deformed shape invariance and exactly solvable hamiltonians with position dependent effective mass known shape invariant potentials for the constant mass schrodinger equation are taken as effective potentials in a position dependent effective mass pdem one the corresponding shape invariance condition turns out to be deformed its solvability imposes the form of both the deformed superpotential and the pdem a lot of new exactly solvable potentials associated with a pdem background are generated in this way a novel and important condition restricting the existence of bound states whenever the pdem vanishes at an end point of the interval is identified in some cases the bound state spectrum results from a smooth deformation of that of the conventional shape invariant potential used in the construction in others one observes a generation or suppression of bound states depending on the mass parameter values the corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials

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

periodic potentials and pt symmetry we enlarge the limited class of analytically solvable one dimensional periodic potentials with a finite number of band gaps by considering the complex pt symmetric generalized associated lame gal potential with four parameters and applying supersymmetry we obtain three duality relations which relate the band edge eigenstates of the twenty four gal potentials obtained by permutation of the four parameters

a new effective mass hamiltonian and associated lame equation bound states a new quantum model with rational functions for the potential and effective mass is proposed in a stretchable region outside which both are constant starting from a generalized effective mass kinetic energy operator the matching and boundary conditions for the envelope wavefunctions are derived it is shown that in a mapping to an auxiliary constant mass schrodinger picture one obtains one period associated lame well bounded by two wells or barriers depending on the values of the ordering parameter the results for bound states of this new solvable model are provided for a wide variation of the parameters

construction of coupled harry dym hierarchy and its solutions from stackel systems in this paper we show how to construct the coupled multicomponent harry dym chd hierarchy from classical stackel separable systems both nonlocal and purely differential parts of hierarchies are obtained we also construct various classes of solutions of chd hierarchy from solutions of corresponding stackel systems

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

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

a quantum quasi harmonic nonlinear oscillator with an isotonic term the properties of a nonlinear oscillator with an additional term kg x2 characterizing the isotonic oscillator are studied the nonlinearity affects to both the kinetic term and the potential and combines two nonlinearities associated to two parameters and kg in such a way that for 0 all the characteristics of the standard isotonic system are recovered the first part is devoted to the classical system and the second part to the quantum system this is a problem of quantization of a system with position dependent mass of the form m x 1 1 x2 with a dependent non polynomial rational potential and with an additional isotonic term the schrodinger equation is exactly solved and the kg dependent wave functions and bound state energies are explicitly obtained for both 0

analytical results of zero gap states in periodic potentials we develop a method to construct various classes of one dimensional periodic potentials with two intersecting energy bands analytical exact results for the zero gap states are presented in an explicit form under certain parameter conditions the position of the energies of these zero gap states in the energy bands is identified numerically

complex periodic potentials with a finite number of band gaps we obtain several new results for the complex generalized associated lame potential v x a a 1 msn2 y m b b 1 msn2 y k m m f f 1 msn2 y k m ik m m g g 1 msn2 y ik m m where y x k m 2 ik m 2 sn y m is the jacobi elliptic function with modulus parameter m and there are four real parameters a b f g first we derive two new duality relations which when coupled with a previously obtained duality relation permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the parameters a b f g second we pose and answer the question how many independent potentials are there with a finite number a of band gaps when a b f g are integers and a b f g 0 for these potentials we clarify the nature of the band edge eigenfunctions we also obtain several analytic results when at least one of the four parameters is a half integer as a by product we also obtain new solutions of heun s differential equation

