rational approximation to the thomas fermi equations abstract we show that a simple and straightforward rational approximation to the thomas fermi equation provides the slope at origin with unprecedented accuracy and that pade approximants of relatively low order are far more accurate than more elaborate approaches proposed recently by other authors we consider both the thomas fermi equation for isolated atoms and for atoms in strong magnetic fields

on number systems with finite degree of ambiguity abstract we show that it is decidable whether or not a given n recognizable set is representable by a number system having finite degree of ambiguity as a corollary we obtain an algorithm for computing the degree of ambiguity of a given set defined by a number system

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

accurate calculation of the solutions to the thomas fermi equations abstract we obtain highly accurate solutions to the thomas fermi equations for atoms and atoms in very strong magnetic fields we apply the pade hankel method numerical integration power series with pade and hermite pade approximants and chebyshev polynomials both the slope at origin and the location of the right boundary in the magnetic field case are given with unprecedented accuracy

on a perturbation treatment of a model for mhd viscous flow we discuss the solution of a nonlinear ordinary differential equation that appears in a model for mhd viscous flow caused by a shrinking sheet we propose an accurate numerical solution and derive simple analytical expressions our results suggest that a recent perturbation treatment of the same problem exhibits a pathological behaviour and conjecture its probable cause

the classical schrodinger equation prufer s spectral algorithm is applied to the classical analogue of schrodinger s equation the discrete spectrum is interpreted as a bifurcation phenomenon caused by two simultaneous classical motions rotation and squeezing the energy eigenvalues coincide with the bifurcation parameters for the classical orbits

fixed point perturbation theory and the potential r2 λr2 1 gr2 ii construction of the solutions for pt i see ibid vol 17 p 3441 1984 constructs the asymptotic power series expansions of the green function and wavefunctions for the lambda r2 1 gr2 anharmonicity oscillator as a new form of the fixed point perturbative formalism the method may be extended to any band matrix hamiltonian in principle the results illustrate its algebraic flexibility as well as a quick numerical convergence

high precision calculation of the eigenvalues for the x2 λx2 1 gx2 potential an analytic continuation procedure using a taylor series is used to produce very accurate wavefunctions and eigenvalues for the one dimensional anharmonic oscillator governed by the potential v x x2 x2 1 gx2

numerical methods for the eigenvalue determination of second order ordinary differential equations an accurate method for the numerical solution of the eigenvalue problem of second order ordinary differential equation using the shooting method is presented the method has three steps firstly initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method secondly the initial value problem is solved using new highly accurate formulas of the linear multistep method thirdly the eigenvalue is properly corrected at the matching point the efficiency of the proposed methods is demonstrated by their applications to bound states for the one dimensional harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential in quantum mechanics

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

bounded delay l codes abstract three natural notions of bounded delay for l codes are investigated all interrelations many of them rather surprising between ordinary codes of bounded delay and l codes of bounded delay of various types are established

the anharmonic oscillator problem a new algebraic solution the second order one dimensional or radial differential schrodinger equation with the potential v r mu r2 nu r4 may be re interpreted as a difference equation of the fourth order indeed the hamiltonian is a pentadiagonal matrix in the standard harmonic oscillator basis mod n n 0 1 thus the authors construct its four independent general solutions by purely algebraic means via expansions in powers of n 1 14 next preserving the analogy between difference and differential equations the physical wavefunctions n mod psi and their energies are determined by the matching of the jost and regular solutions finally using the simplest matching conditions at the origin the resulting numerical algorithm is demonstrated to be both stable and quickly convergent

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

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

direct calculation of accurate siegert eigenvalues the roots of the hankel determinant built with the taylor coefficients of the regularized logarithmic derivative of the eigenfunction give both the bound state and siegert eigenvalues resonances of separable quantum mechanical systems we obtain accurate positions and widths for the resonances of some commonly chosen test examples

