max min lyapunov functions for switched systems and related differential inclusions abstract starting from a finite family of continuously differentiable positive definite functions we study conditions under which a function obtained by max min combinations is a lyapunov function establishing stability for two kinds of nonlinear dynamical systems a differential inclusions where the set valued right hand side comprises the convex hull of a finite number of vector fields and b autonomous switched systems with a state dependent switching signal we investigate generalized notions of directional derivatives for these max min functions and use them in deriving stability conditions with various degrees of conservatism where more conservative conditions are numerically more tractable the proposed constructions also provide nonconvex lyapunov functions which are shown to be useful for systems with state dependent switching that do not admit a convex lyapunov function several examples are included to illustrate the results

exactly solvable potentials of calogero type for q deformed coxeter groups we establish that by parametrizing the configuration space of a one dimensional quantum system by polynomial invariants of q deformed coxeter groups it is possible to construct exactly solvable models of calogero type we adopt the previously introduced notion of solvability which consists of relating the hamiltonian to finite dimensional representation spaces of a lie algebra we present explicitly the gq2 case for which we construct the potentials by means of suitable gauge transformations

duality between the trigonometric bcn sutherland system and a completed rational ruijsenaars schneider van diejen system we present a new case of duality between integrable many body systems where two systems live on the action angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one our investigation utilizes an idea that was exploited previously to provide group theoretic interpretation for several dualities discovered originally by ruijsenaars in the group theoretic framework one applies hamiltonian reduction to two abelian poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space taking the cotangent bundle of u 2n as the upstairs space we demonstrate how this mechanism leads to a new dual pair involving the bcn trigonometric sutherland system thereby we generalize earlier results pertaining to the an trigonometric sutherland system as well as a recent work by pusztai on the hyperbolic bcn sutherland system

linear operators with invariant polynomial space and graded algebra the irreducible finite dimensional representations of the graded algebras osp j 2 j 1 2 3 are expressed in terms of differential operators some quantum deformations of these algebras are shown to admit similar kinds of representations these are formulated in terms of finite difference operators the results are discussed in the framework of the quasi exactly solvable equations

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

the finite toda lattices connection is established between one dimensional toda lattices constructed on the basis of the systems of simple roots of classical and affine lie algebras and other integrable systems of interacting particles that connection allows us to find new lattices differing from the known ones by the interaction of particles near the ends some of the new lattices admit non abelian generalizations

an su 2 analogue of the azbel hofstadter hamiltonian motivated by the recent findings of wiegmann zabrodin and faddeev kashaev concerning the appearance of the quantum symmetry in the problem of a bloch electron on a two dimensional magnetic lattice we introduce a modification of the tight binding azbel hofstadter hamiltonian that is a specific spin s euler top that can be considered as its classical analogue the eigenvalue problem for the proposed model in the coherent state representation is described by the s gap lame equation and thus is completely solvable we observe a striking similarity between the shapes of the spectra of the two models for various values of the spin s

polynomials associated with equilibrium positions in calogero moser systems in a previous paper corrigan sasaki many remarkable properties of classical calogero and sutherland systems at equilibrium are reported for example the minimum energies frequencies of small oscillations and the eigenvalues of lax pair matrices at equilibrium are all integer valued the equilibrium positions of calogero and sutherland systems for the classical root systems ar br cr and dr correspond to the zeros of hermite laguerre jacobi and chebyshev polynomials here we define and derive the corresponding polynomials for the exceptional e6 e7 e8 f4 and g2 and non crystallographic i2 m h3 and h4 root systems they do not have orthogonality but share many other properties with the above mentioned classical polynomials

quadratic algebra associated with rational calogero moser models classical calogero moser models with rational potential are known to be superintegrable that is on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom they possess an additional set of r 1 algebraically and functionally independent globally defined conserved quantities at the quantum level kuznetsov uncovered the existence of a quadratic algebrastructure as an underlying key for superintegrability for the models based on a type root systems here we demonstrate in a universal way the quadratic algebrastructure for quantum rational calogero moser models based on any root systems

integrable spin chain with reflecting end a new integrable spin chain of the haldane shastry type is introduced it is interpreted as the inverse square interacting spin chain with a reflecting end the lattice sites of this model consist of the square roots of the zeros of the laguerre polynomial using the exchange operator formalism the integrals of motion for the model are explicitly constructed

