a nonconfocal generator of involutive systems and levy hierarchy a new nonconfocal generator of involutive systems is obtained and the relation between it and levy equation hierarchy is discussed furthermore the representation of the solution for the levy hierarchy is given

stability of linear time varying delay systems and applications to control problems this paper deals with the stability of a class of linear time varying systems with multiple delays using the lyapunov function method we give sufficient delay dependent conditions for the exponential stability with a given convergence rate which are described in terms of linear matrix inequalities lmi and the solution of riccati differential equations rde the results are applied to the problem of stabilization of linear time varying control systems with multiple delays numerical examples are given to illustrate the results

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

exponential stability and solution bounds for systems with bounded nonlinearities estimation of lyapunov exponents of systems with bounded nonlinearities plays an essential part in their robust control known results in this field are based on the gronwall inequality yielding relatively conservative bounds for lyapunov exponents in this note we obtained more accurate upper bounds for the general lyapunov exponent for systems consisting of a known linear time varying part and an unknown nonlinear component with a bounded lipschitz constant at zero consequently a sufficient condition for exponential stability of this system is formulated the systems are indicated for which the obtained bound is precise i e cannot be improved without additional information on the nonlinear term in the presence of a persisting perturbation an upper bound for the solution norm is derived and expressed in the norm of the solution of the corresponding linear system using the obtained results a condition for exponential stability of a linear time varying control system with a nonlinear feedback is derived numerical results are obtained for a second order time varying system and for the lienard equation in the latter case they are favorably compared with stability conditions previously obtained using the lyapunov function method

re examination of the algebraic scattering theory the foundations of the algebraic scattering theory ast are re examined this revision leads to a modified version of the ast it is proven that the algebraic approach alone does not yield a uniquely determined s matrix however a coordinate representation of the generators of the scattering group allows a unique decision about otherwise arbitrary phases the application of the so 2 3 ast nd the so 1 3 ast to modified coulomb scattering is reconsidered

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

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

quasi exactly solvable models with an inhomogeneous magnetic field let group generators having finite dimensional representation be realized as hermitian linear differential operators without inhomogeneous terms as takes place for example for the so n group then corresponding group hamiltonians containing terms linear in generators along with quadratic ones give rise to quasi exactly solvable models with a magnetic field in a curved space in particular for the so 4 group hamiltonian with isotropic quadratic part the manifold within which a quantum particle moves has the geometry of the einstein universe

a novel approach to exponential stability of nonlinear systems with time varying delays in this paper the stability of nonlinear systems with time varying delays is investigated by means of the concepts of generalized dahlquist constant generalized relative dahlquist constant and relative minimal lipschitz constant in detail two sufficient conditions are derived for the exponential stability of nonlinear systems with time varying delays and the exponential decay of the solutions is also estimated compared with some existing results our stability conditions are less conservative some examples are given to illustrate the effectiveness of the obtained results

bi hamiltonian formulation of the kadomtsev petviashvili and benjamin ono equations it was shown recently that the kadomtsev petviashvili kp equation an integrable equation in 2 1 i e in two spatial and one temporal dimensions admits a bi hamiltonian formulation this was achieved by considering kp as a reduction of a 3 1 dimensional system in the variables x y1 y2 t it is shown here using the kp as a concrete example that equations in 2 1 possess two bi hamiltonian formulations and two recursion operators both hamiltonian operators associated with the x direction are local in contrast only one of the hamiltonian operators associated with the y direction is local furthermore using the benjamin ono equation as a concrete example it is shown that intergrodifferential equations in 1 1 admit an algebraic formulation analogous to that of equations in 2 1

on the families of orthogonal polynomials associated to the razavy potential we show that there are two different families of weakly orthogonal polynomials associated to the quasi exactly solvable razavy potential v x cos 2x m 2 0 m one of these families encompasses the four sets of orthogonal polynomials recently found by khare and mandal while the other one is new these results are extended to the related periodic potential u x cos 2x m 2 for which we also construct two different families of weakly orthogonal polynomials we prove that either of these two families yields the ground state when m is odd and the lowest lying gaps in the energy spectrum of the latter periodic potential up to and including the m 1 th gap and having the same parity as m 1 moreover we show that the algebraic eigenfunctions obtained in this way are the well known finite solutions of the whittaker hill or hill s three term periodic differential equation thus the foregoing results provide a lie algebraic justification of the fact that the whittaker hill equation unlike for instance mathieu s equation admits finite solutions

so 3 algebraic approach to the morse potential we construct so 3 algebra associated with the morse potential and show that these operators obey so 3 commutation relations a so 3 algebraic method is proposed in order to obtain the eigenvalues and eigenfunctions of the morse potential this method exhibits that cartan operator jz the lowering operator j and the raising operator j determine successfully energy eigenvalues the lowest energy eigenfunction and excited energy eigenfunctions respectively

on an estimate of the decay rate for stable linear delay systems in this paper it is shown that for a certain class of stable linear delay systems we can obtain an estimate of the decay rate of the solution by a simple computation

on scattering systems related to the so 2 1 group ii we study a class of solvable hamiltonians h which have strongly broken potential group structure however the scattering matrices of the systems under consideration are also related to the intertwining operator of so 2 1