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

the use of the karhunen loegve procedure for the calculation of linear eigenfunctions abstract it is shown that the karhunen loeve decomposition may be used to determine the eigenfunctions of a general class of linear operators from an ensemble of realizations that are derived from that system given a moderate size data set either numerical or experimental from a linear system good approximations to the eigenfunctions that characterize the underlying equations can be computed by performing the karhunen loeve procedure two numerical examples are presented the vibration of a thin membrane in a rectangular domain and in a stadium these are used to determine the convergence and accuracy of the method it is found that this method yields accurate results for the first few eigenfunctions with relatively few realizations eigenfunctions with less energy are accurately resolved as the size of the ensemble increase the method is shown to be an efficient and practical procedure for determining the eigenfunctions of systems in complex geometries and in cases where the governing equations are not known a priori the effect of random noise contamination of the data set is also investigated and it is found that the karhunen loeve procedure can still achieve accurate results despite the presence of substantial background noise

study of the singular anharmonic potentials by means of the analytic continuation method here we study the singular anharmonic potentials by applying the analytic continuation method of holubec and stauffer 1985 in order to do that we have developed several approximations to the problem because this method cannot be applied when the solution has essential singularities at any point of its domain all the options here shown have the same precision giving us the eigenvalues correct to all the decimal places provided by the computer

effective characteristic polynomials and two point pade approximants as summation techniques for the strongly divergent perturbation expansions of the ground state energies of anharmonic oscillators pad e approximants are able to sum effectively the rayleigh schr odinger perturbation series for the ground state energy of the quartic anharmonic oscillator as well as the corresponding renormalized perturbation expansion e j weniger j ifmmode check c else v c fi i ifmmode check z else v z fi ek and f vinette j math phys 34 571 1993 in the sextic case pad e approximants are still able to sum these perturbation series but convergence is so slow that they are computationally useless in the octic case pad e approximants are not powerful enough and fail on the other hand the inclusion of only a few additional data from the strong coupling domain e j weniger ann phys ny to be published greatly enhances the power of summation methods the summation techniques which we consider are two point pad e approximants and effective characteristic polynomials it is shown that these summation methods give good results for the quartic and sextic anharmonic oscillators and even in the case of the octic anharmonic oscillator which represents an extremely challenging summation problem two point pad e approximants give relatively good results textcopyright 1996 the american physical society

simple systematics in the energy eigenvalues of quantum anharmonic oscillators in an earlier paper we proposed bhattacharya r roy d and bhowmick s 1998 phys lett a 244 9 a remarkably simple scheme for evaluating the ground state energy of x2m quantum anharmonic oscillators in the present paper we extend the scheme to evaluate the excited state energies as well and provide a motivation for the scheme the only information needed in the calculation is the first term of the standard strong coupling expansion for each state of the oscillators we have calculated the first few terms of the standard strong coupling expansion for some excited states for the quartic sextic and octic oscillators and for a given m we provide simple expressions for the strong coupling constants as functions of n which seem to reproduce any excited state energy with a reasonable accuracy moreover for the ground state we propose a simple expression for the first term of the strong coupling expansion which is globally true i e it gives the value of the constant for any m we then calculate a large number of excited state energies for the quartic sextic and octic oscillators over a wide range of values for the coupling parameter for any of the states the problem reduces to one of solving a polynomial equation and the predicted values closely agree with those obtained by other methods

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

definite integrals as solutions for the x2 λx2 1 gx2 potential presents exact solutions and eigenvalues for the schrodinger equation with the x2 x2 1 gx2 interaction on the half axis x or 0 the solutions are given in the form of definite integrals and the eigenvalues by means of a well defined limiting procedure

generalized linear random vibration analysis using autocovariance orthogonal decomposition application of a stationary gaussian random process to describe a nondeterministic forcing function of a linear vibrating system is well studied and documented two algorithms karhunen loeve expansion method and collocation technique for nonstationary and non gaussian forcing processes narrow or broadband but not white noise for linear dynamic systems are developed here in the karhunen loeve expansion method the forcing random process autocovariance is decomposed using the well known karhunen loeve expansion in the karhunen loeve expansion galerkin projection the weighted residual method and collocation technique discretized covariance matrix are used to get the eigenvalues and the eigenfunctions eigenvectors of the autocovariance function numerically the steady state and the transient response of a single degree of freedom system for an exponential autocovariance gaussian random process is obtained using three methods 1 analytical 2 semi analytical and 3 numerical in the semi analytical method the eigenvalues and the eigenfunctions of the exponential autocovariance function are obtained analytically by solving the fredholm integral equation of the second kind and those definitions of the eigenfunctions and the eigenvalues are used to obtain the numerical response of the single degree of freedom system in the case of the steady state response of the single degree of freedom system the convergence of the standard deviation of the response is shown to be a function of the number of karhunen loeve expansion terms used in the expansion of the autocovariance of the forcing function the transient analysis of the same single degree of freedom system is carried out using an exponentially modulated nonstationary process comparison of the proposed methods with respect to the analytical solutions are presented for both the stationary and the nonstationary gaussian excitations the steady state responses as well as the transient responses for non gaussian random processes uniform triangular and beta for the same single degree of freedom system are also presented