equilibrium positions shape invariance and askey wilson polynomials we show that the equilibrium positions of the ruijsenaars schneider van diejen systems with the trigonometric potential are given by the zeros of the askey wilson polynomials with five parameters the corresponding single particle quantum version which is a typical example of discrete quantum mechanical systems with a q shift type kinetic term is shape invariant and the eigenfunctions are the askey wilson polynomials this is an extension of our previous study 1 2 which established the discrete analogue of the well known fact the equilibrium positions of the calogero systems are described by the hermite and laguerre polynomials whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the hermite and laguerre polynomials

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

recent developments on the stability of systems with aperiodic sampling an overview this article presents basic concepts and recent research directions about the stability of sampled data systems with aperiodic sampling we focus mainly on the stability problem for systems with arbitrary time varying sampling intervals which has been addressed in several areas of research in control theory systems with aperiodic sampling can be seen as time delay systems hybrid systems input output interconnections discrete time systems with time varying parameters etc the goal of the article is to provide a structural overview of the progress made on the stability analysis problem without being exhaustive which would be neither possible nor useful we try to bring together results from diverse communities and present them in a unified manner for each of the existing approaches the basic concepts fundamental results converse stability theorems when available and relations with the other approaches are discussed in detail results concerning extensions of lyapunov and frequency domain methods for systems with aperiodic sampling are recalled as they allow to derive constructive stability conditions furthermore numerical criteria are presented while indicating the sources of conservatism the problems that remain open and the possible directions of improvement at last some emerging research directions such as the design of stabilizing sampling sequences are briefly discussed

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

finite element analysis of a quadratic eigenvalue problem arising in dissipative acoustics a quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered the problem is shown to be equivalent to the spectral problem for a noncompact operator and a thorough spectral characterization is given a numerical discretization based on raviart thomas finite elements is analyzed the method is proved to be free of spurious modes and to converge with optimal order implementation issues and numerical experiments confirming the theoretical results are reported

stabilization and control lyapunov functions for language constrained discrete time switched linear systems in this paper the stabilizability of discrete time switched linear systems subject to constraints on the switching law is considered the admissible switching sequences are given by the language generated by a nondeterministic finite state automaton constructive necessary and sufficient conditions for recurrent stabilizability are given and the exact relations with the existence of control lyapunov functions and with general stabilizability are provided the dependence of stabilizability on the automaton initial state is also proved

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

rational eigenvalue problems and applications to photonic crystals we establish new analytic results for a general class of rational spectral problems they arise e g in modelling photonic crystals whose capability to control the flow of light depends on specific

a class of n body problems with nearest and next to nearest neighbour interactions we obtain the exact ground state and part of the excitation spectrum in one dimension on a line and the exact ground state on a circle in the case where the n particles are interacting via nearest and next to nearest neighbour interactions furthermore using the exact ground state we establish a mapping between these n body problems and the short range dyson models introduced recently to model intermediate spectral statistics using this mapping we compute the one and two point functions of a related many body theory in the thermodynamic limit and show the absence of long range order however quite remarkably we prove the existence of an off diagonal long range order in the symmetrized version of the related many body theory generalization of the models to other root systems is also considered besides we also generalize the model on the full line to higher dimensions finally we consider a model in two dimensions in which all the states exhibit novel correlations

quantum versus classical integrability in calogero moser systems calogero moser systems are classical and quantum integrable multiparticle dynamics defined for any root system the quantum calogero systems having 1 q2 potential and a confining q2 potential and the sutherland systems with 1 sin2q potentials have integer energy spectra characterized by the root system various quantities of the corresponding classical systems e g minimum energy frequencies of small oscillations the eigenvalues of the classical lax pair matrices etc at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems to our surprise most of these classical data are also integers or they appear to be quantized to be more precise these quantities are polynomials of the coupling constant s with integer coefficients the close relationship between quantum and classical integrability in calogero moser systems deserves fuller analytical treatment which would lead to better understanding of these systems and of integrable systems in general

