the nearest neighbor resonating valence bond state in a grassmannian form the nearest neighbor resonating valence bond nnrvb state is studied using classical anticommuting grassmann variables the classical partition function corresponding to the self overlap of the nnrvb wavefunction is generated from a local bond hamiltonian expressed in terms of four anticommuting variables it is shown that the one particle per site constraint introduces an interaction term which is a local product of all four variables two approaches are applied to study this hamiltonian i a self consistent field decoupling scheme and ii a systematic perturbation expansion around the unconstrained soluble point bounds on the norm of the wavefunction are derived extensions to the presence of holes long range valence bonds and the introduction of phase fluctuations which violate the marshall sign rule and yield au 1 gauge theory are discussed

some aspects of polya tree distributions for statistical modelling the definition and elementary properties of polya tree distributions are reviewed two theorems are presented showing that polya trees can be constructed to concentrate arbitrarily closely about any desired pdf and that polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy thereby satisfying a consistency condition such theorems are false for dirichlet processes models are constructed combining partially specified polya trees with other information such as monotonicity or unimodality it is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications a numerical example is given a theorem of diaconis and freedman about dirichlet processes is generalized to polya trees allowing polya trees to be the models for errors in regression problems finally empirical bayes models using dirichlet processes are generalized to polya trees an example from berry and christensen is reanalyzed with a polya tree model

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

sextic anharmonic oscillators and orthogonal polynomials under certain constraints on the parameters a b and c it is known that schrodinger s equation d2 dx2 ax6 bx4 cx2 e a 0 with the sextic anharmonic oscillator potential is exactly solvable in this paper we show that the exact wavefunction is the generating function for a set of orthogonal polynomials p t n x in the energy variable e some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi exact solvability we also prove that this set of orthogonal polynomials can be reduced by means of a simple scaling transformation to a remarkable class of orthogonal polynomials pn e p 0 n e recently discovered by bender and dunne

construction of exact solutions to eigenvalue problems by the asymptotic iteration method we apply the asymptotic iteration method aim ciftci hall and saad 2003 j phys a math gen 36 11807 to solve new classes of second order homogeneous linear differential equation in particular solutions are found for a general class of eigenvalue problems which includes schrodinger problems with coulomb harmonic oscillator or poschl teller potentials as well as the special eigenproblems studied recently by bender et al 2001 j phys a math gen 34 9835 and generalized in the present paper to arbitrary dimension

precise determination of the energy levels of the anharmonic oscillator from the quantization of the angle variable using an ansatz motivated by the classical form of el phi where phi is the angle variable we construct operators which satisfy the commutation relations of the creation annihilation operators for the anharmonic oscillator the matrix elements of these operators can be expressed in terms of entire functions in the position complex plane these functions provide solutions of the ricatti equation associated with the time independent schrodinger equation we relate the normalizability of the eigenstates to the global properties of the flows of this equation these exact results yield approximations which complement the wkb approximation and allow an arbitrarily precise determination of the energy levels we give numerical results for the first 10 levels with 30 digits we address the question of the quantum integrability of the system

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

polynomial solutions for a class of second order linear differential equations we analyze the polynomial solutions of the linear differential equation p 2 x y p 1 x y p 0 x y 0 where p j x is a jth degree polynomial we discuss all the possible polynomial solutions and their dependence on the parameters of the polynomials p j x special cases are related to known differential equations of mathematical physics classes of new soluble problems are exhibited general results are obtained for weight functions and orthogonality relations

spin and pseudospin symmetry along with orbital dependency of the dirac hulthen problem abstract the role of the hulthen potential on the spin and pseudospin symmetry solutions is investigated systematically by solving the dirac equation with attractive scalar s r and repulsive vector v r potentials the spin and pseudospin symmetry along with orbital dependency pseudospin orbit and spin orbit dependent couplings of the dirac equation are included to the solution by introducing the hulthen square approximation this effective approach is based on forming the spin and pseudo centrifugal kinetic energy term from the square of the hulthen potential the analytical solutions of the dirac equation for the hulthen potential with the spin orbit and pseudospin orbit dependent couplings are obtained by using the nikiforov uvarov nu method the energy eigenvalue equations and wave functions for various degenerate states are presented for several spin orbital pseudospin orbital and radial quantum numbers under the condition of the spin and pseudospin symmetry

asymptotic iteration method solutions to the relativistic duffin kemmer petiau equation a simple exact analytical solution of the relativistic duffin kemmer petiau equation within the framework of the asymptotic iteration method is presented exact bound state energy eigenvalues and corresponding eigenfunctions are determined for the relativistic harmonic oscillator as well as the coulomb potentials as a nontrivial example the anharmonic oscillator is solved and the energy eigenvalues are obtained within the perturbation theory using the asymptotic iteration method

information measures and bayesian hierarchical models abstract we consider the bayesian statistical models in which the prior distribution of the parameter vector 1 in the distribution of an observable random vector x is to be specified in a hierarchical fashion and one wants to learn about the hyperparameters at each level of this prior distribution it is shown that for a wide class of information measures based on the so called f divergence the information decreases as one moves to higher levels of hyperparameters this result unifies all the theorems in goel and degroot 1981 and provides several other information measures for which the above desirable property holds

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

matrix elements for a generalized spiked harmonic oscillator closed form expressions for the singular potential integrals m x n are obtained with respect to the gol dman and krivchenkov eigenfunctions for the singular potential bx2 a x2 b 0 a 0 the formulas obtained are generalizations of those found earlier by use of the odd solutions of the schrodinger equation with the harmonic oscillator potential aguilera navarro et al j math phys 31 99 1990

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

orientation of particle attachment and local isotropy in diffusion limited aggregation dla we simulate 50 off lattice diffusion limited aggregation dla clusters of one million particles each the probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically for increasing cluster size n the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power law in n from this specific viewpoint asymptotically large dla clusters are locally isotropic

the asymptotic iteration method for the eigenenergies of the schrodinger equation with the potential v r z r gr λr2 the eigenenergies of the schrodinger equation with coulomb potential plus linear and harmonic radial terms are investigated using the perturbation technique within the framework of the asymptotic iteration method the obtained eigenenergies are compared with those obtained from the shifted 1 n expansion method the moment method and also with the exact values obtained from the super symmetric method it is found that asymptotically this method gives accurate results for arbitrary g z and

any l state solutions of the hulthen potential by the asymptotic iteration method in this paper we present the analytical solution of the radial schrodinger equation for the hulthen potential within the framework of the asymptotic iteration method by using an approximation to the centrifugal potential for any l states we obtain the energy eigenvalues and corresponding eigenfunctions for different screening parameters the wavefunctions are physical and energy eigenvalues are in good agreement with the results obtained by other methods for different values in order to demonstrate this the results of the asymptotic iteration method are compared with the results of the supersymmetry numerical integration variational and shifted 1 n expansion methods

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

comment on the asymptotic iteration method for polynomial potentials it is shown that polynomial potentials can be treated by a method which cuts out most of the complications of the popular aim approach several examples are given to show how expectation values and resonance energies can be calculated using the simplified approach

closed form sums for some perturbation series involving hypergeometric functions infinite series of the type are investigated closed form sums are obtained for a positive integer 1 2 3 the limiting case of b after y is replaced with x2 b leads to this type of series appears in the first order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator hamiltonian these results have immediate applications to perturbation series for the energy and wavefunction of the spiked harmonic oscillator hamiltonian

criterion for polynomial solutions to a class of linear differential equations of second order we consider the differential equations y 0 x y s0 x y where 0 x s0 x are c functions we prove i if the differential equation has a polynomial solution of degree n 0 then n nsn 1 n 1sn 0 where n n 1 sn 1 0 n 1andsn s n 1 s0 k 1 n 1 2 conversely ii if n n 1 0 and n 0 then the differential equation has a polynomial solution of degree at most n we show that the classical differential equations of laguerre hermite legendre jacobi chebyshev first and second kinds gegenbauer and the hypergeometric type etc obey this criterion further we find the polynomial solutions for the generalized hermite laguerre legendre and chebyshev differential equations