an accurate method for numerical calculations in quantum mechanics an accurate method for numerical calculations of matrix elements and for solving the eigenvalue problem in quantum mechanics is presented methods for numerical interpolation differentiation and integration provide 15 digit accuracy with double precision arithmetic operations a method for solution of the eigenvalue problem of an ordinary differential equation by using discretization and matrix eigenvalue methods provides 13 to 15 digit accuracy the efficiency of the proposed methods is demonstrated by the applications to bound states for the linear harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential

dynamic group approach to the x2 λx2 1 gx2 potential a modified operator method based upon the so 2 1 dynamic group is applied to the one dimensional anharmonic oscillator potential v x x2 x2 1 gx2 a tilting transformation is carried out in order to improve the rate of convergence of the algebraic perturbation series very accurate results are obtained for the energy eigenvalues and for the wave functions especially in the case of small g values

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

bases and ambiguity of number systems abstract the degree of ambiguity of a given number system can be effectively computed we show also that all degrees of ambiguity do exist both as regards number systems and the sets represented by them in contrast to language theory there are inherently unambiguous sets the paper contains also results concerning the possibility of changing the base of a number system

harmonic oscillations in a quasi relativistic regime a square root anharmonic oscillator potential is analysed as a model which simulates a quasi relativistic squeezing of the harmonic oscillator spectrum several eligible namely perturbative variational hill determinant and riccati pade methods of construction of its bound states are compared

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

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

on generalized dtol systems and their fixed points abstract we study a generalization of dt0l systems obtained by considering noncommutative polynomials together with mappings attaching a morphism to each variable we call the least fixed points of such pairs generalized dt0l languages and study their basic properties

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

semi analytical karhunen loeve representation of irregular waves based on the prolate spheroidal wave functions abstract a new semi analytical approach is presented to solving the matrix eigenvalue problem or the integral equation in karhunen loeve k l representation of random data such as irregular ocean waves instead of direct numerical approach to this matrix eigenvalue problem which may suffer from the computational inaccuracy for big data a pair of integral and differential equations are considered which are related to the so called prolate spheroidal wave functions pswf first the pswf is expressed as a summation of a small number of the analytical legendre functions after substituting them into the pswf differential equation a much smaller size matrix eigenvalue problem is obtained than the direct numerical k l matrix eigenvalue problem by solving this with a minimal numerical effort the pswf and the associated eigenvalue of the pswf differential equation are obtained then the eigenvalue of the pswf integral equation is analytically expressed by the functional values of the pswf and the eigenvalues obtained in the pswf differential equation finally the analytically expressed pswfs and the eigenvalues in the pwsf integral equation are used to form the kernel matrix in the k l integral equation for the representation of exemplary wave data such as ordinary irregular waves it is found that with the same accuracy the required memory size of the present method is smaller than that of the direct numerical k l representation and the computation time of the present method is shorter than that of the semi analytical method based on the sinusoidal functions

a family of complex potentials with real spectrum we consider a two parameter non hermitian quantum mechanical hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions in contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane our model is integrable along a line parallel to the real axis

the riccati pade method for coupled channel equations we present a method for calculating the energies of bound and quasibound states of quantum mechanical problems expressed in terms of coupled channel equations the approach is based on a rational approximation to the logarithmic derivative of the eigenfunction and leads to a simple quantization condition

convergence study of the truncated karhunen loeve expansion for simulation of stochastic processes a random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients karhunen loeve k l series expansion is based on the eigen decomposition of the covariance function its applicability as a simulation tool for both stationary and non stationary gaussian random processes is examined numerically in this paper the study is based on five common covariance models the convergence and accuracy of the k l expansion are investigated by comparing the second order statistics of the simulated random process with that of the target process it is shown that the factors affecting convergence are a ratio of the length of the process over correlation parameter b form of the covariance function and c method of solving for the eigen solutions of the covariance function namely analytical or numerical comparison with the established and commonly used spectral representation method is made k l expansion has an edge over the spectral method for highly correlated processes for long stationary processes the spectral method is generally more efficient as the k l expansion method requires substantial computational effort to solve the integral equation the main advantage of the k l expansion method is that it can be easily generalized to simulate non stationary processes with little additional effort copyright 2001 john wiley sons ltd

