to produce the squeezing by adequately selecting the initial state of the system is given

a family of complex potentials with real spectrum we consider a two parameter non hermitian quantum mechanical hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions in contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane our model is integrable along a line parallel to the real axis

almost sharp quantum effects quantum effects are represented by operators on a hilbert space satisfying 0 a i and sharp quantum effects are represented by projection operators we say that an effect a is almost sharp if a pqp for projections p and q we give simple characterizations of almost sharp effects we also characterize effects that can be written as longer products of projections for generality we first work in the formalism of von neumann algebras we then specialize to the full operator algebra b h and to finite dimensional hilbert spaces

nonbinary kasami sequences over gf p the correlation values and the distribution of these correlation values for the small set of nonbinary kasami sequences over gf p p prime are presented the correlation results are an extension of the binary results and have p 2 correlation levels this nonbinary kasami set is asymptotically optimum with respect to its correlation properties these sequences are obtained as in the binary case from a large primitive polynomial of degree n 2 m and a small primitive polynomial of degree m that yields a sequence length of p sup n 1 and maximum nontrivial correlation value of 1 p sup m nonbinary kasami sequences are directly implemented using shift registers and are applicable for code division multiple access systems

products of projections in von neumann algebras we describe the elements of von neumann algebras which can be represented as products of orthogonal projections and idempotents and estimate the minimal number of terms in the product

exactly solvable hydrogen like potentials and the factorization method a set of factorization energies is introduced giving rise to a generalization of the schrodinger or infeld and hull factorization for the radial hydrogen like hamiltonian an algebraic intertwining technique involving such factorization energies leads us to derive n parametric families of potentials which in general are almost isospectral to the hydrogen like radial hamiltonians the construction of susy partner hamiltonians with ground state energies greater than the corresponding ground state energy of the initial hamiltonian is also explicitly performed

higher order susy linearized nonlinear heisenberg algebras and coherent states using an iterative construction of the first order intertwining technique we find k parametric families of exactly solvable anharmonic oscillators whose spectra consist of a part isospectral to the oscillator plus k additional levels at arbitrary positions below e0 it is seen that the natural ladder operators for these systems give place to polynomial nonlinear algebras and it is shown that these algebras can be linearized the coherent states construction is performed in the nonlinear and linearized cases

nonlinear chains and painleve equations abstract ode systems which define a periodically closed sequence of backlund transformations bt and which are equivalent to the painleve equations p2 p6 are presented transformation properties of painleve equations can be easily derived from the discrete symmetries of these systems

two step rational extensions of the harmonic oscillator exceptional orthogonal polynomials and ladder operators the type iii hermite xm exceptional orthogonal polynomial family is generalized to a double indexed one with m1 even and m2 odd such that m2 m1 and the corresponding rational extensions of the harmonic oscillator are constructed by using second order supersymmetric quantum mechanics the new polynomials are proved to be expressible in terms of mixed products of hermite and pseudo hermite ones while some of the associated potentials are linked with rational solutions of the painlev iv equation a novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial heisenberg algebra of order m2 m1 1 which may alternatively be interpreted in terms of a special type of m2 m1 2 th order shape invariance property

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

fourth order superintegrable systems separating in cartesian coordinates i exotic quantum potentials a study is presented of two dimensional superintegrable systems separating in cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta all quantum mechanical potentials that do not satisfy any linear differential equation are found they do however satisfy nonlinear odes we show that these equations always have the painleve property and integrate them in terms of known painleve transcendents or elliptic functions

susy partners of the truncated oscillator painleve transcendents and backlund transformations in this work the supersymmetric technique is applied to the truncated oscillator to generate hamiltonians ruled by second and third order polynomial heisenberg algebras which are connected to the painleve iv and painleve v equations respectively the aforementioned connection is exploited to produce particular solutions to both nonlinear differential equations and the backlund transformations relating them

coherent states for isospectral oscillator hamiltonians coherent states for a family of isospectral oscillator hamiltonians are derived from a suitable choice of annihilation and creation operators the fock bargmann representation is also considered

generalized deformed oscillator and nonlinear algebras the harmonic oscillator is deformed arbitrarily and the properties of the deformed oscillator algebra are studied the eigenstates and eigenvalues of the deformed oscillator are calculated this generalized deformed oscillator algebra is used to create nonlinear deformation of the classical su 2 algebra the deformed boson realizations of an arbitrary nonlinear su 2 algebra are calculated the algebra of the q deformed oscillator is a special case of the generalized deformed oscillator algebra