the asymptotic iteration method for the angular spheroidal eigenvalues the asymptotic iteration method is applied to calculate the angular spheroidal eigenvalues ml c it is shown that this method asymptotically gives accurate results over the full range of parameter values l m and c

local sensitivity diagnostics for bayesian inference we investigate diagnostics for quantifying the effect of small changes to the prior distribution over a k dimensional parameter space we show that several previously suggested diagnostics such as the norm of the frechet derivative diverge at rate n k 2 if the base prior is an interior point in the class of priors under the density ratio topology diagnostics based on divergences exhibit similar asymptotic behavior we show that better asymptotic behavior can be obtained by suitably restricting the classes of priors we also extend the diagnostics to see how various marginals of the prior affect various marginals of the posterior

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

local sensitivity of posterior expectations we investigate the degree to which posterior expectations are sensitive to prior distributions using a local method based on functional differentiation invariance considerations suggest a family of norms which can be used to measure perturbations to the prior the sensitivity measure is seen to depend heavily on the choice of norm asymptotic results suggest which norm will yield the most useful results in practice we find that to maintain asymptotically sensible behaviour it is necessary to reduce the richness of the class of prior perturbations as the dimension of the parameter space increases jeffreys prior is characterized as the prior to which inference is least sensitive

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

loop coverings of two dimensional square lattices abstract in a two dimensional square lattice where all sites are covered by loops in the past few years a classical statistical mechanics model of dense loop gas was considered for its relevance for the evaluation of observables of the quantum antiferromagnetic heisenberg model on the resonating valence bond state by use of transfer matrices we evaluate several observables for narrow strips with a width of two or three sites a simple extrapolation consistent with rigorous bounds is conjectured for its possible use in a variational ansatz in the hubbard model

on an iteration method for eigenvalue problems we discuss a recently proposed asymptotic iteration method for eigenvalue problems we analyse its rate of convergence the use of adjustable parameters to improve it and the relationship with an alternative method based on the same ideas

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

exact solutions of deformed schrodinger equation with a class of non central physical potentials in this paper we present exact solutions of schrodinger equation for a class of non central physical potentials within the formalism of position dependent effective mass the energy eigenvalues and eigenfunctions of the bound states for the schrodinger equation are obtained analytically by means of asymptotic iteration method and easily calculated through a new generalized decomposition of the effective potential allowing easy separation of the coordinates our results are in excellent agreement with other works in the literature

on the solution of a class of the reduced wave equation and the riccati differential equation in this paper we present an iteration technique for a class of differential equation having the form z email protected where l is a function in c we show that we can construct not only the general solution of the reduced wave equation but also the general solution of the riccati differential equation by using this iteration technique if the given function l is satisfies the condition l 2 n email protected 2 n email protected 2 n email protected 2 n email protected email protected then we give a simple application of this technique to inverse conductive scattering problem for an inhomogeneous spherical medium

generalized bethe ansatz equations for hofstadter problem the problem of diagonalization of the quantum mechanical hamiltonian governing dynamics of an electron on a two dimensional triangular or square lattice in external uniform magnetic field applied perpendicularly to the lattice plane the flux through lattice cell divided by the elementary quantum flux being a rational number is reduced to the generalized bethe ansatz like equations on the high genus algebraic curve our formulae for the trigonometric case where the genus of the curve vanishes contain as a particular case a recent result of wiegmann and zabrodin

non relativistic treatment of diatomic molecules interacting with a generalized kratzer potential in hyperspherical coordinates we investigate solutions of a non relativistic wave equation in hyperspherical coordinates for a diatomic molecule system interacting with a generalized kratzer potential rovibrational eigenvalues and corresponding wavefunctions of non relativistic diatomic molecules have been determined within the framework of the asymptotic iteration method certain fundamental conditions for the applications of the asymptotic iteration method such as a suitable asymptotic form for the wave function and the termination condition for the iteration process are discussed n dimensional bound state eigenfunction solutions used in studying the dynamical variables of diatomic molecules are obtained in terms of a confluent hypergeometric function and a generalized laguerre polynomial this systematic approach is tested by calculating the rovibrational energy spectra of hydrogen and sodium chloride molecules

energy spectrum of a generalized scarf potential using the asymptotic iteration method and the tridiagonal representation approach the well known trigonometric scarf potential is generalized by adding a sinusoidal term and then treated using the asymptotic iteration method aim and the tridiagonal representation approach tra the energy spectrum of the associated bound states is computed for the aim we have improved convergence of the quantization condition that terminates the iterations asymptotically this is accomplished by looking for the range of initial values of the space variable in the terminating condition that produces stable results plateau of convergence we have shown that with increasing iteration this plateau of convergence grows up rapidly to an optimal iteration number and then shrinks slowly to a point the value of this point or points may depend on the physical parameters the numerical results have been compared favorably with those resulting from the tra the well known trigonometric scarf potential is generalized by adding a sinusoidal term and then treated using the asymptotic iteration method aim and the tridiagonal representation approach tra the energy spectrum of the associated bound states is computed for the aim we have improved convergence of the quantization condition that terminates the iterations asymptotically this is accomplished by looking for the range of initial values of the space variable in the terminating condition that produces stable results plateau of convergence we have shown that with increasing iteration this plateau of convergence grows up rapidly to an optimal iteration number and then shrinks slowly to a point the value of this point or points may depend on the physical parameters the numerical results have been compared favorably with those resulting from the tra

analytical solutions of a generalized non central potential in n dimensions we present that n dimensional non relativistic wave equation for the generalized non central potential with arbitrary angular momentum is analytically solvable in the hyperspherical coordinates asymptotic iteration method as a different approach is applied to obtain n dimensional energy eigenvalues and the corresponding eigenfunctions in hyperspherical coordinates the wave function solutions are obtained in terms of hypergeometric functions and jacobi polynomials the bound states of quantum systems under consideration for some special cases such as hartmann and makarov potentials have been discussed in n dimensions

physical applications of second order linear differential equations that admit polynomial solutions in this paper conditions for the second order linear differential equation to have polynomial solutions are given several applications of these results to schrodinger s equation are discussed conditions under which the confluent biconfluent and general heun equation yields polynomial solutions are explicitly given some new classes of exactly solvable differential equations are also discussed the results of this work are expressed in such a way as to allow direct use without preliminary analysis

the klein gordon equation with ring shaped potentials asymptotic iteration method this study presents the solutions of three dimensional klein gordon equation for the spherically and non spherically harmonic oscillatory ring shaped potentials within the framework of asymptotic iteration method using the method of variable separation this study obtains the radial and angular equations and then the bound states energy eigenvalues and corresponding eigenfunctions are obtained analytically

asymptotic iteration technique for second order dynamic equations on time scales an asymptotic iteration method for solving second order homogeneous linear dynamic equation of the form y a0 t y b0 t y is introduced where a0 t and b0 t are crdm t functions and the positive integer m depends on the differentiability of a0 t b0 t and iteration steps

a solution of the bosonic and algebraic hamiltonians by using an aim we apply the notion of an asymptotic iteration method aim to determine the eigenvalues of the bosonic hamiltonians that include a wide class of quantum optical models we consider the solutions of the hamiltonians which are even the polynomials of the fourth order with respect to the boson operators we also demonstrate the applicability of the method for obtaining the eigenvalues of the simple lie algebraic structures the eigenvalues of the multi boson hamiltonians have been obtained by transforming them into the form of the single boson hamiltonian in the framework of the aim

variational sturmian approximation a nonperturbative method of solving time independent schrodinger equation a variationally improved sturmian approximation for solving time independent schrodinger equation is developed this approximation is used to obtain the energy levels of a quartic anharmonic oscillator a quartic potential and a gaussian potential the results are compared with those of the perturbation theory the wkb approximation and the accurate numerical values