exact time dependence of solutions to the time dependent schrodinger equation solutions of the schrodinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time independent operators using time dependent unitary transformations exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time dependent hamiltonians in d dimensions including non hermitean symmetric hamiltonians hamiltonians are constructed using time dependent unitary spatial transformations comprising dilatations translations and rotations and solutions are found in several forms as eigenfunctions of a quadratic invariant as coherent state eigenfunctions of boson operators as plane wave solutions from which the general solution is obtained as an integral transform by means of the fourier transform and as distributional solutions for which the initial wavefunction is the dirac function for the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d 1 including barut girardello and perelomov coherent states i e vector coherent states which are shown to be related to eigenfunctions of the quadratic invariant by the transformation this transformation which leaves the ermakov equation invariant implements su 1 1 transformations on linear dynamical invariants coherent states are derived also for the time dependent linear potential exact solutions are found for hamiltonians with electromagnetic interactions in which the time dependent magnetic and electric fields are not necessarily spatially uniform as an example it is shown how to find exact solutions of the time dependent schrodinger equation for the dirac magnetic monopole in the presence of time dependent magnetic and electric fields of a specified form

an exactly solvable schrodinger equation with finite positive position dependent effective mass the solution of the one dimensional schrodinger equation is discussed in the case of position dependent mass the general formalism is specified for potentials that are solvable in terms of generalized laguerre polynomials and mass functions that are positive and bounded on the whole real x axis the resulting four parameter potential is shown to belong to the class of implicit potentials closed expressions are obtained for the bound state energies and the corresponding wave functions including their normalization constants the constant mass case is obtained by a specific choice of the parameters it is shown that this potential contains both the harmonic oscillator and the morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials possible generalizations of the method are outlined

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

remarks on the solution of the position dependent mass schrodinger equation an approximate method is proposed to solve the position dependent mass pdm schrodinger equation the procedure suggested here leads to the solution of the pdm schrodinger equation without transforming the potential function to the mass space or vice versa the method based on the asymptotic taylor expansion of the function produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the pdm schrodinger equation the results show that the pdm and constant mass schrodinger equations are not isospectral the calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm

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

ordering ambiguity versus representation in this work we show that the ordering ambiguity on quantization depends on the representation choice this property is then used to solve unambiguously some particular systems finally we speculate on the consequences for more involved cases

from stackel systems to integrable hierarchies of pde s benenti class of separation relations we propose a general scheme of constructing of soliton hierarchies from finite dimensional stackel systems and related separation relations in particular we concentrate on the simplest class of separation relations called benenti class i e certain stackel systems with quadratic in momenta integrals of motion

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

a new class of exact solutions for coupled scalar field equations the authors report a class of exact solutions valid in new regimes of parameter values for a generic system of nonlinear coupled scalar field equations which is the stationary reduction of the schrodinger boussinesq or korteweg de vries system and which extends the henon heiles system to hamiltonians with indefinite kinetic energy the results are applied to co propagating coupled nonlinear waves in magnetized plasmas

energy eigenvalues for the systems with position dependent effective mass based on the analytical transfer matrix method atmm the universal dispersion equation determining the energy spectrum for an arbitrary potential function experienced by a particle with arbitrary position dependent effective mass distribution is obtained as an illustrative example for a given mass distribution m x and a potential function v x our predicted results are in complete agreement with the exact ones the energy spectrum of a harmonic oscillator with nonconstant mass is obtained since the exact solutions to schrodinger equation with nonconstant effective mass are a valuable means for checking and improving models and numerical methods in the study of electronic properties the atmm will find wide applications in material science and condensed matter

lame equation sl 2 algebra and isospectral deformations it is proven that the n zone lame equation is equivalent to a spectral problem for a quadratic element of a universal enveloping sl 2 algebra the riemannian surface corresponding to the eigenstates in a parametric space forms the 2n 1 sheet surface which splits in four subsurfaces three of them contain the same number of sheets

elliptic solutions and blow up in an integrable henon heiles system we consider an integrable case of the henon heiles system and use an isomorphism with the two gap kdv flow to construct families of real elliptic trajectories which are associated with two gap elliptic solitons of the kdv equation some of these solutions exhibit blow up in finite time

