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

coherent states associated with the wavefunctions and the spectrum of the isotonic oscillator classes of coherent states are presented by replacing the labelling parameter z of klauder perelomov type coherent states by confluent hypergeometric functions with specific parameters temporally stable coherent states for the isotonic oscillator hamiltonian are presented and these states are identified as a particular case of the so called mittag leffler coherent states

on the use of reducible functional differential equations in biological models biological models often lead to systems of delay or functional differential equations and to questions concerning the stability of equilibrium solutions of such equations the monographs by cushing 5 and macdonald 1 1 discuss a number of examples of such models which describe phenomena from population dynamics ecology and physiology the above cited work of macdonald is mainly devoted to the analysis of models leading to functional differential equations which are reducible to systems of ordinary differential equations for example consider the system

construction of solvable quantum mechanical potentials a method of generating some quantum mechanical exactly solvable potentials esps using the properties of classical orthogonal polynomials cops is presented it is illustrated using hypergeometric polynomial utilization of other cops to get more esps is indicated

the generalized quantum isotonic oscillator recently it has been proved that a nonlinear quantum oscillator generalization of the isotonic one is exactly solvable for certain values of its parameters here we show that the schrodinger equation for such an oscillator can be transformed into a confluent heun equation we give a very simple and efficient algorithm to solve it numerically no matter what the values of the parameters are algebraic quasi polynomial solutions for particular values of the parameters are found

l error bound for the phase matching approximation the one step at a time hankel norm model reduction version parallel to what was done for the balanced model reduction version of the phase approximation problem see li and jonckheere 1987 this paper provides an l bound on the phase approximation error when one step at a time optimal hankel norm model reduction is used to reduce the stable projection of the phase function matrix the one step at a time hankel norm procedure allows for an l phase error bound twice as small as that of the balanced version

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

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

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

analytic solutions darboux transformation operators and supersymmetry for a generalized one dimensional time dependent schrodinger equation abstract in this paper analytically investigated is a generalized one dimensional time dependent schrodinger equation using darboux transformation operator technique we construct the first order darboux transformation and the real valued condition of transformed potential for the generalized schrodinger equation to prove the equivalence of the supersymmetry formalism and the darboux transformation we investigate the relationship among first order darboux transformation supersymmetry and factorization of the corresponding effective mass hamiltonian furthermore the nth order darboux transformations are constructed by means of different method finally by using darboux transformation some analytical solutions are generated in a recursive manner for some examples of the schrodinger equation

special linear combinations of orthogonal polynomials abstract the paper lists a number of problems that motivate consideration of special linear combinations of polynomials orthogonal with the weight p x on the interval a b we study properties of the polynomials as well as the necessary and sufficient conditions for their orthogonality the special linear combinations of chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations

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

master symmetries non hamiltonian symmetries and superintegrability of the generalized smorodinsky winternitz system the first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach it is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries in the second part the properties of dynamical but non hamiltonian symmetries are studied it is proved that the higher order superintegrability of the generalized smorodinsky winternitz system is a consequence of the existence of symplectic symmetries not preserving the hamiltonian function

fisher information and shannon entropy of position dependent mass oscillators abstract we calculate the fisher information and the shannon entropy for three position dependent mass oscillators these systems can be seen as deformed harmonic oscillators in the sense that when the deformation parameter goes to zero they are identical to the constant mass harmonic oscillator for two out of the three oscillators we observe that as increases the position fisher information f x increases while the momentum fisher information f p decreases on the other hand the shannon entropy always increases for the three systems with increasing discussion about squeezing effect in either position or momentum due to the variation and a relation between the product of fisher information and the shannon entropy are also presented

an observation of quadratic algebra dual family of nonlinear coherent states and their non classical properties in the generalized isotonic oscillator in this paper we construct nonlinear coherent states for the generalized isotonic oscillator and study their non classical properties in detail by transforming the deformed ladder operators suitably which generate the quadratic algebra we obtain heisenberg algebra from the algebra we define two non unitary and one unitary displacement type operators while the action of one of the non unitary type operators reproduces the original nonlinear coherent states the other one fails to produce a new set of nonlinear coherent states dual pair we show that these dual states are not normalizable for the nonlinear coherent states we evaluate the parameter a3 and examine the non classical nature of the states through quadratic and amplitude squared squeezing effect further we derive analytical formula for the p function q function and the wigner function for the nonlinear coherent states all of them confirm the non classicality of the nonlinear coherent states in addition to the above we obtain the