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

polynomials associated with equilibria of affine toda sutherland systems an affine toda sutherland system is aquasi exactly solvable multi particle dynamics based on an affine simple root system it is a cross between two well known integrable multi particle dynamics an affine toda moleculeand a sutherland system polynomials describing the equilibrium positions of affine toda sutherland systems are determined for all affine simple root systems

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

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

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

quasi exact solvability of the a _ 2 g _ 2 elliptic model algebraic forms sl 3 g 2 hidden algebra and polynomial eigenfunctions the potential of the a2 quantum elliptic model three body calogero moser elliptic model is defined by the pairwise three body interaction through the weierstrass function and has a single coupling constant a change of variables has been found which are a2 elliptic invariants such that the potential becomes a rational function while the flat space metric as well as its associated vector are polynomials in two variables it is shown that the model possesses the hidden algebra the hamiltonian is an element of the universal enveloping algebra for the arbitrary coupling constant thus it is equivalent to the quantum euler arnold top the integral in a form of the third order differential operator with polynomial coefficients is constructed explicitly being also an element of it is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions up to a non singular gauge factor occurs for these values of the coupling constants there exists a particular integral it commutes with the hamiltonian in action on the space of polynomial eigenfunctions and the hamiltonian is invariant with respect to two dimensional projective transformations it is shown that the a2 model has another hidden algebra introduced in rosenbaum et al 1998 int j mod phys a 13 3885 the potential of the g2 quantum elliptic model three body wolfes elliptic model is defined by the pairwise and three body interactions through the weierstrass function and has two coupling constants a change of variables has been found which are g2 elliptic invariants such that the potential becomes a rational function while the flat space metric as well as its associated vector are polynomials in two variables it is shown the model possesses the hidden algebra it is shown that there exists a discrete family of the coupling constants for which a finite number of polynomial eigenfunctions up to a non singular gauge factor occurs for these values of the coupling constants there exists a particular integral

stability of discrete time switching systems with constrained switching sequences we introduce a novel framework for the stability analysis of discrete time linear switching systems with switching sequences constrained by an automaton the key element of the framework is the algebraic concept of multinorm which associates a different norm per node of the automaton and allows to exactly characterize stability building upon this tool we develop the first arbitrarily accurate approximation schemes for estimating the constrained joint spectral radius that is the exponential growth rate of a switching system with constrained switching sequences more precisely given a relative accuracy r 0 the algorithms compute an estimate of within the range 1 r these algorithms amount to solve a well defined convex optimization program with known time complexity and whose size depends on the desired relative accuracy r 0

approximation of the constrained joint spectral radius via algebraic lifting this paper studies the linear constrained switching system which is a discrete time switched linear system whose switching sequences are constrained by a deterministic finite automaton the stability of a constrained switching system is characterized by its constrained joint spectral radius that is known to be difficult to compute or approximate using the semi tensor product of matrices the matrix form expression of a constrained switching system is shown to be equivalent to that of a lifted arbitrary switching system then the constrained joint generalized spectral radius of a constrained switching system is proved to be equal to the joint generalized spectral radius of its lifted arbitrary switching system which can be approximated by off the shelf algorithms

the lax pairs for elliptic cn and bcn ruijsenaars schneider models and their spectral curves we study the elliptic c n and bcn ruijsenaars schneider models which are elliptic generalization of systems given in previous paper by the present authors chen j math phys 41 8132 2000 the lax pairs for these models are constructed by hamiltonian reduction technology we show that the spectral curves can be parametrized by the involutive integrals of motion for these models taking nonrelativistic limit and scaling limit we verify that they lead to the systems corresponding to calogero moser and toda types c 2001 american institute of physics

