the extended krylov subspace method and orthogonal laurent polynomials abstract the need to evaluate expressions of the form f a v where a is a large sparse or structured symmetric matrix v is a vector and f is a nonlinear function arises in many applications the extended krylov subspace method can be an attractive scheme for computing approximations of such expressions this method projects the approximation problem onto an extended krylov subspace k l m a span a l 1 v a 1 v v a v a m 1 v of fairly small dimension and then solves the small approximation problem so obtained we review available results for the extended krylov subspace method and relate them to properties of laurent polynomials the structure of the projected problem receives particular attention we are concerned with the situations when m l and m 2 l

a new algebraization of the lameequation we develop a new way of writing the lamehamiltonian in lie algebraic form this yields in a natural way an explicit formula for both the lamepolynomials and the classical non meromorphic lame functions in terms of chebyshev polynomials and of a certain family of weakly orthogonal polynomials

the quantum superalgebra uq osp 1 2n deformed para bose operators and root of unity representations we recall the relation between the lie superalgebra osp 1 2n and para bose operators the quantum superalgebra uq osp 1 2n defined as usual in terms of its chevalley generators is shown to be isomorphic to an associative algebra generated by so called pre oscillator operators satisfying a number of relations from these relations and the analogue with the non deformed case one can interpret these pre oscillator operators as deformed para bose operators some consequences for uq osp 1 2n cartan weyl basis poincare birkhoff win basis and its hopf subalgebra uq gl n are pointed out finally using a realization in terms of q commuting q bosons we construct an irreducible finite dimensional unitary fock representation of uq osp 1 2n and its decomposition in terms of uq gl n representations when q is a root of unity

quantal problems with partial algebraization of the spectrum we discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly solvable problems e g harmonic oscillator and all others the problems belonging to this class are distinguished by the fact that a part of the eigenvalues and eigenfunctions can be found algebraically but not the whole spectrum the reason explaining the existence of the quasi exactly solvable problems is a hidden dynamical symmetry present in the hamiltonian for one dimensional motion this hidden symmetry issl 2 r it is shown that other groups lead to a partial algebraization in multidimensional quantal problems in particular sl 2 r sl 2 r so 3 andsl 3 r are relevant to two dimensional motion inducing a class of quasi exactly solvable two dimensional hamiltonians typically they correspond to systems in a curved space but sometimes the curvature turns out to be zero graded algebras open the possibility of constructing quasi exactlysolvable hamiltonians acting on multicomponent wave functions for example with a non minimal superextension ofsl 2 r we get a hamiltonian describing the motion of a spinor particle

quantum deformation of bose parastatistics a q deformation of the transformation of the chevalley basis to an odd basis of generators of the universal enveloping of the lie superalgebra g n osp 1 2n is presented it is shown that one thus obtains a reasonable quantum deformation of the algebra of n para bose oscillators

supersymmetry and algebraic darboux transformations we describe a class of algebraically solvable susy models by considering the deformation of invariant polynomial flags by means of the darboux transformation the algebraic deformations corresponding to the addition of a bound state to a shape invariant potential are particularly interesting the polynomial flags in question are indexed by a deformation parameter m 1 2 and lead to new algebraically solvable models we illustrate these ideas by considering deformations of the hyperbolic p schl teller potential

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

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

on the solvability of the quantum rabi model and its 2 photon and two mode generalizations we study the solvability of the time independent matrix schrodinger differential equations of the quantum rabi model and its 2 photon and two mode generalizations in bargmann hilbert spaces of entire functions we show that the rabi model and its 2 photon and two mode analogs are quasi exactly solvable we derive the exact closed form expressions for the energies and the allowed model parameters for all the three cases in the solvable subspaces up to a normalization factor the eigenfunctions for these models are given by polynomials whose roots are determined by systems of algebraic equations

quantization of uq osp 1 2n with deformed para bose operators the observation that n pairs of para bose pb operators generate the universal enveloping algebra of the orthosymplectic lie superalgebra osp 1 2n is used in order to define deformed pb operators it is shown that these operators are an alternative to the chevalley generators on this background uq osp 1 2n its cartan weyl generators and their supercommutation relations are written down entirely in terms of deformed pb operators an analogue of the poincare birkhoff witt theorem is formulated