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

the rayleigh ritz method still competitive abstract an extended rayleigh ritz method for computing two sided eigenvalue bounds of the one dimensional schrodinger equation is presented the method is based on the b spline approximation over the truncated domain for the low lying eigenvalues and problems with rapidly increasing spectrum the corresponding lower bounds are much more accurate than the known temple bounds numerical results for problems exhibiting different kinds of the eigenfunction behaviour are presented

a moment equation reformulation of rayleigh ritz theory we present a moment equation rayleigh ritz rr variational theory for the ground state energy an extended fourier transform space can be defined through the moment equation corresponding to a given schrodinger hamiltonian within this extended space we can implement a variational ansatz with respects to configurations of the form in which the energy dependent functions i satisfy the momentum space schrodinger equation ii are uniquely prescribed iii are most likely non integrable and iv yield the physical solution for physical e and values on the basis of this representation one can minimize the energy expectation value with respects to the e and missing moment variables the proposed approach is in sharp contrast to traditional configuration space rr implementations in which the selection of a variational basis is not manifest a priori particularly for problems of spatial dimension greater than one the analysis of one and two dimension problems is presented

schrodinger equation as recurrences iii partitioning approach for pt ii see ibid vol 17 no 8 p 1603 10 1984 in the generalised lanczos basis the author partitions the hamiltonian to tridiagonal form and mutatis mutandis apply the ideas of i and ii it solves the schrodinger eigenvalue problem again the resulting method modifies and complements the formalism and simplifies its use and interpretation as an example of application the unique asymptotical effective hamiltonian is derived also for the anharmonic oscillators the convergence of its continued fractional definition is proved and the compact formula for the higher order asymptotic corrections is found

a finite difference approach for the calculation of perturbed oscillator energies a simple numerical method for calculating eigenvalues and corresponding eigenvectors of the schrodinger equation for a perturbed oscillator is described the derived results are compared with previously derived numerical data and with available exact values

splitting in a double minimum potential with almost twofold degenerate lower levels an analytic continuation procedure using taylor series is utilised to produce very accurate wavefunctions and eigenvalues for a double minimum potential having almost degenerate eigenvalues comparison is made with results obtained by other means

potential r2 λr2 1 gr2 and the analytic continued fractions the author constructs the exact wavefunctions and an analytic continued fractional green function for the one and three dimensional schrodinger equation with the potential v r h r2 r2 lambda r2 1 gr2 h 1 4 lambda g 0 in the numerical application of the formulae the energy and norm of the boundary state psi may be approximated by both their lower and upper estimates

finite difference calculations of eigenvalues for various potentials the energy levels of the schrodinger equation for the potentials v x sigma 2n 2 lambda 2nx2n v x z2x2 x2n and v x x2 lambda x2n 1 gx2 with 2n 4 6 8 18 20 have been calculated by using a finite difference method for various values of lambda and the quantum number n the obtained results are compared with previously available results

l codes and number systems abstract the notion of an l code is introduced as a generalization of the customary notion of a code attention is focused on unary l codes where the target alphabet consists of one letter only unary l codes can be identified with such number systems where the representation is unique surprisingly many unary morphisms are l codes furthermore even for non l codes there is only one exceptional length we also give a complete classification for the binary case

eigenenergies of the r2 lambda r2 1 gr2 and 1 2 r2 or gr4 1 g alpha r2 potentials obtained by the hypervirial method the energy levels of the schrodinger equation involving the potentials v r r2 lambda r2 1 gr2 and v or r 1 2 r2 or gr4 1 g alpha r2 are calculated by using hypervirial and pade approximant methods

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

on lindenmayerian algebraic sequences abstract we define and study lindenmayerian algebraic sequences these sequences are a generalization of algebraic sequences k regular sequences and automatic sequences

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

the eigenvalue problem for infinite complex symmetric tridiagonal matrices with application abstract we consider an infinite complex symmetric not necessarily hermitian tridiagonal matrix t whose diagonal elements diverge to in modulus and whose off diagonal elements are bounded we regard t as a linear operator mapping a maximal domain in the hilbert space l2 into l2 assuming the existence of t 1 we consider the problem of approximating a given simple eigenvalue of t by an eigenvalue n of tn and nth order principal submatrix of t let x x 1 x 2 t be an eigenvector corresponding to assuming xtx 0 and f n 1 x n 1 x n 0 as n we show that there exists a sequence n of tn such that n f n 1 x n 1 1 o 1 x t x 0 where fn 1 represents the n n 1 element of t application to the following problems is included a solve jv z 0 for v given z 0 and b compute the eigenvalues of the mathieu equation fortunately the existence of t 1 need not be verified for these examples since we may show that t i with taken appropriately has an inverse

the strong coupling expansion for anharmonic oscillators we accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm the method applies to any state and to any anharmonicity degree by means of the perturbation coefficients we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion

ambiguity and decision problems concerning number systems the representation of integers in arbitrary number systems is considered the main emphasis is on problems concerning ambiguity completeness and equivalence a rather general automata theoretic method for solving such in essence purely number theoretic problems is developed the method seems to be applicable in a variety of different situations

reply to comments on a new efficient method for calculating perturbation energies using functions which are not quadratically integrable in a previous paper sk la and c zek 1996 a new approach to calculating perturbation energies of bound states based on the use of functions which are not quadratically integrable was suggested this method has been analysed in a comment by guardiola and ros 1996 the present comment is the reply to this previous comment the aim is to give a simple and straightforward proof of the method and clarify some other points important for its application

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

schrodinger equation as recurrences ii general solutions and their physical asymptotics for pt i see ibid vol 16 no 17 p 4001 11 1983 the general schrodinger equation is converted into equivalent 2t 1 term recurrences by the generalised lanczos construction of the basis then the exact bound states are given by the t parametric determinantal formula finally the values of the free parameters are shown to follow from the hill type normalisability requirement

tight bounds to the schrodinger equation eigenvalues increasing tight upper and lower bounds to the schrodinger equation eigenvalues are obtained from the riccati equation for the logarithmic derivative of the wavefunction the solution of this non linear equation is written as a pade approximant and the bounds are given by the roots of a sequence of determinants numerical results are shown for the anharmonic and quartic oscillators

energy levels of a nonpolynomial oscillator using finite difference technique abstract the energy levels of a one dimensional system are calculated for the rational potentials v 1 x n g x 2 x 2n 1 gx 2 and v 2 x i n m g x 2i gx 2n 1 g x 2m using the finite difference approach over wide range of values of the perturbation parameters g i n m and for various state numbers n the numerical results are compared with previous numerical data and with available exact values

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

large order behaviour of the strong coupling perturbation expansion for anharmonic oscillators a new formula describing the large order behaviour of the strong coupling perturbation coefficients for the anharmonic oscillators with the hamiltonian is suggested a new method for the accurate calculation of the square root branch points of the energy from the numerical values of the coefficients is also suggested the branch points and the related minimal values of the coupling constant for which the expansion converges are calculated for the ground state of the quartic sextic octic and decadic oscillators

calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential we choose the squeezed vacuum state as a one parameter trial wavefunction to minimize the energy of an anharmonic oscillator with a polynomial perturbation potential using the optimal vacuum state obtained we generate the optimal generalized number state basis and set up the hamiltonian in the normal ordered form then truncating the basis we produce finite dimensional matrices in a simple manner and find their eigenvalues by standard methods we will apply our method to the morse potential whose exact eigenvalues are known as a check on the accuracy of the results that we have derived two models of the double well and the quartic potential are considered as examples

hankel hadamard analysis of quantum potential x2 λx2 1 gx2 the hankel hadamard moment determinant analysis of handy and bessis 1985 is applied to the potential x2 lambda x2 1 gx2 rapidly convergent lower and upper bounds to the ground state energy and first excited state are obtained application of a novel type of pade analysis allows the determination of all other excited states through an orthogonality quantisation prescription

