on the schrodinger equation for the interaction x2 λx2 1 gx2 the method of obtaining the explicit expressions of all the even odd parity exact solutions of the schrodinger equation for the interaction x2 lambda x2 1 gx2 is discussed when the couplings lambda and g satisfy some specific relations in the general case a simple equation for approximating the energy eigenvalues has been developed

scaling generator coordinate method and the eigensolutions of the schrodinger equation an extremely simple approximate variational method for the study of the eigensolutions of the one dimensional schrodinger equation is introduced based upon the idea of employing scaled eigenstates of the harmonic oscillator as generating functions for the generator coordinate method the approach is successfully applied to a variety of situations

on the existence of a kazantzis kravaris luenberger observer we state sufficient conditions for the existence on a given open set of the extension to nonlinear systems of the luenberger observer as it has been proposed by kazantzis and kravaris we prove it is sufficient to choose the dimension of the system giving the observer less than or equal to 2 twice the dimension of the state to be observed we show that it is sufficient to know only an approximation of the solution of a pde needed for the implementation we establish a link with high gain observers finally we extend our results to systems satisfying an unboundedness observability property

modular design of nonlinear observers for state and disturbance estimation this work studies the problem of design of nonlinear observers in the presence of exogenous disturbances in particular the present work proposes a systematic design method for nonlinear state and disturbance observers in modular form consisting of an observer for the disturbance free part of the system along with a disturbance observer and a state estimate corrector the modular observer is first defined and characterized in a general setting and then a systematic design method is developed on the basis of exact linearization with eigenvalue assignment necessary and sufficient conditions for feasibility of the proposed nonlinear modular observer design are derived the proposed modular nonlinear observer is tested through simulation studies in an illustrative example involving a biochemical reactor

on the schrodinger equation for the interaction x2 λx2 1 gx2 the lowest even and odd parity energy levels of schrodinger s equation for the interaction x2 lambda x2 1 gx2 for arbitrary positive values of lambda and g are obtained accurately from a first order variational perturbation treatment

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

on the hill determinant method the conditions of applicability of the hill determinant method to the potentials v r v1r v2r2 and v x x2 lambda x2 1 gx2 are investigated it is shown that the procedure leads to the actual eigenvalues in the former case provided the ansatz is properly chosen but it completely fails to yield the right answer in the latter one

fixed point perturbation theory and the potential r2 λr2 1 gr2 i analysis of convergence the simplest illustration of the recently suggested asymptotic perturbative approach to the band matrix hamiltonians is found in the harmonic oscillator complemented by the non polynomial anharmonicity lambda r2 1 gr2 in the paper the detailed construction of the effective hamiltonian is given and the convergence of its fixed point expansion is shown

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

perturbed harmonic oscillator ladder operators eigenenergies and eigenfunctions for the x2 λx2 1 gx2 interaction the perturbed ladder operators method is applied to the resolution of the perturbed harmonic oscillator wave equation for the case where the perturbation is expandable in a convergent series of hermite polynomials hk when v x b2x2 sigma kckhk b12 x it is found that the use of a hermite polynomials basis together with the use of binomial coefficient functions in the quantum number greatly simplifies the determination of the perturbed ladder and factorisation functions thus one obtains analytical expressions of the eigenenergies and eigenfunctions up to any order of the perturbation without increasing intricacy thorough calculation has been given for a perturbing potential v x function even in x as an illustrative application of the procedure the resolution of the schrodinger equation with a potential function v x x2 lambda x2 1 gx2 g 0 is reinvestigated

branching processes in the annni model the mean field equations of the simple cubic annni model are studied on finite lattices the results are consistent with the sequence of distinct commensurate phases 2k 13 k 1 2 3 springing from the multiphase point found using low temperature series expansions moreover evidence for new structure combination branching processes is presented which generate phases of type 2l3 m 2l 13 n or 23l m 23l 1 n where l m and n are integers