the pseudospin symmetric solution of the morse potential for any κ state in this paper the energy eigenvalues and the corresponding spinors of the dirac particles are obtained and the pseudospin symmetric solution of the attractive scalar and repulsive vector morse potential for any spin orbit quantum number is provided by using the pekeris approximation within the framework of the asymptotic iteration method

the asymptotic iteration method revisited the asymptotic iteration method is a technique for solving analytically and approximately the linear second order differential equation especially the eigenvalue problems that frequently appear in theoretical and mathematical physics the analysis and mathematical justifications of the success and failure of the asymptotic iteration method are detailed in this work a theorem explaining why the asymptotic iteration method works for the eigenvalue problem is presented as a byproduct a new procedure to generate unlimited classes of exactly solvable differential equations is also introduced

a hill series method for resonances a simple algorithm is given to treat perturbed oscillator bound states and resonances the method is applied to a bound state problem for which the energy is a non analytic function of a perturbation parameter and also to two resonance problems one of which has a spectrum with an unusual dual nature

an approach to robust bayesian analysis for multidimensional parameter spaces abstract in some multiparameter bayesian problems the prior is more naturally described as a joint density for all of the parameters and in others it is more naturally described as a product of marginal and conditional densities although ordinary quantile e contamination and density ratio classes of priors may be sensible choices for robust bayesian analyses in the former case they may be less satisfactory in the latter this article shows how to construct product classes of priors for use in the second case using one class of priors for the marginal and another for each conditional then the article introduces density bounded classes as an alternative to the more familiar classes for representing uncertainty about the prior it is often of interest to find tight bounds on posterior expectations as the prior ranges over the class of priors an apparently difficult nonlinear optimization problem regardless of which class of priors is used this article describes a technique for turning the problem into

comment on an application of the asymptotic iteration method to a perturbed coulomb model we discuss a recent application of the asymptotic iteration method aim to a perturbed coulomb model contrary to what was argued before we show that the aim converges and yields accurate energies for that model we also consider alternative perturbation approaches and show that one of them is equivalent to that recently proposed by another author

reply to the comment on a solution of the bosonic and algebraic hamiltonians by using an aim we reply to the comment made by fernandez 2011 j phys a math theor 44 228001 on our recent paper koc et al 2010 j phys a math theor 43 395208 we have shown that the main equations are consistent and the results can be applied for solving a significant number of problems we admit that one or two steps missing in the original paper may lead to a different interpretation

variational and perturbative schemes for a spiked harmonic oscillator a variational analysis of the spiked harmonic oscillator hamiltonian operator d2 dx2 x2 l l 1 x2 x where is a real positive parameter is reported in this work the formalism makes use of the functional space spanned by the solutions of the schrodinger equation for the linear harmonic oscillator hamiltonian supplemented by a dirichlet boundary condition and a standard procedure for diagonalizing symmetric matrices the eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system valid for positive and relatively large values of the coupling parameter additionally a large coupling perturbative expansion is carried out and the contributions up to fourth order to the ground state energy are explicitly evaluated numerical results are compared for the special case 5 2

large order shifted 1 n expansions through the asymptotic iteration method the perturbation technique within the framework of the asymptotic iteration method is used to obtain large order shifted 1 n expansions where n is the number of spatial dimensions this method is contrary to the usual rayleigh schrodinger perturbation theory no matrix elements need to be calculated the method is applied to the schrodinger equation and the non polynomial oscillator potential is discussed as an illustrative example

arbitrary l state solutions of the rotating morse potential by the asymptotic iteration method for non zerovalues we present an analytical solution of the radial schr odinger equation for the rotating morse potential using the pekeris approximation within the framework of the asymptotic iteration method the bound state energy eigenvalues and corresponding wavefunctions are obtained for a number of diatomic molecules and the results are compared with the findings of the super symmetry the hypervirial perturbation the nikiforov uvarov the variational the shifted 1 n and the modified shifted 1 n expansion methods

asymptotic iteration method for eigenvalue problems an asymptotic iteration method for solving second order homogeneous linear differential equations of the form y 0 x y s0 x y is introduced where 0 x 0 and s0 x are c functions applications to schrodinger type problems including some with highly singular potentials are presented