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

so 2 1 lie algebra the jacobi matrix and the scattering states of the morse oscillator work exploring the connection between so 2 1 lie algebra and the jacobi matrix method for scattering is extended to the morse potential regular and irregular solutions of the three term recursion associated with the jacobi matrix are specified by their asymptotic behaviour and closed form expressions in terms of generalised hypergeometric functions are obtained this example further illustrates the connection between scattering theory and orthogonal polynomials

classical liouville completely integrable systems associated with the solutions of boussinesq burgers hierarchy two new finite dimensional completely integrable systems in the liouville sense are obtained the solutions of boussinesq burgers hierarchy are generated by using involutive solutions of the commutable flows in the completely integrable systems

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

complex confocal involutive systems associated with the solutions of the akns evolution equations in this paper the complex involutive systems associated with the ablowitz kaup newell and segur akns evolution equations are discussed by means of the complex representations of the standard symplectic form and the standard poisson bracket on the real space r2n using the equivalent relations between the hamiltonian canonical equations it is proved that the complex involutive systems are exactly finite dimensional completely integrable systems in the liouville sense therefore by mapping nonlinearly from the involutive solutions of the compatible hamiltonian canonical equations to the akns evolution equations the representations of the solutions of the akns evolution equations are obtained

on solvable potentials related to so 2 2 scattering problems in one dimension associated with the so 2 2 group are studied the s matrices for the systems under consideration are calculated by using the theory of intertwining operators for semisimple lie groups the wave functions are expressed in terms of matrix elements of principal series representation of the so 2 1 group

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

multicomponent integrable reductions in the kadomtsev petviashvilli hierarchy new types of reductions of the kadomtsev petviashvili kp hierarchy are considered on the basis of sato s approach within this approach the kp hierarchy is represented by infinite sets of equations for potentials u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp hierarchy was studied under constraints of the following type ni 1 i i x s x where s x are symmetries for the kp equation and i i i i are eigenfunctions with eigenvalue i it is shown that for the first three cases 2 3 4 these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 1 n 1 n bi hamiltonian structures for the new hierarchies are presented

on algebraic classification of hermitian quasi exactly solvable matrix schrodinger operators on line we construct six multi parameter families of hermitian quasi exactly solvable matrix schrodinger operators in one variable the method for finding these operators relies heavily upon a special representation of the lie algebra o 2 2 sl 2 sl 2 whose representation space contains an invariant finite dimensional subspace furthermore we select those quasi exactly solvable matrix models that have square integrable eigenfunctions on these models are in direct analogy with the quasi exactly solvable scalar schrodinger operators obtained by turbiner and ushveridze

on algebraic classification of quasi exactly solvable matrix models we suggest a generalization of the lie algebraic approach for constructing quasi exactly solvable one dimensional schrodinger equations this generalization is based on representations of lie algebras by first order matrix differential operators we have classified inequivalent representations of the lie algebras of dimensions up to three by first order matrix differential operators in one variable next we describe invariant finite dimensional subspaces of the representation spaces of the one two dimensional lie algebras and of the algebra these results enable us to construct multiparameter families of first and second order quasi exactly solvable models in particular we have obtained two classes of quasi exactly solvable matrix schrodinger equations

equations of motion and wave functions in spherical coordinates for an integrable hamiltonian system the classical equations of motion for the hamiltonian h n 0 y2 2 u2 x2 where x2 1 y is the conjugate momentum to x and u is constant are solved by separation of variables in spherical coordinates in the hamilton jacobi equation this flow is related to the one obtained from the projection of geodesic free flow on the sphere s2n 1 wave functions for the quantum case together with the level degeneracy for the generic case u 0 are also given

completely integrable relativistic hamiltonian systems and separation of variables in hermitian hyperbolic spaces the hamilton jacobi and laplace beltrami equations on the hermitian hyperbolic space hh 2 are shown to allow the separation of variables in precisely 12 classes of coordinate systems the isometry group of this two complex dimensional riemannian space su 2 1 has four mutually nonconjugate maximal abelian subgroups these subgroups are used to construct the separable coordinates explicitly all of these subgroups are two dimensional and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones the symmetry reduction of the free hh 2 hamiltonian by a maximal abelian subgroup of su 2 1 reduces this hamiltonian to one defined on an o 2 1 hyperboloid and involving a nontrivial singular potential separation of variables on hh 2 and more generally on hh n thus provides a new method of generating nontrivial completely integrable relativistic hamiltonian systems

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

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

finite band solutions of the classical boussinesq burgers equations the classical boussinesq burgers hierarchy is decomposed into two systems of solvable ordinary differential equations with the help of the lax representations of stationary evolution equations from which the finite band solutions of the higher order classical boussinesq burgers equations are obtained

symmetry scattering for su 2 2 with applications in the framework of symmetry scattering su 2 2 invariant differential equations on the homogeneous space x su 2 2 s u 2 u 2 are studied the radial schrodinger equation for a family of one or two dimensional potentials or for two particles arise from the asymptotic behavior of the solutions exact partial wave scattering amplitudes are derived

on polynomial solutions of differential equations a general method of obtaining linear differential equations having polynomial solutions is proposed the method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some lie algebra in the projectivized representation possessing an invariant subspace and the spectral problem for a certain linear differential operator with variable coefficients it is shown in general that polynomial solutions of partial differential equations occur in the case of lie superalgebras there are polynomial solutions of some matrix differential equations quantum algebras give rise to polynomial solutions of finite difference equations particularly known classical orthogonal polynomials will appear when considering sl 2 r acting on rp1 as examples some polynomials connected to projectivized representations of sl2 r sl2 r q osp 2 2 and so3 are briefly discussed

on euclidean connections for su 1 1 suq 1 1 and the algebraic approach to scattering we obtain a general euclidean connection for the su 1 1 suq 1 1 algebras our euclidean connection allows an algebraic derivation for the s matrix these algebraic s matrices reduce to the known ones in suitable circumstances we also obtain a map between the su 1 1 and suq 1 1 representations

lie algebras of vector fields in the real plane finite dimensional real analytic lie algebras of vector fields on r 2 are completely classified up to changes of local coordinates

improved exponential stability for time varying systems with nonlinear delayed perturbations in this paper a new sufficient delay dependent exponential stability condition for a class of linear time varying systems with nonlinear delayed perturbations is derived by using an improved lyapunov krasovskii functional the proposed exponential stability conditions are formulated in terms of the solution of lyapunov differential equations the approach allows for computation of the bounds that characterize the exponential stability rate of convergence of the solution compared with existing results our conditions are shown to be less conservative numerical examples are given to illustrate the effectiveness of the obtained conditions

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

on scattering systems related to the group scattering systems related to the noncompact groups g in the sense that the hamiltonian of the system can be written as a function of the casimir operator of g are considered the s matrix for such systems are defined in terms of an intertwining operator of underling symmetry group g the s matrices for one dimensional scattering systems with symmetry group are classified

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

on algebraic models of relativistic scattering in this paper we develop an algebraic technique for building relativistic models in the framework of the direct interaction theories the interacting mass operator m in the bakamjian thomas construction is related to a quadratic casimir operator c of a non compact group g as a consequence the s matrix can be gained from an intertwining relation between weyl equivalent representations of g the method is illustrated by explicit application to a model with so 3 1 dynamical symmetry

two finite dimensional completely integrable hamiltonian systems associated with the solutions of the mkdv hierarchy in this paper the equations of the mkdv hierarchy and their lax pairs are obtained from zero curvature equations using trace formulas of the spectral problem and the lax pairs two finite dimensional completely integrable systems are found the solving equations of the mkdv hierarchy are transformed into the solving hamiltonian canonical equations

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

path integral on s2 the rosen morse oscillator guided by the group chain so 3 contains implies o 2 the authors construct an angular path integral for the symmetric rosen morse oscillator on s2 by explicit path integration they obtain the normalised energy eigenfunction as well as the exact energy spectrum

a class of graded lie algebras of vector fields and first order differential operators finite dimensional lie algebras of polynomial vector fields on rn that contain the elements xi and xi xi for i 1 n were studied to any lie algebra l of this class an n valued n n matrix a and a set of special elements s 1 n are associated it is proven that the pair a s necessarily satisfies two properties conversely to any pair a s satisfying those two properties is associated a lie algebra l a s such that l a s is maximal in the class of all l with matrix a and special elements s for the lie algebras l a s the possible extensions to first order differential operators and its modules of c functions are discussed 1994 american institute of physics