character formulas for irreducible modules of the lie superalgebras sl m n kac distinguished between typical and atypical finite dimensional irreducible representations of the lie superalgebras sl m n and provided an explicit character formula appropriate to all the typical representations here the range of validity of some character formulas for atypical representations that have been proposed are discussed several of them are of the kac weyl type but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl 3 4 having ruled out therefore all such formulas a completely new extension of the kac weyl character formula is proposed the validity of this formula in the case of all covariant tensor irreducible representations is proved and some evidence in support of the conjecture that it covers all irreducible representations of sl m n is presented

a description of the quantum superalgebrauq sl n 1 m via creation and annihilation generators a description of the quantum superalgebra uq sl n 1 m and in particular of the special linear superalgebra sl n 1 m via creation and annihilation generators cags is given it provides an alternative to the canonical description of uq sl n 1 m in terms of chevalley generators a conjecture that the fock representations of the cags provide microscopic realizations of exclusion statistics is formulated

n fold supersymmetry and quasi solvability associated with x2 laguerre polynomials we construct a new family of quasi solvable and n fold supersymmetric quantum systems where each hamiltonian preserves an exceptional polynomial subspace of codimension 2 we show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the x2 laguerre polynomials of the second kind in addition we find that the two kinds of the x2 laguerre polynomials are ingeniously connected with each other by the n fold supercharge

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

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

quasi exactly solvable extensions of the lame equation quasi exactly solvable equations are spectral equations which possess a finite number of algebraic eigenvectors for a few of these equations such as the lame equation the number of algebraic solutions is larger than predicted by the general theory a class of quasi exactly solvable equations in one variable is considered and the conditions under which a rich algebraization occurs is discussed families of equations which by themselves are not quasi exactly solvable but can be transformed into systems of coupled quasi exactly solvable equations are discussed these results suggest a scheme for the classification of quasi exactly solvable systems in one variable

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

probabilistic bounds for the matrix condition number with extended lanczos bidiagonalization reliable estimates for the condition number of a large sparse real matrix a are important in many applications to get an approximation for the condition number kappa a an approximation for the smallest singular value is needed standard krylov subspaces are usually unsuitable for finding a good approximation to the smallest singular value therefore we study extended krylov subspaces which turn out to be ideal for the simultaneous approximation of both the smallest and largest singular value of a matrix first we develop a new extended lanczos bidiagonalization method with this method we obtain a lower bound for the condition number moreover the method also yields probabilistic upper bounds for kappa a the user can select the probability with which the upper bound holds as well as the ratio of the probabilistic upper bound and the lower bound

quasi exactly solvable lie algebras of differential operators in two complex variables the authors completely classify all quasi exactly solvable lie algebras of first order differential operators in two complex variables applications to quasi exactly solvable quantum problems are indicated

multiple algebraisations of an elliptic calogero sutherland model recently gomez ullate et al phys lett b 511 112 2001 have studied an n particle quantum problem with elliptic function potentials they have shown that the hamiltonian operator preserves a finite dimensional space of functions and as such is quasi exactly solvable qes in this article we show that other types of invariant function spaces exist which are in close relation to the algebraic properties of the elliptic functions accordingly series of new algebraic eigenfunctions can be constructed

families of quasi exactly solvable extensions of the quantum oscillator in curved spaces we introduce two new families of quasi exactly solvable qes extensions of the oscillator in a d dimensional constant curvature space for the first three members of each family we obtain closed form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the bethe ansatz method we prove that the first member of each family has a hidden sl 2 symmetry and is connected with a qes equation of the first or second type respectively one dimensional results are also derived from the d dimensional ones with d 2 thereby getting qes extensions of the mathews lakshmanan nonlinear oscillator

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

jacobson generators of the quantum superalgebra uq sl n 1 m and fock representations as an alternative to chevalley generators we introduce jacobson generators for the quantum superalgebra uq sl n 1 m the expressions of all cartan weyl elements of uq sl n 1 m in terms of these jacobson generators become very simple we determine and prove certain triple relations between the jacobson generators necessary for a complete set of supercommutation relations between the cartan weyl elements fock representations are defined and a substantial part of this paper is devoted to the computation of the action of jacobson generators on basis vectors of these fock spaces it is also determined when these fock representations are unitary finally dyson and holstein primakoff realizations are given not only for the jacobson generators but for all cartan weyl elements of uq sl n 1 m

quasi exact solvability and the direct approach to invariant subspaces we propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of this approach is used here to construct new exactly solvable and quasi exactly solvable quantum hamiltonians on the line which are not lie algebraic it is also applied to generate potentials with multiple algebraic sectors we discuss two illustrative examples of these two applications we show that the generalized lame potential possesses four algebraic sectors and describe a quasi exactly solvable deformation of the morse potential which is not lie algebraic