orbital behaviour transition from the he non heiles to the three particle toda lattice hamiltonian abstract we examine the orbital behaviour in successive n order approximations 3 n 10 of the three particle toda lattice n 3 corresponding to the henon heiles galactic hamiltonian and in transition models between the 3rd and 4th orders of this expansion by using the poincare consequent method and by computing the e folding rate of the main unstable periodic orbits we locate and quantify the most important instabilities responsible for chaos in systems corresponding to even and odd order expansions

generalized stackel transform and reciprocal transformations for finite dimensional integrable systems we present a multiparameter generalization of the stackel transform the latter is also known as the coupling constant metamorphosis and show that under certain conditions this generalized stackel transform preserves liouville integrability noncommutative integrability and superintegrability the corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form and we investigate the properties of this reciprocal transformation finally we show that the hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized stackel transform

mapping of two dimensional schrodinger equation under the point transformation for the two dimensional schrodinger equation the general form of the point transformations such that the result can be interpreted as a schrodinger equation with effective i e position dependent mass is studied a wide class of such models with different forms of mass function is obtained in this way starting from the solvable two dimensional model the variety of solvable partner models with effective mass can be built several illustrating examples not amenable to conventional separation of variables are given

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

an exactly soluble schrodinger equation with smooth position dependent mass the one dimensional generalized schrodinger equation for a system with smooth potential and mass step is resolved exactly the wave function depends on the heun s function which is a solution of a second order fuchsian equation with four singularities the behavior of the transmission coefficient as a function of energy is compared to that of the case of an abrupt potential and mass step two limiting cases are also studied when the width of the mass step is vanishing and when the smooth potential and mass step tend to an abrupt potential and mass step

hierarchy of qm susys on a bounded domain we systematically formulate a hierarchy of isospectral hamiltonians in one dimensional supersymmetric quantum mechanics on an interval and on a circle in which two successive hamiltonians form supersymmetry we find that boundary conditions compatible with supersymmetry are severely restricted in the case of an interval a hierarchy of at most three isospectral hamiltonians is possible with unique boundary conditions while in the case of a circle an infinite tower of isospectral hamiltonians can be constructed with a two parameter family of boundary conditions

a systematic study on the exact solution of the position dependent mass schrodinger equation an algebraic method of constructing potentials for which the schrodinger equation with position dependent mass can be solved exactly is presented a general form of the generators of su 1 1 algebra has been employed with a unified approach to the problem our systematic approach reproduces a number of earlier results and also leads to some novelties we show that the solutions of the schrodinger equation with position dependent mass are free from the choice of parameters for position dependent mass two classes of potentials are constructed that include almost all exactly solvable potentials

a delta well with a mass jump in this paper it is shown that a one dimensional hamiltonian with an attractive delta potential at the origin plus a mass jump at the same point cannot have a bound state as is the case with the ordinary attractive delta potential with constant mass unless another term is added into the potential in the form of a derivative of the dirac delta at the origin the consistency of this singular potential with two terms is guaranteed by choosing suitable matching conditions at the singular point for the wavefunctions furthermore it is proved that the self adjointness of the hamiltonian with both singular interactions determines the coefficient of the derivative of the delta in a unique manner under these conditions the bound state and its energy are obtained and it is checked that the correct results in the limit of equal masses are obtained

stackel systems generating coupled kdv hierarchies and their finite gap and rational solutions we show how to generate coupled kdv hierarchies from stackel separable systems of benenti type we further show that the solutions of these stackel systems generate a large class of finite gap and rational solutions for ckdv hierarchies most of these solutions are new

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 classes of quasi solvable potentials their exactly solvable limit and related orthogonal polynomials we have generated using an sl 2 r lie algebraic formalism several new classes of quasi solvable elliptic potentials which in the appropriate limit go over to the exactly solvable forms we have obtained exact solutions of the corresponding spectral problem for some real values of the potential parameters we have also given explicit expressions of the families of associated orthogonal polynomials in the energy variable

associated lame and various other new classes of elliptic potentials from sl 2 r and related orthogonal polynomials using representations of sl 2 r generators which yield associated lame hamiltonians we obtain new classes of elliptic potentials we explicitly calculate eigenstates and spectra for these potentials and construct the associated orthogonal polynomials we show that in the proper limit these potentials reduce to well known exactly solvable potentials