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

stability of perturbed systems with time varying delays abstract this paper gives easily verifiable sufficient conditions of robust asymptotic stability of linear time delay systems subject to parametric unstructured or highly structured perturbations the criteria given in this paper are delay independent or delay dependent the considered delay may be time varying an estimation of the transient behaviour of the studied systems is also provided exponential rate of convergence scalar or vectorial inequalities involving hurwitz matrices matrix measures and norms constitute the mathematical foundations of the exposed results

a group theoretical calculation of the s matrix for the dirac coulomb problem recently developed group theoretic techniques for calculating the s matrix are extended to the relativistic scattering regime the dirac coulomb problem is treated in detail phase shifts for an approximate coulomb dirac hamiltonian which has so 3 1 symmetry are obtained for the first time again using a group theoretical approach the methods developed here may be useful in the construction of algebraic models for proton nucleus scattering

global stabilization for linear continuous time varying systems in this paper stabilization problem via static output feedback controls for linear time varying systems is investigated based on the lyapunov function techniques we show that for linear time varying systems the global null controllability guarantees the output feedback stabilization the verifiable stabilizability conditions and output feedback control design are stated the result can be applicable to the output feedback stabilizability of a class of nonlinear time varying systems numerical examples illustrated the conditions are given

group classification of heat conductivity equations with a nonlinear source we suggest a systematic procedure for classifying partial differential equations pdes invariant with respect to low dimensional lie algebras this procedure is a proper synthesis of the infinitesimal lie method the technique of equivalence transformations and the theory of classification of abstract low dimensional lie algebras as an application we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms we have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry it is shown that there are 3 7 28 and 12 inequivalent classes of pdes of the type considered that are invariant under the one two three and four dimensional lie algebras correspondingly furthermore we prove that any pde belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation this classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach furthermore a number of new invariant equations are constructed which have rich symmetry properties and therefore may be used for mathematical modelling of say nonlinear heat transfer processes

algebraic exact solvability of trigonometric type hamiltonians associated to root systems in this article we study and settle several structural questions concerning the exact solvability of the olshanetsky perelomov quantum hamiltonians corresponding to an arbitrary root system we show that these operators can be written as linear combinations of certain basic operators admitting infinite flags of invariant subspaces namely the laplacian and the logarithmic gradient of invariant factors of the weyl denominator the coefficients of the constituent linear combination become the coupling constants of the final model we also demonstrate the l2 completeness of the eigenfunctions obtained by this procedure and describe a straightforward recursive procedure based on the freudenthal multiplicity formula for constructing the eigenfunctions explicitly

integrable systems based on su p q homogeneous manifolds the general theory of the separation of variables in hamilton jacobi and laplace beltrami equations on the su p q hyperboloid is used to introduce completely integrable hamiltonian systems on o p q hyperboloids each of the q 1 different cartan subalgebras of su p q leads to a different integrable o p q potential different complete sets of integrals of motion are obtained for each of the integrable systems

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

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

abelian integrals and the reduction method for an integrable hamiltonian system a classical finite dimensional integrable hamiltonian system corresponding to the motion of a particle constrained to an n dimensional sphere n 0 x2 1 with the hamiltonian h 1 2 y2 u2 x2 e x2 where u and e are constants and y are the momenta conjugate to x is integrated using several different methods these are the following 1 the projection of geodesic free flow on a larger space namely the sphere s2n 1 for e 0 the flow is obtained in terms of elementary functions 2 separation of variables in the hamilton jacobi equation in elliptic coordinates or alternatively the use of a complete set of integrals of motion in involution to reduce hamilton s equations to quadratures the flow is obtained in terms of abelian integrals which are then inverted in terms of generalized functions the relation between the different methods and results is clarified using methods of algebraic geometry in particular the geometry of quadrics

algebraic integrability of the confluent neumann system in this paper we study the neumann system which describes the harmonic oscillator of arbitrary dimension constrained to the sphere in particular we will consider the confluent case where two eigenvalues of the potential coincide which implies that the system has s1 symmetry we will prove complete algebraic integrability of the confluent neumann system and show that its flow can be linearized on the generalized jacobian torus of some singular algebraic curve the symplectic reduction of s1 action will be described and we will show that the general rosochatius system is a symplectic quotient of the confluent neumann system where all the eigenvalues of the potential are double

convergence of bounded solutions for a class of systems of delay differential equations abstract in this paper the convergence of a class of systems of delay differential equations is considered we show that every bounded solution of such systems tends to an equilibrium under certain hypotheses our results extend some corresponding ones already known

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

models with sl 2 c symmetry and their s matrices we study quantum systems whose scattering modes are governed by unitary representation of the algebra the s matrices of the systems under consideration are defined from intertwining relations for the weyl equivalent representations of the group sl 2 c or its lie algebra

reductions of 2 1 dimensional integrable systems via mixed potential eigenfunction constraints new types of reductions of 2 1 dimensional integrable systems which are associated with mixed potential eigenfunction constraints are discussed necessary and sufficient conditions for the admissibility of such constraints are presented classification results for integrable equations will be used for applying these conditions several typical examples are considered including the kp mkp 2 1 dimensional savada kotera kaup kupershmidt harry dym and nizhnik veselov novikov equations