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

spectral approximation for polynomial eigenvalue problems abstract in this paper we extent the classical spectral approximation theory for compact and bounded operators to general linear operators and then apply it to polynomial eigenvalue problems pep we also study the essential spectrum in peps and prove that this spectrum is stable under relatively compact perturbations based on this analysis we give some suggestions to make algorithms for solving peps more efficient

matrices differential operators and polynomials it is noted that the matrix z of order n defined in terms of the n arbitrary numbers xj by the formula zjk jk j l 1 xj xl 1 1 jk xj xk 1 may be considered in an appropriate framework to correspond to the differential operator d dx there follow prescriptions to construct explicit matrices of arbitrary order n in terms of n or more arbitrary numbers or of the n roots of given polynomials matrices whose eigenvalues and eigenvectors are given fully or in part by very simple formulas novel representations of the classical polynomials hermite laguerre lagrange gegenbauer jacobi are also obtained such as the formula for hermite polynomials hn x 2n det xi h where i is the unit matrix of order n and the matrix h of order n is defined by hjk 2n 1 2 jk n 1 exp 2i j 1 2 exp 2i j 1 jk exp 2i j 2i sin j k 1 exp i j k with j j n arbitrary

graph lyapunov function for switching stabilization and distributed computation abstract this paper studies stabilization of discrete time switched linear systems slss using the notion of graph control lyapunov functions gclfs a gclf is a set of lyapunov functions defined on a weighted digraph where each lyapunov function is represented by a node in the digraph and there is a lyapunov inequality associated with each subgraph consisting of a node and its out neighbors the weight of a directed edge indicates the decay or growth rate of the lyapunov functions it is proved that an sls is switching stabilizable if and only if there exists a gclf the main benefits of gclfs are reduced computational cost and conservatism for stabilizability tests besides we show that the proposed gclf framework unifies several control lyapunov functions and the related stabilization theorems moreover we propose a distributed algorithm to evaluate the stabilizability with reduced computational costs by taking benefits of the graph structure of gclfs several examples are given to demonstrate the efficiency of the algorithm

integrability of difference calogero moser systems a general class of n particle difference calogero moser systems with elliptic potentials is introduced besides the step size and two periods the hamiltonian depends on nine coupling constants we prove the quantum integrability of the model for n 2 and present partial results for n 3 in degenerate cases rational hyperbolic or trigonometric limit the integrability follows for arbitrary particle number from previous work connected with the multivariable q polynomials of koornwinder and macdonald liouville integrability of the corresponding classical systems follows as a corollary limit transitions lead to various well known models such as the nonrelativistic calogero moser systems associated with classical root systems and the relativistic calogero moser system

difference calogero moser systems and finite toda chains limits of a recently introduced n particle difference calogero moser system with elliptic potentials are studied we obtain hyperbolic and rational difference calogero moser systems with an eight parameter external field and finite difference toda chains with four parameter potentials acting on the boundary particles hamiltonians for a number of known integrable n particle systems such as ruijsenaars relativistic calogero moser and toda models and their generalizations associated with classical root systems can be seen as special cases of the hamiltonians considered in this paper

failure of the hill determinant method for the sextic anharmonic oscillator an extended analysis of eigenvalues and wavefunctions resulting from the hill determinant method is carried out for the one dimensional sextic anharmonic oscillator it is shown that in spite of a seemingly natural ansatz of the method irrelevant wavefunctions may appear leading to incorrect eigenvalues the domain of applicability of the method is limited and its practical use demands a priori investigation before concrete calculations of spectral characteristics effective variational calculations based on adequate trial functions are performed and comparison of the results of both approaches is presented

affine toda sutherland systems a cross between two well known integrable multi particle dynamics an affine toda molecule and a sutherland system is introduced for any affine root system though it is not completely integrable but partially integrable or quasi exactly solvable it inherits many remarkable properties from the parents the equilibrium position is algebraic i e proportional to the weyl vector the frequencies of small oscillations near equilibrium are proportional to the affine toda masses which are essential ingredients of the exact factorizable s matrices of affine toda field theories some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained an affine toda calogero system with a corresponding rational potential is also discussed

rodrigues formulas for the macdonald polynomials we present formulas of rodrigues type giving the macdonald polynomials for arbitrary partitions through the repeated application of creation operatorsbk k 1 l on the constant 1 three expressions for the creation operators are derived one from the other when the last of these expressions is used the associated rodrigues formula readily implies the integrality of the q t kostka coefficients the proofs given in this paper rely on the connection between affine hecke algebras and macdonald polynomials

the 1d interacting bose gas in a hard wall box we consider the integrable one dimensional function interacting bose gas in a hard wall box which is exactly solved via the coordinate bethe ansatz the ground state energy including the surface energy is derived from the lieb liniger type integral equations the leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete bethe equations and the integral equations this allows the investigation of both finite size and boundary effects in the integrable model we also study the luttinger liquid behaviour by calculating luttinger parameters and correlations the hard wall boundary conditions are seen to have a strong effect on the ground state energy and phase correlations in the weak coupling regime enhancement of the local two body correlations is shown by application of the hellmann feynman theorem