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

an l error bound for the phase approximation problem given a random process spectral factor w cdot the phase approximation algorithm initiated by jonckheere and helton 1 constructs a reduced order spectral factor hat w cdot such that parallel w w ast hat w hat w ast parallel is small in the hankel norm sense in this note we derive the l infty error bound parallel w w ast hat w hat w ast parallel infty leq 4 sigma k 1 sigma n where the s are the canonical correlation coefficients a similar result holds in the multivariable case

spectral properties of a confined nonlinear quantum oscillator in one and three dimensions we analyze the spectral behaviour of a nonlinear quantum oscillator model under confinement the underlying potential is given by a harmonic oscillator interaction plus a nonlinear term that can be weakened or strengthened through a parameter numerical eigenvalues of the model in one and three dimensions are presented the asymptotic behaviour of the eigenvalues for confinement relaxation and for vanishing nonlinear term in the potential is investigated our findings are compared with existing results

quantization of the lienard ii equation and jacobi s last multiplier in this paper the role of jacobi s last multiplier in mechanical systems with a position dependent mass is unveiled in particular we map the lienard ii equation to a position dependent mass system the quantization of the lienard ii equation is then carried out using the point canonical transformation method together with the von roos ordering technique finally we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated laguerre and exceptional laguerre functions

generalized laguerre polynomials with position dependent effective mass visualized via wigner s distribution functions we construct analytically and numerically the wigner distribution functions for the exact solutions of the position dependent effective mass schrodinger equation for two cases belonging to the generalized laguerre polynomials using a suitable quantum canonical transformation expectation values of position and momentum operators are obtained analytically in order to verify the universality of heisenberg s uncertainty principle

on a novel supersymmetric connections between harmonic and isotonic oscillators abstract beginning from the harmonic oscillator we obtain a one parameter family of potentials whose spectra coincide with that of the isotonic oscillator diverse arguments concerning supersymmetric transformations factorization procedures and riccati equations are used in a systematic way in particular we take advantage of the non standard factorization of the schrodinger equation which appears when considering the energy of the first excited level as reference to be more specific supersymmetry is neither exact nor spontaneously broken but realizes itself acting on wave functions vanishing at x 0

a report on the nonlinear squeezed states and their non classical properties of a generalized isotonic oscillator we construct nonlinear squeezed states of a generalized isotonic oscillator potential we demonstrate the non existence of a dual counterpart of nonlinear squeezed states in this system we investigate the statistical properties exhibited by the squeezed states in particular mandel s parameter second order correlation function photon number distributions and parameter a3 in detail we also examine the quadrature and amplitude squared squeezing effects finally we derive the expression for the s parameterized quasi probability distribution function of these states all this information about the system is new to the literature

new solvable singular potentials we obtain three new solvable real shape invariant potentials starting from the harmonic oscillator poschl teller i and poschl teller ii potentials on the half axis and extending their domain to the full line while taking special care to regularize the inverse square singularity at the origin the regularization procedure gives rise to a delta function behaviour at the origin our new systems possess underlying nonlinear potential algebras which can also be used to determine their spectra analytically

a generalized cole hopf transformation for nonlinear odes we introduce a hybrid cole hopf darboux transformation to relate the solutions of nonlinear and linear second order differential equations and derive classification and sufficient condition for this correspondence we explore physical applications of this correspondence to nonlinear oscillations the duffing equation and a nonlinear form of the schr dinger equation for the nonrelativistic hydrogen atom in addition we show that solutions of some nonlinear second order equations are related to the special functions of mathematical physics through this transformation these nonlinear equations can be viewed as the class of special nonlinear equations which correspond to the linear differential equations which define the special functions of mathematical physics

disconjugacy regularity of multi indexed rationally extended potentials and laguerre exceptional polynomials the power of the disconjugacy properties of second order differential equations of schrodinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re examining the extensions of the isotonic oscillator or radial oscillator potential derived in kth order supersymmetric quantum mechanics or multistep darboux backlund transformation method the function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term whose value is calculated by induction over k the sign of this term being the same as that of the already known highest degree term the potential denominator has the same sign at both extremities of the definition interval a property that is shared by the seed eigenfunction used in the potential construction by virtue of disconjugacy such a property implies the nodeless character of both the eigenfunction and the resulting potential

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

integrability of the one dimensional schrodinger equation we present a definition of integrability for the one dimensional schrodinger equation which encompasses all known integrable systems i e systems for which the spectrum can be explicitly computed for this we introduce the class of rigid functions built as liouvillian functions but containing all solutions of rigid differential operators in the sense of katz and a notion of natural of boundary conditions we then make a complete classification of rational integrable potentials many new integrable cases are found some of them physically interesting

simple strong glass forming models mean field solution with activation we introduce simple models inspired by previous models for froths and covalent glasses with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems these models are also a generalization of backgammon or urn models to a non constant number of particles where entropic barriers are replaced by energy barriers allowing for the existence of activated processes we formulate a mean field version of the models which keeps most of the features of the finite dimensional ones and solve analytically the out of equilibrium dynamics in the low temperature regime where activation plays an essential role