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

the schrodinger equation with position dependent mass it is shown that the traditional hartree second order shooting method for eigenvalues can be modified to give accurate energy levels for a schrodinger equation involving a position dependent mass some numerical applications are given to demonstrate the effectiveness of the method

restricted flows of soliton hierarchies coupled kdv and harry dym case restricted flows of soliton hierarchies associated with the energy dependent schrodinger spectral problem are determined explicitly a remarkable connection with separable potentials is used for proving complete integrability of the restricted flows a previously unknown lagrangian and hamiltonian formulation of the neumann system is found whole families of generalizations of the neumann and garnier systems are given

regularized scarf potentials energy band structure and supersymmetry the singular one dimensional periodic scarf potential is regularized by means of one parameter square well counter terms it is shown that the regularized spectrum converges formally to the conventional scarf energy bands for specific values of the parameter the behaviour of the regularizations under supersymmetric transformations is also investigated this is a key point for the algebraic solvability of the regularized scarf potential

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

l2 series solutions of the dirac equation for power law potentials at rest mass energy we obtain solutions of the three dimensional dirac equation for radial power law potentials at rest mass energy as an infinite series of square integrable functions these are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the dirac operator is tridiagonal the wave equation results in a three term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials these are modified versions of the meixner pollaczek polynomials and of the continuous dual hahn polynomials the choice depends on the values of the angular momentum and the power of the potential

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 the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

a unified treatment of exactly solvable and quasi exactly solvable quantum potentials by exploiting the hidden algebraic structure of the schrodinger hamiltonian namely the sl 2 we propose a unified approach of generating both exactly solvable and quasi exactly solvable potentials we obtain in this way two new classes of quasi exactly solvable systems one of which is of periodic type while the other hyperbolic

a squeeze like operator approach to position dependent mass in quantum mechanics we provide a squeeze like transformation that allows one to remove a position dependent mass from the hamiltonian methods to solve the schrodinger equation may then be applied to find the respective eigenvalues and eigenfunctions as an example we consider a position dependent mass that leads to the integrable morse potential and therefore to well known solutions

new quasi exactly solvable periodic potentials we present two new classes of quasi exactly solvable periodic potentials it is shown that their solutions are given in terms of a known special function when the parameters in the two periodic potentials satisfy a certain condition some of the bloch waves and associated energies can be found exactly and in closed form at either the center or the edge of the first brillouin zone two fundamental sets of solutions have been constructed to calculate all the bloch waves and energy bands

spectrum generating algebras for position dependent mass oscillator schrodinger equations the interest of quadratic algebras for position dependent mass schrodinger equations is highlighted by constructing spectrum generating algebras for a class of d dimensional radial harmonic oscillators with d 2 and a specific mass choice depending on some positive parameter via some minor changes the one dimensional oscillator on the line with the same kind of mass is included in this class the existence of a single unitary irreducible representation belonging to the positive discrete series type for d 2 and of two of them for d 1 is proved the transition to the constant mass limit 0 is studied and deformed su 1 1 generators are constructed these operators are finally used to generate all the bound state wavefunctions by an algebraic procedure

generalized korteweg de vries equation induced from position dependent effective mass quantum models and mass deformed soliton solution through inverse scattering transform we consider one dimensional stationary position dependent effective mass quantum model and derive a generalized korteweg de vries kdv equation in 1 1 dimension through lax pair formulation one being the effective mass schrodinger operator and the other being the time evolution of wave functions we obtain an infinite number of conserved quantities for the generated nonlinear equation and explicitly show that the new generalized kdv equation is an integrable system inverse scattering transform method is applied to obtain general solution of the nonlinear equation and then n soliton solution is derived for reflectionless potentials finally a special choice has been made for the variable mass function to get mass deformed soliton solution the influence of position and time dependence of mass and also of the different representations of kinetic energy operator on the nature of such solitons is investigated in detail the remarkable features of such solitons are demonstrated in several interesting figures and are contrasted with the conventional kdv soliton associated with constant mass quantum model