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

multivariable continuous hahn and wilson polynomials related to integrable difference systems multivariable generalizations of the continuous hahn and wilson polynomials are introduced as eigenfunctions of rational ruijsenaars type difference systems with an external field

quantum calogero moser models integrability for all root systems the issues related to the integrability of quantum calogero moser models based on any root systems are addressed for the models with degenerate potentials i e the rational with without the harmonic confining force the hyperbolic and the trigonometric we demonstrate the following for all the root systems i construction of a complete set of quantum conserved quantities in terms of a total sum of the lax matrix l i e sigma mu nu is an element ofr l n mu nu in which r is a set of r r vectors invariant under the action of the coxeter group they form a single coxeter orbit ii proof of liouville integrability iii triangularity of the quantum hamiltonian and the entire discrete spectrum generalized jack polynomials are defined for all root systems as unique eigenfunctions of the hamiltonian iv equivalence of the lax operator and the dunkl operator v algebraic construction of all excited states in terms of creation operators these are mainly generalizations of the results known for the models based on the a series i e su n type root systems

the intersection of vibronic levels the basic one dimensional vibronic problem has been solved exactly in an analytical form for degenerate configurations the solution using the intermediate p representation and the heun functions expansion on the gauss basis set has indicated the existence of dynamic symmetry the direct solution in x representation has specified an effective method for the calculation of vibronic problems conical intersection parameters have been obtained and analyzed for a few lower levels

universal lax pairs for spin calogero moser models and spin exchange models for any root system and a set of vectors which form a single orbit of the reflection weyl group g generated by a spin calogero moser model can be defined for each of the potentials rational hyperbolic trigonometric and elliptic for each member of to be called a site we associate a vector space v whose element is called a spin its dynamical variables are the canonical coordinates of a particle in rr r rank of and spin exchange operators which exchange the spins at the sites and s here s is the reflection generated by for each and a spin exchange model can be defined the hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only it is obtained by freezing the canonical variables at the equilibrium point of the corresponding classical calogero moser model for ar and set of vector weights it reduces to the well known haldane shastry model universal lax pair operators for both spin calogero moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials

stability and stabilizability of discrete time dual switching systems with application to sampled data systems in this paper stability and stabilizability of discrete time dual switching linear systems is investigated the switched systems under consideration have two switching variables one of them is stochastic described by an underlying markov chain the other one can be regarded either as a deterministic disturbance or as a control input leading to stability or stabilizability problems respectively for the considered class of systems sufficient conditions for mean square stability with or without control gain synthesis and mean square stabilizability are provided in terms of matrix inequalities when the stochastic switching is driven by an independent identically distributed sequence we establish simpler conditions without additional conservatism then it is shown how the proposed framework can be used to study aperiodic sampled data systems with stochastic computation times the results are illustrated on examples borrowed from the literature

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

the dn ruijsenaars schneider model the lax pair of the ruijsenaars schneider model with interaction potential of trigonometric type based on dn lie algebra is presented we give a general form for the lax pair and prove partial results for small n liouville integrability of the corresponding system follows a series of involutive hamiltonians generated by the characteristic polynomial of the lax matrix the rational case appears as a natural degeneration and the non relativistic limit exactly leads to the well known calogero moser system associated with dn lie algebra

exact results for the potentials v ax2s bx4s 2 the condition for a double well potential v ax2s bx4s 2 s 1 2 3 to have a negative energy branch is derived for an arbitrary geometry of the well the total number of negative levels is determined some general features of the spectrum are pointed out the results are based on a sequence of exactly obtainable zero energy solutions