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

conditional symmetry and spectrum of the one dimensional schrodinger equation we develop an algebraic approach to studying the spectral properties of the stationary schrodinger equation in one dimension based on its high order conditional symmetries this approach makes it possible to obtain in explicit form representations of the schrodinger operator by n n matrices for any n n and thus to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant n n matrices the connection to so called quasiexactly solvable models is discussed it is established in particular that the case when conditional symmetries reduce to high order lie symmetries corresponds to exactly solvable schrodinger equations a symmetry classification of schrodinger equation admitting nontrivial high order lie symmetries is carried out which yields a hierarchy of exactly solvable schrodinger equations exact solutions of these are constructed in explicit form possible applications of the technique developed to multidimensional linear and one dimensional nonlinear schrodinger equations are briefly discussed

exceptional orthogonal polynomials and the darboux transformation we adapt the notion of the darboux transformation to the context of polynomial sturm liouville problems as an application we characterize the recently described xm laguerre polynomials in terms of an isospectral darboux transformation we also show that the shape invariance of these new polynomial families is a direct consequence of the permutability property of the darboux crum transformation

type b 3 fold supersymmetry and non polynomial invariant subspaces we obtain the most general type b 3 fold supersymmetry by solving directly the intertwining relation we then show that it is a necessary and sufficient condition for a second order linear differential operator to have three linearly independent local analytic solutions we find that there are eight linearly independent non trivial linear differential operators of this kind as a by product we find new quasi solvable second order operators preserving a monomial or polynomial subspace one in type b two in type c and four in type x2 all of which have been missed in the existing literature in addition we show that type a type b and type c 3 fold supersymmetries are connected continuously via one parameter a few new quasi solvable models are also presented

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

new quasi exactly solvable hamiltonians in two dimensions quasi exactly solvable schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods such operators lie in the enveloping algebra of a finite dimensional lie algebra of first order differential operators the hidden symmetry algebra in this paper we develop some general techniques for constructing quasi exactly solvable operators our methods are applied to provide a wide variety of new explicit two dimensional examples on both flat and curved spaces of quasi exactly solvable hamiltonians corresponding to both semisimple and more general classes of lie algebras

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

quasi exactly solvable spin 1 2 schrodinger operators the algebraic structures underlying quasi exact solvability for spin 1 2 hamiltonians in one dimension are studied in detail necessary and sufficient conditions for a matrix second order differential operator preserving a space of wave functions with polynomial components to be equivalent to a schrodinger operator are found systematic simplifications of these conditions are analyzed and are then applied to the construction of new examples of multi parameter qes spin 1 2 hamiltonians in one dimension

invariant modules and the reduction of nonlinear partial differential equations to dynamical systems abstract we completely characterize all nonlinear partial differential equations leaving a given finite dimensional vector space of analytic functions invariant existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi exactly solvable quantum hamiltonians these results rely on a useful extension of the classical wronskian determinant condition for linear independence of functions in addition new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented

perturbation results of critical elliptic equations of caffarelli kohn nirenberg type abstract we find for small e positive solutions to the equation div x 2a u x 2 1 a u 1 ek x u p 1 x bp in r n which branch off from the manifold of minimizers in the class of radial functions of the corresponding caffarelli kohn nirenberg type inequality moreover our analysis highlights the symmetry breaking phenomenon in these inequalities namely the existence of non radial minimizers

criterion for polynomial solutions to a class of linear differential equations of second order we consider the differential equations y 0 x y s0 x y where 0 x s0 x are c functions we prove i if the differential equation has a polynomial solution of degree n 0 then n nsn 1 n 1sn 0 where n n 1 sn 1 0 n 1andsn s n 1 s0 k 1 n 1 2 conversely ii if n n 1 0 and n 0 then the differential equation has a polynomial solution of degree at most n we show that the classical differential equations of laguerre hermite legendre jacobi chebyshev first and second kinds gegenbauer and the hypergeometric type etc obey this criterion further we find the polynomial solutions for the generalized hermite laguerre legendre and chebyshev differential equations

a classification of generalized quantum statistics associated with basic classical lie superalgebras generalized quantum statistics such as para statistics is usually characterized by certain triple relations in the case of para fermi statistics these relations can be associated with the orthogonal lie algebra bn so 2n 1 in the case of para bose statistics they are associated with the lie superalgebra b 0 n osp 1 2n in a previous paper a mathematical definition of a generalized quantum statistics associated with a classical lie algebra g was given and a complete classification was obtained here we consider the definition of a generalized quantum statistics associated with a basic classical lie superalgebra g just as in the lie algebra case this definition is closely related to a certain z grading of g we give in this paper a complete classification of all generalized quantum statistics associated with the basic classical lie superalgebras a m n b m n c n and d m n

two novel classes of solvable many body problems of goldfish type with constraints two novel classes of many body models with nonlinear interactions of goldfish type are introduced they are solvable provided the initial data satisfy a single constraint in one case in the other two constraints i e for such initial data the solution of their initial value problem can be achieved via algebraic operations such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials entirely isochronous versions of some of these models are also exhibited i e versions of these models whose nonsingular solutions are all completely periodic with the same period

recursion relations for the extended krylov subspace method abstract the evaluation of matrix functions of the form f a v where a is a large sparse or structured symmetric matrix f is a nonlinear function and v is a vector is frequently subdivided into two steps first an orthonormal basis of an extended krylov subspace of fairly small dimension is determined and then a projection onto this subspace is evaluated by a method designed for small problems this paper derives short recursion relations for orthonormal bases of extended krylov subspaces of the type k m mi 1 a span a m 1 v a 1 v v a v a mi v m 1 2 3 with i a positive integer and describes applications to the evaluation of matrix functions and the computation of rational gauss quadrature rules

unified derivation of exact solutions for a class of quasi exactly solvable models we present a unified treatment of exact solutions for a class of four quantum mechanical models namely the anharmonic singular potential the generalized quantum isotonic oscillator the soft core coulomb potential and the non polynomially modified oscillator we show that all four cases are reducible to the same basic ordinary differential equation which is quasi exactly solvable a systematic and closed form solution to the basic equation is obtained via the bethe ansatz method using the result general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations a hidden sl 2 algebraic structure is also discovered in these models

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

new algebraic quantum many body problems we develop a systematic procedure for constructing quantum many body problems whose spectrum can be partially or totally computed by purely algebraic means the exactly solvable models include rational and hyperbolic potentials related to root systems in some cases with an additional external field the quasi exactly solvable models can be considered as deformations of the previous ones which share their algebraic character

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

a jacobi davidson type svd method we discuss a new method for the iterative computation of a portion of the singular values and vectors of a large sparse matrix similar to the jacobi davidson method for the eigenvalue problem we compute in each step a correction by approximately solving a correction equation we give a few variants of this jacobi davidson svd jdsvd method with their theoretical properties it is shown that the jdsvd can be seen as an accelerated inexact newton scheme we experimentally compare the method with some other iterative svd methods

computing fundamental matrix decompositions accurately via the matrix sign function in two iterations the power of zolotarev s functions the symmetric eigenvalue decomposition and the singular value decomposition svd are fundamental matrix decompositions with many applications conventional algorithms for computing these decomposi

leibniz algebras of heisenberg type we introduce and provide a classification theorem for the cl ass of heisenberg fock leibniz algebras this category of algebras is formed by those leibniz algebras l whose corresponding lie algebras are heisenberg algebras hn and whose hn modules i where i denotes the ideal generated by the squares of elements of l are isomorphic to fock modules we also consider the three dimensional heisenberg algebra h3 and study three classes of leibniz algebras with h3 as corresponding lie algebra by taking certain generalizations of the fock module moreover we describe the class of leibniz algebras with hn as corresponding lie algebra and such that the action i hn i gives rise to a minimal faithful representation of hn the classification of this family of leibniz algebras for the case of n 3 is given ams subject classifications 2010 17a32 17b30 17b10

some exact results for mid band and zero band gap states of associated lame potentials applying certain known theorems about one dimensional periodic potentials we show that the energy spectrum of the associated lame potentials a a 1 msn2 x m b b 1 m cn2 x m dn2 x m consists of a finite number of bound bands followed by a continuum band when both a and b take integer values further if a and b are unequal integers we show that there must exist some zero band gap states i e doubly degenerate states with the same number of nodes more generally in case a and b are not integers but either a b or a b is an integer a b we again show that several of the band gaps vanish due to degeneracy of states with the same number of nodes finally when either a or b is an integer and the other takes a half integral value we obtain exact analytic solutions for several mid band states

lie algebras of differential operators and lie algebraic potentials abstract an explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a finite dimensional lie algebra of first order differential operators is found solving a problem arising in the lie algebraic approach to scattering theory and molecular dynamics one dimensional potentials corresponding to these lie algebras are explicitly classified which include the harmonic oscillator morse one soliton poschl teller mathieu lame confluent hypergeometric and bessel potentials

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

pseudo orthogonal groups and integrable dynamical systems in two dimensions integrable systems in low dimensions constructed through the symmetry reduction method are studied using phase portrait and variable separation techniques in particular invariant quantities and explicit periodic solutions are determined widely applied models in physics are shown to appear as particular cases of the method

estimating extremal eigenvalues and condition numbers of matrices the paper describes probabilistic algorithms which may be used to make rough estimates of the largest and smallest eigenvalues of a positive definite matrix and the condition number of a nonsingular matrix in the 2 norm given varepsilon 0 and a prescribed relative error the algorithms compute estimates which with probability at least 1 varepsilon have relative errors less than that prescribed in particular the method gives a reliable way to estimate the condition number of a matrix of large degree

on the construction of quasi exactly solvable schrodinger operators on homogeneous spaces the closure conditions for a quasi exactly solvable operator is the requirement that a second order lie algebraic differential operator be equivalent up to scale change to a schrodinger operator on curved space the present work begins with an invariant characterization of the closure conditions and after a series of steps gives a reformulation of the closure conditions in terms of finite dimensional representations of the underlying lie algebra these techniques are used to give a complete solution to the closure condition for two different planar realizations of sl 2 along the way two theorems about homogeneous solutions to the closure conditions are introduced and proved also introduced is the class of abelian solutions to the closure conditions such solutions appear on flat spaces and give rise to a schrodinger operator with zero potential the concluding remarks highlight remaining questions and give references to research that address said questions

quasi exact solution of the rabi hamiltonian a method is suggested to obtain the quasi exact solution of the rabi hamiltonian it is conceptually simple and can be easily extended to other systems the analytical expressions are obtained for eigenstates and eigenvalues in terms of orthogonal polynomials it is also demonstrated that the rabi system in a particular case coincides with the quasi exactly solvable poschl teller potential

some spectral equivalences between schrodinger operators spectral equivalences of the quasi exactly solvable sectors of two classes of schrodinger operators are established using gaudin type bethe ansatz equations in some instances the results can be extended leading to full isospectrality in this manner we obtain equivalences between pt symmetric problems and hermitian problems we also find equivalences between some classes of hermitian operators

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

exclusion statistics operator algebras and fock space representations we study exclusion statistics within the second quantized approach we consider operator algebras with positive definite fock space and restrict them in a such a way that certain state vectors in fock space are forbidden ab initio we describe three characteristic examples of such exclusion namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space e g the calogero sutherland type of exclusion statistics exclusion in the single oscillator fock space where some states in single oscillator fock space are forbidden e g the gentile realization of exclusion statistics and a combination of these two exclusions e g green s realization of para fermi statistics for these types of exclusions we discuss the extended haldane statistics parameters g recently introduced by two of us in 1996 mod phys lett a 11 3081 and associated counting rules within these three types of exclusions in fock space the original haldane exclusion statistics cannot be realized

quasi exact solvability of the pauli equation we present a general procedure for determining possible nonuniform magnetic fields such that the pauli equation becomes quasi exactly solvable qes with an underlying sl 2 symmetry this procedure makes full use of the close connection between qes systems and supersymmetry of the ten classes of sl 2 based one dimensional qes systems we have found that nine classes allow such construction

classification of generalized quantum statistics associated with the exceptional lie super algebras generalized quantum statistics gqs associated with a lie algebra or lie superalgebra extends the notion of para bose or para fermi statistics such gqs have been classified for all classical simple lie algebras and basic classical lie superalgebras in the current paper we finalize this classification for all exceptional lie algebras and superalgebras since the definition of gqs is closely related to a certain z grading of the lie super algebra g our classification reproduces some known z gradings of exceptional lie algebras for exceptional lie superalgebras such a classification of z gradings has not been given before

fock representations of the superalgebra sl n 1 m its quantum analogue uq sl n 1 m and related quantum statistics fock space representations of the lie superalgebra sl n 1 m and of its quantum analogue uq sl n 1 m are written down the results are based on a description of these superalgebras via creation and annihilation operators the properties of the underlying statistics are briefly discussed

fock representations of quantum fields with generalized statistics we develop a rigorous framework for constructing fock representations of quantum fields obeying generalized statistics the main features of these representations are investigated various aspects of the underlying mathematical structure are illustrated by means of explicit examples

solutions for confluent and double confluent heun equations this paper examines some solutions for confluent and double confluent heun equations in the first place we review two leaver s solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation leaver e w j math phys 27 1238 1986 and introduce an additional expansion in series of irregular confluent hypergeometric functions then we find the conditions under which one of these solutions can be written as a linear combination of the others in the second place by means of limiting procedures we generate solutions for the double confluent equation as well as for special limits of both the confluent and double confluent equations finally we present some problems which are ruled by one or the other of these four equations

a study of quasi exactly solvable models within the quantum hamilton jacobi formalism a few quasi exactly solvable models are studied within the quantum hamilton jacobi formalism by assuming a simple singularity structure of the quantum momentum function we show that the exact quantization condition leads to the condition for quasi exact solvability

jacobson generators fock representations and statistics of sl n 1 the properties of a statistics related to the class a of simple lie algebras t d palev preprint jinr e17 10550 1977 hep th 9705032 are further investigated the description of each sl n 1 is carried out via generators a1 an which we call jacobson generators with respect to these generators the definition of a fock space of sl n 1 is given it is proved that the fock spaces wp p n are the simple symmetric finite dimensional modules of sl n 1 the pauli principle of the underlying statistics is formulated within each wp operators b p i ai p i 1 n called quasi bose creation and annihilation operators caos are defined then limp b p i bi where bi are ordinary bose caos therefore a statistics appears as an approximation of bose statistics with caos acting in finite dimensional state spaces we indicate that the p 1 quasi bose operators b 1 1 b 1 n are natural operators for the description of hard core bose models and of the related heisenberg spin models we argue th

the darboux transformation and algebraic deformations of shape invariant potentials we investigate the backward darboux transformations addition of the lowest bound state of shape invariant potentials on the line and classify the subclass of algebraic deformations those for which the potential and the bound states are simple elementary functions a countable family m 0 1 2 of deformations exists for each family of shape invariant potentials we prove that the m th deformation is exactly solvable by polynomials meaning that it leaves invariant an infinite flag of polynomial modules p m m p m 1 m where p n m is a codimension m subspace of in particular we prove that the first m 1 algebraic deformation of the shape invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules p n 1 by construction these algebraically deformed hamiltonians do not have an sl 2 hidden symmetry algebra structure

new quasi exactly solvable sextic polynomial potentials a hamiltonian is said to be quasi exactly solvable qes if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in a closed form an entirely new class of qes hamiltonians having sextic polynomial potentials is constructed these new hamiltonians are different from the sextic qes hamiltonians in the literature because their eigenfunctions obey symmetry rather than hermitian boundary conditions these new hamiltonians present a novel problem that is not encountered when the hamiltonian is hermitian it is necessary to distinguish between the parametric region of unbroken symmetry in which all of the eigenvalues are real and the region of broken symmetry in which some of the eigenvalues are complex the precise location of the boundary between these two regions is determined numerically using the extrapolation techniques and analytically using the wkb analysis

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

gl 3 c invariance of type b 3 fold supersymmetric systems type b 3 fold supersymmetry is a necessary and sufficient condition for a quantum hamiltonian to admit three linearly independent local solutions in closed form we show that any such a system is invariant under gl 3 c homogeneous linear transformations in particular we prove explicitly that the parameter space is transformed as an adjoint representation of it and that every coefficient of the characteristic polynomial appeared in 3 fold superalgebra is algebraic invariants in the type a case it includes as a subgroup the gl 2 c projective transformation studied in the literature we argue that any n fold supersymmetric system has a gl n c invariance for an arbitrary integral n

on the solvability of the generalized hyperbolic double well models we present exact solutions for the schrodinger equation with the hyperbolic double well potential vqp x v0sinhp x coshq x we show that the model preserves a finite dimensional polynomial space for some p and q thus using the bethe ansatz method we obtain closed form expressions for the spectrum and wavefunction as well as the allowed parameter for the class v6p x which is contrary to a report in a recent article c a downing j math phys 54 072101 2013 we also discuss the hidden sl2 algebraic structure of the class