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

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

spectral factor reduction by phase matching the continuous time single input single output case a procedure for reducing the order of a random process spectral factor without altering its stable minimum phase property is presented it is based upon the balanced or the hankel norm approximation of a certain all pass function characterizing the phase of the process spectral factor the motivation for this particular methodology for model order reduction stems from the canonical correlation analysis of akaike 1975 further bode s gain phase relation and nehari s extension used in conjunction provide a functional analytic justification of the reduction procedure the net result of this procedure is a stable minimum phase reduced order spectral factor that matches the original one fairly well the paper draws heavily on the early discrete time paper of jonckheere and helton 1984 in the paper we treat the continuous time case and extend the results of jonckheere and helton in several directions

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

minimal unitary representation of su 2 2 and its deformations as massless conformal fields and their supersymmetric extensions we study the minimal unitary representation minrep of so 4 2 over a hilbert space of functions of three variables obtained by quantizing its quasiconformal action on a five dimensional space the minrep of so 4 2 which coincides with the minrep of su 2 2 similarly constructed corresponds to a massless conformal scalar in four space time dimensions there exists a one parameter family of deformations of the minrep of su 2 2 for positive negative integer values of the deformation parameter one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in 0 2 2 0 representation of the sl 2 c subgroup we construct the supersymmetric extensions of the minrep of su 2 2 and its deformations to those of su 2 2 n the minimal unitary supermultiplet of su 2 2 4 in the undeformed case simply corresponds to the massless n 4 yang mills supermultiplet in four dimensions for each given nonzero integer value of one obtains a uniq

power spectrum reduction by optimal hankel norm approximation of the phase of the outer spectral factor a new approach to stochastic processes power spectrum reduction based on akaike s canonical correlation between the process past and future is developed as a first result the correlation between the canonical past and the canonical future is given its precise functional analytic interpretation as the hankel operator generated lay the all pass function characterizing the phase of the process outer spectral factor the canonical correlation coefficients turn out to he nothing else but the singular values of this hankel operator next the all pass function characterizing the phase of the outer spectral factor is reduced using the optimal hankel norm procedure of adamjan arov and krein the reduction of the all pass phase function is implemented with the safeguards necessary for the magnitude information to be carried all the way through this involves bode s relations between the gain and the phase of an outer function nehari s extension and the winding number of the all pass phase function which is itself shown to be related to the definition of the past and the future finally the net result is a reduced order outer spectral factor that matches the high order spectral factor fairly well

algebraic solution for the natanzon confluent potentials an algebraic method is used to solve the bound state problem for the most general natanzon potential for which the schrodinger equation can be reduced to the confluent hypergeometric form hence confluent potentials the solution is obtained straightforwardly a simple argument is given to sustain that the natanzon potential is the most general confluent potential

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

algebraic solutions of shape invariant position dependent effective mass systems keeping in view the ordering ambiguity that arises due to the presence of position dependent effective mass in the kinetic energy term of the hamiltonian a general scheme for obtaining algebraic solutions of quantum mechanical systems with position dependent effective mass is discussed we quantize the hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass initially proposed by von roos and levy leblond the algebraic method used to obtain the solutions is based on the concepts of supersymmetric quantum mechanics and shape invariance in order to exemplify the general formalism a class of non linear oscillators has been considered this class includes the particular example of a one dimensional oscillator with different position dependent effective mass profiles explicit expressions for the eigenenergies and eigenfunctions in terms of generalized hermite polynomials are presented moreover properties of these modified herm

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

simple bound state energy spectra solutions of dirac equation for hulthen and eckart potentials within the framework of simple similarity transformation method we present simple schrodinger like equation for dirac equation with hulthen and eckart lorentz vectors potentials using the asymptotic iteration method we have managed to obtain the explicit analytical solution of the bound state energy spectra as well as the eigenfunctions for both potentials discussion related to the eigenfunction for both cases is also considered we have shown that the wave function is sensitive for both potentials as well as the parameter

solvable potentials generated by sl 2 r canonical transformations are used to build realisations of sl 2 r in terms of the basic quantum mechanical operators q and p the results are used to construct solvable potentials in the framework of one dimensional quantum mechanics