renyi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems we discuss some properties of the generalized entropies called renyi entropies and their application to the case of continuous distributions in particular it is shown that these measures of complexity can be divergent however their differences are free from these divergences thus enabling them to be good candidates for the description of the extension and the shape of continuous distributions we apply this formalism to the projection of wave functions onto the coherent state basis i e to the husimi representation we also show how the localization properties of the husimi distribution on average can be reconstructed from its marginal distributions that are calculated in position and momentum space in the case when the phase space has no structure i e no classical limit can be defined numerical simulations on a one dimensional disordered system corroborate our expectations

applications of the differentiability of eigenvectors and eigenvalues to a perturbed harmonic oscillator the potential interaction x2 1 gx2 g 0 of the harmonic oscillator h0 d2 dx2 x2 considered as an operator in the space l2 is bounded this together with the nondegeneracy of the eigenvalues implies that the eigenvectors of the perturbed harmonic oscillator as functions of the parameters and g are strongly differentiable the eigenvalues are therefore differentiable functions for every real and every real g 0 in particular the first eigenvalue e1 as a function of is strictly concave e 1 0 emphasis is given to the inequality that follows from the strict concavity of the function e1

shifted 1 n expansion approach to the interaction v r r2 λr2 1 gr2 the authors use the shifted 1 n expansion method to calculate the energy eigenvalues of the non polynomial oscillator v r r2 r2 1 gr2 the results thus obtained are in excellent agreement with exact numerical results it is also shown that the non polynomial interaction can be made supersymmetric and in this case also the results obtained by the shifted 1 n method matches extremely well with the exact analytic results given by supersymmetry

double exponential sinc collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function we show that the double exponential sinc collocation method provides an efficient uniformly accurate solution to the one dimensional time independent schrodinger equation for a general class of rational potentials of the form v x p x q x the derived algorithm is based on the discretization of the hamiltonian of the schrodinger equation using sinc expansions this discretization results in a generalized eigenvalue problem the eigenvalues of which correspond to approximations of the energy values of the starting hamiltonian a systematic numerical study is conducted beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

on the x2 λx2 1 gx2 interaction presents an infinite set of exact solutions of odd parity for the one dimensional schrodinger equation with the interaction x2 lambda x2 1 gx2 these complement the exact solutions of even parity that have been obtained recently by flessas phys lett a vol 84 p 121 1981

interaction λx2 1 gx2 revisited abstract it is shown that the refined non perturbative hill s method solves in a very neat way the schrodinger equation e x 2 x 2 1 gx 2 0 the refinement of the standard hill s method consists in finding a renormalized frequency to optimize the calculation this method appears to be preferable to the variational one recently used by a k mitra in the same context

the hill determinant method in application to the sextic oscillator limitations and improvement using the sextic anharmonic oscillator a chi 2 b chi 4 c chi 6 as a testing ground it is shown why the hill determinant method has a limited range of applicability a possible improvement is suggested and discussed

on exact solutions of the schrodinger equation in the one and three dimensional schrodinger equation a broad class of the regular potentials rational functions of the rational powers of r may admit exact bound state solutions in the generalised harmonic oscillator elementary form psi r rsigma polynomial exp polynomial the necessary and sufficient conditions of this phenomenon are derived in the form of coupled algebraic equations the methods of their solution and a few examples are discussed in particular the well known coulombic and oscillator solvability and the similar recent results of singh et al 1978 and whitehead et al 1982 are reproduced as the two simplest special cases

nonlinear observer design in the siegel domain we extend the method of kazantzis and kravaris systems control lett 34 1998 pp 241 247 for the design of an observer to a larger class of nonlinear systems the extended method is applicable to any real analytic observable nonlinear system it is based on the solution of a first order singular nonlinear pde this solution yields a change of state coordinates which linearizes the error dynamics under very general conditions the existence and uniqueness of the solution is proved lyapunov s auxiliary theorem and siegel s theorem are obtained as corollaries the technique is constructive and yields a method for constructing approximate solutions

on the schrodinger equation for the interaction x2 λx2 1 gx2 by forming the 6 6 pade approximants to the energy perturbation series for the interaction x2 lambda x2 1 gx2 as obtained from the hypervirial relations the ground and the first three excited state energy levels are calculated for g or 2 the results of the present calculation are in good agreement with those of the variational calculations the authors also present another set of exact solutions to the schrodinger equation for the interaction x2 lambda x2 1 gx2