on realizations of nonlinear lie algebras by differential operators we study realizations of polynomial deformations of the lie algebra in terms of differential operators strongly related to bosonic operators we also distinguish their finite and infinite dimensional representations the linear quadratic and cubic cases are explicitly visited but the method works for arbitrary degrees in the polynomial functions multi boson hamiltonians are studied in the context of these nonlinear lie algebras and some examples dealing with quantum optics are pointed out

connection between quantum systems involving the fourth painleve transcendent and k step rational extensions of the harmonic oscillator related to hermite exceptional orthogonal polynomial the purpose of this communication is to point out the connection between a 1d quantum hamiltonian involving the fourth painleve transcendent piv obtained in the context of second order supersymmetric quantum mechanics and third order ladder operators with a hierarchy of families of quantum systems called k step rational extensions of the harmonic oscillator and related with multi indexed xm1 m2 mk hermite exceptional orthogonal polynomials of type iii the connection between these exactly solvable models is established at the level of the equivalence of the hamiltonians using rational solutions of the fourth painleve equation in terms of generalized hermite and okamoto polynomials we also relate the different ladder operators obtained by various combinations of supersymmetric constructions involving darboux crum and krein adler supercharges their zero modes and the corresponding energies these results will demonstrate and clarify the relation observed for a particular case in previous papers

third order differential ladder operators and supersymmetric quantum mechanics hierarchies of one dimensional hamiltonians in quantum mechanics admitting third order differential ladder operators are studied each hamiltonian has associated three step darboux pseudo cycles and painleve iv equations as a closure condition the whole hierarchy is generated applying some operations on the cycles these operations are investigated in the frame of supersymmetric quantum mechanics and mainly involve algebraic manipulations a consistent geometric representation for the hierarchy and cycles is built that also helps in understanding the operations three kinds of hierarchies are distinguished and a realization based on the harmonic oscillator hamiltonian is supplied giving an interpretation for the spectral properties of the hamiltonians of each hierarchy

polynomial heisenberg algebras polynomial deformations of the heisenberg algebra are studied in detail some of their natural realizations are given by the higher order susy partners and not only by those of first order as is already known of the harmonic oscillator for even order polynomials here it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd moreover it will be proved that the general systems ruled by such kinds of algebras in the quadratic and cubic cases involve painleve transcendents of types iv and v respectively

non hermitian susy hydrogen like hamiltonians with real spectra it is shown that the radial part of the hydrogen hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant t he 1 susy approach is used to construct non hermitian operators with hydrogen spectra other non hermitian hamiltonians are shown to admit an extra complex energy at n ew self adjoint hydrogen like hamiltonians are also derived by using a 2 susy transformation with complex conjugate pairs

phase operators temporally stable phase states mutually unbiased bases and exactly solvable quantum systems we introduce a one parameter generalized oscillator algebra that covers the case of the harmonic oscillator algebra and discuss its finite and infinite dimensional representations according to the sign of the parameter we define an hamiltonian operator associated with and examine the degeneracies of its spectrum for the finite when 0 and the infinite when 0 representations of we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators to overcome the difficulties related to the phase operator in the infinite dimensional case and to avoid the degeneracy problem for the finite dimensional case we introduce a truncation procedure which generalizes the one used by pegg and barnett for the harmonic oscillator this yields a truncated generalized oscillator algebra where s denotes the truncation order we construct two types of temporally stable states for as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of two applications are considered in this paper the first concerns physical realizations of and in the context of one dimensional quantum systems with finite morse system or infinite poschl teller system discrete spectra the second deals with mutually unbiased bases used in quantum information

painleve equations from darboux chains i piii pv we show that the painleve equations piii pvi can be derived in a unified way from a periodic sequence of darboux transformations for a schrodinger problem with quadratic eigenvalue dependence the general problem naturally divides into three different branches each described by an infinite chain of equations the painleve equations are obtained by closing the chain periodically at the lowest nontrivial level s the chains provide symmetric forms for the painleve equations from which hirota bilinear forms and lax pairs are derived in this paper part i we analyse in detail the cases piii pv while pvi will be studied in part ii

supersymmetry factorisation of the schrodinger equation and a hamiltonian hierarchy the author presents a systematic procedure for constructing a hierarchy of non relativistic hamiltonians with the property that the adjacent members of the hierarchy are supersymmetric partners i e they share the same eigenvalue spectrum except for the missing ground state and the eigenvectors are simply related