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

quantization of quadratic lienard type equations by preserving noether symmetries abstract the classical quantization of a family of a quadratic lienard type equation lienard ii equation is achieved by a quantization scheme nucci 2011 28 that preserves the noether point symmetries of the underlying lagrangian in order to construct the schrodinger equation this method straightforwardly yields the schrodinger equation as given in choudhury and guha 2013 6

factorization solution of a family of quantum nonlinear oscillators in a recent paper carinena j f perelomov a m ranada m f and santander m 2008 j phys a math gen 41 085301 analyzed a non polynomial one dimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators in particular they proved that it is schrodinger soluble and explicitly obtained the wavefunctions and energies of the bound states in this paper we show that these results can be obtained much more simply by noting that this potential is a supersymmetric partner potential of the harmonic oscillator we then use this observation to generate an infinite set of potentials which can exactly be solved in a similar manner

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

solvable rational extensions of the morse and kepler coulomb potentials we show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential this is made by using darboux backlund transformations based on unphysical regular riccati schrodinger functions which are obtained from specific symmetries associated with the considered family of potentials

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

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

an exact invariant for the time dependent double well anharmonic oscillators lie theory and quasi invariance groups the author derives an exact energy type invariant for the double well anharmonic oscillators this invariant holds for certain time dependent potential parameters a nonlinear superposition principle is used to solve the nonlinear constraint equations which appear in the resolution

quantum solvability of a general ordered position dependent mass system mathews lakshmanan oscillator in position dependent mass pdm problems the quantum dynamics of the associated systems have been understood well in the literature for particular orderings however no efforts seem to have been made to solve such pdm problems for general orderings to obtain a global picture in this connection we here consider the general ordered quantum hamiltonian of an interesting position dependent mass problem namely the mathews lakshmanan oscillator and try to solve the quantum problem for all possible orderings including hermitian and non hermitian ones the other interesting point in our study is that for all possible orderings although the schrodinger equation of this mathews lakshmanan oscillator is uniquely reduced to the associated legendre differential equation their eigenfunctions cannot be represented in terms of the associated legendre polynomials with integral degree and order rather the eigenfunctions are represented in terms of associated legendre polynomials with non integral degree and orde

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

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

single variable realisation of the su 1 1 spectrum generating algebra and discrete eigenvalue spectra of a class of potentials a wide range of operators can be expressed in terms of the generators of the su 1 1 algebra the spectral properties of which are well known we attempt to find a realisation of the algebra in its most general form and thus evolve a unified approach to the problem of finding the spectra of hamiltonians amenable to the technique rather than taking each case separately we extend the analysis to obtain the spectra of potentials whose coordinate dependence is implicit

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 new potential with the spectrum of an isotonic oscillator with the method of factorisation a new potential with free parameters is generated this potential possesses exactly the same energy spectrum as the isotonic oscillator harmonic oscillator with centripetal barrier the wavefunctions and shift operators of the new system are also presented

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

bi hamiltonian structure of the bi dimensional superintegrable nonlinear isotonic oscillator the higher order superintegrability of the two dimensional isotonic oscillator noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator with rational ratio of frequencies is directly related with the existence of some complex functions with interesting poisson bracket properties first the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi hamiltonian complex structure in the second part several real symplectic structures are obtained and the properties of the recursion operators are studied

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

d dimensional generalization of the point canonical transformation for a quantum particle with position dependent mass the d dimensional generalization of the point canonical transformation for a quantum particle endowed with a position dependent mass in the schrodinger equation is described illustrative examples including the harmonic oscillator coulomb spiked harmonic kratzer morse oscillator p schl teller and hulthen potentials are used as reference potentials to obtain exact energy eigenvalues and eigenfunctions for target potentials at different position dependent mass settings

clusters of near degenerate levels dominate negative moments of spectral determinants the negative moments of spectral determinants n 1 e i en k diverge when 0 as k for a spectrum with equally distributed levels the exponent k k 1 for random matrix ensembles with parameters 1 orthogonal 2 unitary 4 symplectic we argue that the divergences for each k are determined by competitions between near degenerate level clusters whose sizes depend on k and we conjecture that k int k 1 1 k 1 int k 1 1 for poisson distributed levels unrestricted clustering leads to the divergence of the moments increasing with the number n of levels in the interval considered and k n k 1

on the solutions of the position dependent effective mass schrodinger equation of a nonlinear oscillator related with the isotonic oscillator we study exact solutions of the position dependent effective mass schrodinger equation by considering the new solvable nonlinear oscillator that relates to the isotonic oscillator through the method of point canonical transformations we construct exactly solvable potentials by choosing physically important position dependent mass distributions we also provide the energy spectrum of the bound states and the wavefunctions of the solvable potentials