bound state method with elementary product wavefunctions we propose a new version of the multipoint pade technique to solve schrodinger equations in their first order riccati form our compact product representation of wavefunctions interpolates or matches available approximants at several points the mechanism of matching determines the energies and eliminates spurious solutions on v x mu 2x2 lambda x2 1 gx2 its efficiency is illustrated numerically

accurate eigenvalues and eigenfunctions for quantum mechanical anharmonic oscillators the representation of the taylor expansion of the logarithmic derivative of the wavefunction by means of a pade approximant followed by an appropriate quantization condition proves a powerful way of obtaining accurate eigenvalues of the schrodinger equation in this paper we investigate in detail some of the interesting features of this approach termed riccati pade method rpm by means of its application to anharmonic oscillators we analyse the occurrence of many roots in the neighborhoods of the physical eigenvalues in the weak coupling regime and also obtain accurate coefficients of the strong coupling expansion we finally investigate the global and the local accuracy of the rpm eigenfunctions

quasi relativistic harmonic oscillator bound states the quasi relativistic harmonic oscillator bound states constructed by znojil 1996 j phys a math gen 29 2905 are investigated via a new methodical proposal compared with those obtained by an anonymous referee from a direct numerical integration method of znojil s paper our results appear to be more favourable than those obtained by znojil via quasi perturbative variational hill determinant and riccati pade methods bound states with larger angular momenta l are also constructed

the quantum harmonic oscillator on the sphere and the hyperbolic plane κ dependent formalism polar coordinates and hypergeometric functions a nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates r by making use of a curvature dependent formalism the curvature is considered as a parameter and then the radial schrodinger equation becomes a dependent gauss hypergeometric equation the energy spectrum and the wave functions are exactly obtained in both the sphere s2 0 and the hyperbolic plane h2 0 a comparative study between the spherical and the hyperbolic quantum results is presented

link between the strong coupling and the weak coupling asymptotic perturbation expansions for the quartic anharmonic oscillator for eigenvalues of the hamiltonian of the quartic anharmonic oscillator we give a representation which reproduces two well known limiting cases from the theory of the anharmonic oscillator expansions the weak coupling asymptotic perturbation and the convergent strong coupling we give an estimation of the radius of convergence of the strong coupling expansion for the ground state eigenvalue

on exact solutions of the schrodinger equation a new approach has been used to show what kind of potentials can be used to obtain a series of exact solutions in closed forms of the schrodinger equation for bound states in particular the well known coulombic and oscillator solutions and others are reproduced as special cases

numeration systems linear recurrences and regular sets a numeration system based on a strictly increasing sequence of positive integers u0 1 u1u2 expresses a non negative integer n as a sum n j 0 i ajuj in this case we say the string a i a i 1 a1 a0 is a representation for n

on the improvement of convergence of hill determinants we propose a simple approach to analytically determine a parameter of a convergence factor exp gx2 4 in the hill determinant method it provides much more rapid convergence of the method than the other analytical approaches currently used for the g determination

application of the eigenvalue moment method to important one dimensional quantum systems the authors study the effectiveness of the eigenvalue moment method as applied to various one dimensional quantum problems which have appeared in the literature of particular interest are the radial potential problems including angular momentum effects ze2 r beta and r2 lambda r2 1 gr2 studied by de meyer and vanden berghe 1990 and more recently by fernandez 1991 and witwit 1991 respectively they also examine the potentials 1 2x2 or gx4 1 alpha gx2 exhaustively studied by auberson and boissiere 1983 finally they examine the one dimensional regulated bohr atom with potential z mod x mod alpha well studied by loudon 1959 and haines and roberts 1969

on d0l systems with immigration abstract we study dol systems with immigration we show that sequence and growth equivalence are decidable we establish regularity and decidability results concerning degrees of ambiguity as a consequence of results about subword complexity we show that regularity and regularity are decidable for languages generated by growing systems

efficient solution of differential equations by analytic continuation a new method is described for the automatic solution of linear ordinary differential equations by the use of taylor series this new method is shown to be superior in speed and accuracy to conventional methods the method is illustrated for coulomb wavefunctions confluent hypergeometric functions zeros of bessel functions and s wave phaseshift for e h scattering in the static approximation