on some solutions to generalized spheroidal wave equations and applications expansions in series of coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations gswes are analysed and written together in pairs each pair consists of a solution in series of hypergeometric functions and another in series of coulomb wavefunctions and has the same recurrence relations for the series coefficients but the solutions inside it present different radii of convergence expansions without a phase parameter are derived by truncating the series with a phase parameter for the whittaker hill equation solutions are found by treating that equation as a particular case of gswe while for the confluent gswe solutions with and without a phase parameter are given as pairs of series of coulomb wavefunctions amongst the applications there are equations for the time dependence of dirac test fields in some nonflat friedmann robertson walker spacetimes the radial schrodinger equation for an electron in the field of two coulombian centres and the schrodinger equation for the razavy type quasi exactly solvable potentials for these problems it is possible to find wavefunctions in terms of infinite series regular and convergent over the entire range of the independent variable by matching expansions belonging to one or more of the above pairs the infinite series solutions for the razavy type potentials in addition to the polynomial ones suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi classical methods of approximation

quantum versus classical integrability in ruijsenaars schneider systems the relationship resemblance and or contrast between quantum and classical integrability in ruijsenaars schneider systems which are one parameter deformations of calogero moser systems is addressed many remarkable properties of classical calogero and sutherland systems based on any root system at equilibrium are reported in a previous paper corrigan sasaki for example the minimum energies frequencies of small oscillations and the eigenvalues of lax pair matrices at equilibrium are all integer valued in this paper we report that similar features and results hold for the ruijsenaars schneider type of integrable systems based on the classical root systems

common algebraic structure for the calogero sutherland models we investigate a common algebraic structure for the rational and trigonometric calogero sutherland models by using the exchange operator formalism we show that the set of jack polynomials whose arguments are dunkl type operators provides an orthogonal basis for the rational case

hidden algebras of the super calogero and sutherland models we propose to parametrize the configuration space of one dimensional quantum systems of n identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates it is shown that in this parametrization the hamiltonians of the an bcn bn cn and dn calogero and sutherland models as well as their supersymmetric generalizations can be expressed for arbitrary values of the coupling constants as quadratic polynomials in the generators of a borel subalgebra of the lie algebra gl n 1 or the lie superalgebra gl n 1 n for the supersymmetric case these algebras are realized by first order differential operators this fact establishes the exact solvability of the models according to the general definition given by turbiner and implies that the calogero and jack sutherland polynomials as well as their supersymmetric generalizations are related to finite dimensional irreducible representations of the lie algebra gl n 1 and the lie superalgebra gl n 1 n

perturbations around the zeros of classical orthogonal polynomials starting from degree n solutions of a time dependent schrodinger like equation for classical orthogonal polynomials a linear matrix equation describing perturbations around the n zeros of the polynomial is derived the matrix has remarkable diophantine properties its eigenvalues are independent of the zeros the corresponding eigenvectors provide the representations of the lower degree 0 1 n 1 polynomials in terms of the zeros of the degree n polynomial the results are valid universally for all the classical orthogonal polynomials including the askey scheme of hypergeometric orthogonal polynomials and its q analogues

joint spectral radius and path complete graph lyapunov functions we introduce the framework of path complete graph lyapunov functions for approximation of the joint spectral radius the approach is based on the analysis of the underlying switched system via inequalities imposed among multiple lyapunov functions associated to a labeled directed graph inspired by concepts in automata theory and symbolic dynamics we define a class of graphs called path complete graphs and show that any such graph gives rise to a method for proving stability of the switched system this enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic common sum of squares path dependent quadratic and maximum minimum of quadratics lyapunov functions we compare the quality of approximation obtained by certain classes of path complete graphs including a family of dual graphs and all path complete graphs with two nodes on an alphabet of two matrices we derive approximation guarantees for several families of path complete graphs such as the de bruijn graphs this provides worst case performance bounds for path dependent quadratic lyapunov functions and a constructive converse lyapunov theorem for maximum minimum of quadratics lyapunov functions

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

algebraic linearization of dynamics of calogero type for any coxeter group calogero moser systems can be generalized for any root system including the noncrystallographic cases the algebraic linearization of the generalized calogero moser systems and of their quadratic respectively quartic perturbations are discussed

equilibria of discrete integrable systems and deformation of classical orthogonal polynomials the ruijsenaars schneider systems are discrete version of the calogero moser cm systems in the sense that the momentum operator p appears in the hamiltonians as a polynomial in e p is a deformation parameter instead of an ordinary polynomial in p in the hierarchies of c m systems we determine the polynomials describing the equilibrium positions of the rational and trigonometric ruijsenaars schneider systems based on classical root systems these are deformation of the classical orthogonal polynomials the hermite laguerre and jacobi polynomials which describe the equilibrium positions of the corresponding calogero and sutherland systems the orthogonality of the original polynomials is inherited by the deformed ones which satisfy three term recurrence and certain functional equations the latter reduce to the celebrated second order differential equations satisfied by the classical orthogonal polynomials

elliptic quantum many body problem and double affine knizhnik zamolodchikov equation the elliptic matrix quantum olshanetsky perelomov problem is introduced for arbitrary root systems by means of an elliptic version of the dunkl operators its equivalence with the double affine generalization of the knizhnik zamolodchikov equation in the induced representations is established

a moments analysis of quasi exactly solvable systems a new perspective on the sextic potential gx 6 bx 4 mx 2 beta over x 2 a quantum system is quasi exactly solvable qes when a subset of its eigenvalues can be obtained in closed form or they are the roots of closed form expressions in one dimension qes states assume the form involving a known positive reference function and a polynomial p x so that the taylor expansion of truncates at a finite order for non qes states this truncation procedure for suitable reference functions can provide approximate values for the eigenvalues it corresponds to the hill determinant method which is known to be unstable when assumes the controlling asymptotic form of the physical state for this reason it cannot be used to simultaneously generate exact qes states while approximating non qes states to address these limitations the orthogonal polynomial projection quantization oppq method was developed handy and vrinceanu 2013 j phys a math theor 46 135202 2013 j phys b atom mol opt phys 46 115002 it is demonstrated in this paper that oppq can directly and transparently provide at the same time both algebraic solutions for qes states and convergent numerically stable approximations for non qes states within the oppq analysis for the wavefunction representation the bender dunne energy orthogonal polynomials correspond exactly up to a numerical factor to the energy dependent power moments p dx xp x within this perspective the existence of qes states is associated with an anomalous kink behavior in the order of the finite difference moment equation corresponding to the s suggesting a change in the number of degrees of freedom this was first noted through the implementation of the eigenvalue moment method the first application of semidefinite programming analysis to quantum operators handy and bessis 1985 phys rev lett 55 931 this moments perspective also reveals additional properties for the non qes states of the same symmetry as the qes states their lower order moments must be zero we demonstrate our results in the context of the two sextic potentials vsa x x6 mx2 bx4 and vss x x6 mx2 b x2

heisenberg algebra umbral calculus and orthogonal polynomials umbral calculus can be viewed as an abstract theory of the heisenberg commutation relation p m 1 in ordinary quantum mechanics p is the derivative and m the coordinate operator here we shall realize p as a second order differential operator and m as a first order integral one we show that this makes it possible to solve large classes of differential and integrodifferential equations and to introduce new classes of orthogonal polynomials related to laguerre polynomials these polynomials are particularly well suited for describing the so called flatenned beams in laser theory

reply to comment on the darboux transformation and algebraic deformations of shape invariant potentials we reply to the comment on our recent paper made by drs sinha and roy we agree that the backward darboux transformation method used in our paper is equivalent to the approach based on ces potentials but we stress that the emphasis and the results of our paper are different

explicit solutions of the classical calogero and sutherland systems for any root system explicit solutions of the classical calogero rational with without harmonic confining potential and sutherland trigonometric potential systems is obtained by diagonalization of certain matrices of simple time evolution the method works for calogero sutherland systems based on any root system it generalizes the well known results by olshanetsky and perelomov for the a type root systems explicit solutions of the rational and trigonometric higher hamiltonian flows of the integrable hierarchy can be readily obtained in a similar way for those based on the classical root systems

on path complete lyapunov functions geometry and comparison we study optimization based criteria for the stability of switching systems known as path complete lyapunov functions and ask the question can we decide algorithmically when a criterion is less conservative than another our contribution is twofold first we show that a path complete lyapunov function which is a multiple lyapunov function by nature can always be expressed as a common lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it geometrically our results provide for each path complete criterion an implied invariant set second we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given path complete lyapunov functions