periodic potentials and supersymmetry we review the current status of one dimensional periodic potentials and also present several new results it is shown that using the formalism of supersymmetric quantum mechanics one can considerably enlarge the limited class of analytically solvable one dimensional periodic potentials further using the landen transformations as well as cyclic identities for jacobi elliptic functions discovered by us recently it is shown that a linear superposition of lame as well as associated lame potentials is also analytically solvable finally using anti isospectral transformations we also obtain a class of analytically solvable complex pt invariant periodic potentials having real band spectra

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

elliptic baker akhiezer functions and an application to an integrable dynamical system an ansatz for elliptic baker akhiezer functions is used to find new solutions for various spectral problems the sturm liouville problem with four and five gap lame potentials and two third order problems one of the third order problems is the classical halphen equation with a spectral parameter the eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized henon heiles system

a quantum exactly solvable nonlinear oscillator related to the isotonic oscillator a nonpolynomial one dimensional quantum potential representing an oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator harmonic oscillator with a centripetal barrier is studied first the general case that depends on a parameter a is considered and then a particular case is studied with great detail it is proven that it is schr dinger solvable and then the wavefunctions n and the energies en of the bound states are explicitly obtained finally it is proven that the solutions determine a family of orthogonal polynomials related to the hermite polynomials and such that i every is a linear combination of three hermite polynomials and ii they are orthogonal with respect to a new measure obtained by modifying the classic hermite measure

cyclic identities involving jacobi elliptic functions we derive a number of local identities involving jacobi elliptic functions and use them to obtain several new results first we present an alternative simpler derivation of the cyclic identities discovered by us recently along with an extension to several new cyclic identities second we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp 2i s wheres is any integer third we systematize the local identities by deriving four local master identities analogous to the master identities for the cyclic sums discussed by us previously fourth we point out that many of the local identities can be thought of as exact discretizations of standard non linear differential equations satisfied by the jacobi elliptic functions finally we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving jacobi elliptic functions

symmetric effective mass schrodinger equations we outline a general method of obtaining exact solutions of symmetric schrodinger equations with a position dependent effective mass using this method exact solutions of some symmetric potentials have been obtained we have also discussed the construction of isospectral symmetric potentials

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

separable systems with quadratic in momenta first integrals the separability theory of hamiltonian systems on riemannian manifolds is reviewed and developed particular attention is paid to the systems generated by the so called special conformal killing tensors i e benenti systems then infinitely many new classes of separable systems are constructed by appropriate deformations of benenti class systems

quasi exactly solvable potentials on the line and orthogonal polynomials in this paper we show that a quasi exactly solvable normalizable or periodic one dimensional hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three term recursion relation in particular we prove that normalizable exactly solvable one dimensional systems are characterized by the fact that their associated polynomials satisfy a two term recursion relation we study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one dimensional quasi exactly solvable hamiltonian showing in particular that its associated stieltjes measure is supported on a finite set from this we deduce that the corresponding moment problem is determined and that the kth moment grows like the kth power of a constant as k tends to infinity we also show that the moments satisfy a constant coefficient linear difference equation and that this property actually characterizes weakly orthogonal polynomial 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

bifurcations of invariant manifolds in the generalized henon heiles system abstract the generalized henon heiles hamiltonian h 1 2 q dot 1 2 q dot 2 2 aq 1 2 bq 2 2 q 1 2 q 2 e 3 q 2 3 in the integrable case e 6 is studied from topological point of view the topology of the real level sets for all generic values of the constants of the motion and all generic bifurcations of the liouville tori and cylinders are described by making use of the reach algebraic structure of the problem

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures