vectorial continued fractions and an algebraic construction of effective hamiltonians in the haromonic oscillator basis the effective hamiltonians are constructed for the polynomial interactions v r ti 1gir2i and t 4 their form factorized in terms of the vectorial continued fractions is simpler in a semirelativistic c reformulation its degeneracy in the c limit confluence of the physical and unphysical solutions is removed by solving a coupled set of the nonlinear algebraic equations in reduce

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

potential v r μr2 νr4 and mutual equivalence of the two recent algebraic approaches to schrodinger equations the authors consider the anharmonic oscillator hamiltonian and solve the corresponding schrodinger equation by means of the recently proposed fixed point fp perturbation theory the fp formalism resembles an introduction of the creation and annihilation operators uses a vectorial generalisation of continued fractions introduces the inverse model space dimension as a natural and controllably small expansion parameter and is shown to define a smooth exact effective truncation of the hamiltonian matrix it also reproduces the recent n 1 asymptotic formulae for the wavefunctions

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

perturbation theory applied to sound propagation in flowing media confined by a cylindrical waveguide first order perturbation theory is employed to examine sound propagation in flowing media confined by a cylindrical waveguide the use of perturbation theory allows examination of mode phase speed changes due to any radially dependent flow w r as long as the flow magnitude is sufficiently small the condition to be fulfilled is satisfied in the flow range 0 0 3 m s for the specific values of cylinder radius ultrasound frequency and sound speed analyzed in the present work in the general case however the condition in eq 1 of the present work must be fulfilled this freedom of choice i e the possibility to handle any radial flow profile is used to analyze two flow profile cases 1 where w r is a linear combination of a laminar flow profile and a flat profile corresponding to turbulent flow and 2 where w r is a linear combination of a laminar flow profile and a more realistic logarithmic dependent turbulent flow profile in both cases it is shown that large errors may result in ultraso

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

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

comments on a paper concerning the analytic continued fraction technique for bound states the author shows that the treatment of bound and confined states for a class of confinement potentials by application of the analytic continued fraction method as presented in a recent paper dalta et al ibid vol 13 p 3161 1980 is erroneous and may lead to wrong results for the eigenvalues of the relevant schrodinger equation

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

on the lattice structure of a nonlinear generator with modulus 2 a abstract nonlinear congruential pseudorandom number generators based on inversions have been introduced and analysed recently these generators do not show the simple lattice structure of the widely used linear congruential generators which are too regular for certain simulation purposes in the present paper a nonlinear congruential generator based on inversions with respect to a power of two modulus is considered it is shown that the set of points formed by consecutive pseudorandom numbers has a more complicated lattice structure it forms a superposition of shifted lattices the corresponding lattice bases are explicitly determined and analysed

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

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

chain models and their effective hamiltonians we develop a new formalism suitable for diagonalization of hamiltonians with 2t 1 nonzero diagonals chain models in a systematic perturbation like way we get the expansions of green and or wave functions they are generated by solution of a sequence of fixed point quadratic equations for t t dimensional matrices for t 1 and t 2 this solution is feasible by elementary means so that the respective tridiagonal and pentadiagonal chain models may be considered exactly solvable in this context in an alternative first order variational perturbation formulation the method may provide the simultaneous upper and lower energy bounds for any t

on the αx2 βx4 interaction by applying the laplace transform method the authors obtain a class of exact solutions for the schrodinger equation with the potential v x alpha x2 beta x4 x 0 alpha or 0 beta 0 the corresponding energy spectrum being continuous a particular feature of these eigen solutions is that they may not be accessible to perturbation theory the properties of the solutions obtained are investigated and their relation to the case where x or 0 is included is discussed

exact analytical eigenfunctions for the x2 λ x2 1 gx2 interaction the author discusses an ansatz for the eigenfunctions of the x2 x2 1 gx2 interaction introduced by blecher and leach 1987 the author shows that the ansatz provides pairs of solutions in five cases having and g connected by 6g2 4g and conjectures the possible existence of an infinite number of such solution pairs

asymmetric anharmonic oscillators in the hill determinant picture a new version of the hill determinant construction of bound states is proposed it is based on the matching of two suitable power series ansatze for the wave function for the asymmetric well potential v x ax bx2 cx3 dx4 the convergence of the approximants to the physical solutions is proved for b bminimal the taylor coefficients for this anharmonic oscillator s wave function are specified by four term recurrences

harmonic oscillator well with a screened coulombic core is quasi exactly solvable in the quantization scheme which weakens the hermiticity of a hamiltonian to its mere t invariance the superposition v x x2 ze2 x of the harmonic and coulomb potentials is defined at the purely imaginary effective charges ze2 if and regularized by a purely imaginary shift of x this model is quasi exactly solvable we show that at each excited n 1 th harmonic oscillator energy e 2n 3 there exists not only the well known harmonic oscillator bound state at the vanishing charge f 0 but also a normalizable n 1 plet of the further elementary sturmian eigenstates n x at eigencharges f f n 0 n 0 1 n beyond the smallest multiplicities n we recommend perturbative methods for their construction

a simplified fixed point perturbation theory ii an application to the first few pade potentials for pt i see ibid vol 20 p 907 1987 in the radial schrodinger equation for bound states the central potentials v r vcoulombb vshort range are considered with vshort range replaced by its p j p pade approximants p j or 1 the earlier method is completed and a few coefficients of the fixed point perturbation theory asymptotic expansions of the effective hamiltonians are evaluated in reduce for p or 3

a generalized morse asymmetric potential and multiplets of its non numerical exact bound states the solvable morse model with its asymmetric phenomenological potential v r a 1 exp mu r ralpha 2 is generalized via a computer assisted algebraic construction we show that for certain three component phenomenological potentials v r a 1 exp mu r ralpha 2 b 1 exp mu r rbeta 3 c 1 exp mu r rgamma 4 the closed and exact bound states may exist not only in singlets but also in doublets and triplets

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

rd oscillators with arbitrary d 0 and perturbation expansions with sturmians in contrast to widespread belief the current rayleigh schrodinger perturbation theory may provide an easy description of double well oscillators and or of the strongly anharmonic forces with an arbitrary power law asymptotical growth one has only to work in a suitable sturmian basis the feasibility and numerical efficiency of the construction is illustrated on a few one dimensional one body examples

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

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

extended numerov method for computing eigenvalues of specific schrodinger equations numerov s method and an extended version of it are introduced for computing eigenvalues of schrodinger equations with potentials v x which are even functions with respect to x furthermore it is assumed that the wavefunctions tend to zero for x to or infinity the derived results are compared with previously derived numerical data and with available exact values

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

on a nonlinear congruential pseudorandom number generator a nonlinear congruential pseudorandom number generator with modulus m 2 w is proposed which may be viewed to comprise both linear as well as inversive congruential generators the condition for it to generate sequences of maximal period length is obtained it is akin to the inversive one and bears a remarkable resemblance to the latter

an accurate finite difference method for the numerical solution of the schro dinger equation abstract an accurate finite difference approach for computing eigenvalues of schrodinger equations is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x we investigate the well known potential of the onedimensional anharmonic oscillator the symmetric double well potential the razavy potential and the doubly anharmonic oscillator potential ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient than previously derived methods

a nonlinear congruential pseudorandom number generator with power of two modulus a nonlinear congruential pseudorandom number generator is studied where the modulus is a power of two investigation of this generator was suggested by knuth 7 a simple necessary and sufficient condition is given for this generator to have the maximal period length

pseudorandom vector generation by the compound inversive method pseudorandom vectors are of importance for parallelized simulation methods in this paper a detailed analysis of the compound inversive method for the generation of k dimensional uniform pseudorandom vectors a vector analog of the compound inversive method for pseudorandom number generation is carried out in particular periodicity properties and statistical independence properties of the generated sequences are studied based on the discrete discrepancy of s tuples of successive terms in the sequence the results show that the generated sequences have attractive statistical independence properties for pseudorandom vectors of dimensions k 4

a new method in the many body problem ii the earlier proposed method for exact determination of the energy spectra of hamiltonians h and the quantities where w is a finite dimensional perturbation is extended to treat a still larger class of hamiltonians in particular some two and three dimensional problems became amenable to this approach in addition the expressions for the density and local densities of states are deduced as an example an impurity problem is considered in a simple cubic lattice with short range and in a linear chain with long range interactions

inversive congruential pseudorandom numbers avoid the planes nonlinear congruential pseudorandom number generators based on inversions have recently been introduced and analyzed these generators do not show the lattice structure of the widely used linear congruential method in the present paper it is proved that the points formed by d consecutive pseudorandom numbers of an inversive congruential generator with prime modulus possess an even stronger property any hyperplane in space contains at most d of these points that is to say the hyperplane spanned by d arbitrary points of an inversive congruential generator contains no further points this feature makes the inversive congruential method particularly attractive for simulation problems where linear structures within the generated points should be avoided

a new finite difference scheme with minimal phase lag for the numerical solution of the schrodinger equation a new finite difference approach for the numerical solution of the eigenvalue problem of the schrodinger equations is developed in this paper the new approach is based on the maximization of the algebraic order and the minimization of the phase lag we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient when compared with previously derived methods

a new approach to inversion of surface wave dispersion relation for determination of depth distribution of non uniform stresses in elastic materials ultrasonic surface rayleigh waves become dispersive when propagating on non uniformly stressed media in light of this the acoustoelastic effect on their propagation in deformed but initially isotropic materials has been investigated in the past in order to determine the surface stress and gradients of stress with depth an energy perturbation approach considerably reduces the complexity in the treatment of the acoustoelastic effect and inversion of the perturbation relation offers an advantageous route to obtaining the stress gradients this paper presents a new mechanism for effecting this inversion which tries to overcome the effects of the ill posed nature of the problem preliminary simulation results for commonly occurring stress profiles are presented

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

exact solutions of the harmonic oscillator plus non polynomial interaction the exact solutions to a one dimensional harmonic oscillator plus a non polynomial interaction ax2bx2 1cx2 a 0 c 0 are given by the confluent heun functions hc z the minimum value o

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

singular anharmonicities and the analytic continued fractions the potential v r 2r2 2r 4 is investigated and its bound states are constructed by a generalized hill determinant method it is shown that the binding energy e and another free parameter in the underlying laurent series representation of the wave function may be obtained from a coupled pair of hill determinant equations

further polynomial type eigenfunctions the method developed previously by the author for obtaining eigenfunctions in the form polynomial exponential polynomial for a linear second order differential equation in normal form is extended to embrace the general case when the interaction potential or square of the refractive index profile has the form polynomial polynomial various situations arise in the development of the theory some of which can be concluded algebraically but the majority require numerical calculations regarding the rank of quite a general matrix containing many unknown parameters

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

generalized schwinger duality in bound state theory it is shown that an exactly solvable bound state problem is the generator of a nonterminable sequence of partially exactly solvable problems the reversible passage from exact to partial solvability is realized through a class of admissible nonlinear coordinate transformations of which the parabolic schwinger transformation that relates the coulomb and oscillator problems is a particular case interesting spectral features of a novel set of partially solvable problems that emerge through the present considerations are also pointed out

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

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

the schrodinger equation for the f x g x interaction a natural extension of the omega 2x2 lambda x2 1 gx2 interaction is omega 2x2 f x g x where f and g are polynomials in x2 the method of obtaining quasi exact solutions of the schrodinger equation is outlined and some simple examples are given solutions which are valid for part of the real line are obtained when g x has real zeros

on the classical theory of ordinary linear differential equations of the second order and the schrodinger equation for power law potentials the power law potentials in the schrodinger equation solved recently are shown to come from the classical treatment of the singularities of a linear second order differential equation this allows one to enlarge the class of solvable power law potentials

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

some observations on the nature of solutions for the interaction v x x2 λx2 1 gx2 the authors study the nature of exact solutions for the nonpolynomial interaction in particular it is shown that all the solutions are supersymmetric it is also shown that this problem admits a dynamical su 2 symmetry so that an arbitrarily large part of the spectrum but not the whole spectrum can be determined exactly

accurate calculation of the eigenvalues of the x2 λx2 1 gx2 potential pour calculer les valeurs propres de h on calcule trois quantites qui convergent vers ces valeurs propres la difference entre 2 de ces quantites est une mesure de l erreur faite dans le calcul

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

stress distribution determination in isotropic materials via inversion of ultrasonic rayleigh wave dispersion data the possibility of using measurements of the dispersion of rayleigh surface waves propagating on pre stressed initially isotropic materials to determine the surface stress and gradients in such materials is analytically investigated using a first order perturbation formula for the description of the acoustoelastic effect on rayleigh waves it is shown that knowledge of the frequency dependence of the change in phase or phase velocity of the rayleigh wave after propagating a certain distance can in theory be used to determine the stress level at the surface of the medium as well as its derivatives up to a given order basic questions concerning the uniqueness of the inversion are addressed and a formal method of inverting inhomogeneous bi axial stress distributions is presented the effect of the range of frequencies included in the measurements is discussed and illustrated with a numerical example

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

elementary bound states for the power law potentials the author shows how the exact polynomial exp polynomial bound state solutions to the radial schrodinger equation may be constructed for any potential of the form v r sigma delta in d gdelta rdelta where d is a finite set of arbitrary rational numbers and some of the couplings gdelta are not independent

exact solutions of the schrodinger equation for a class of anharmonic oscillators abstract a mathematical basis for the existence of exact solutions to the time independent schrodinger equation for a class of quantum anharmonic oscillators is given a natural extension of that basis provides a broader class than that given by flessas some numerical examples of the calculation of eigenvalues are given a brief comparison is made with the earlier work of hioe macmillen and montroll

pairs of analytical eigenfunctions for the x2 λ x2 1 gx2 interaction a general method is discussed for deriving pairs of exact analytical eigenfunctions of the x2 x2 1 gx2 interaction recently discussed by blecher and leach and commented on by gallas it is shown that those pairs always consist of an odd and an even parity solution in each case and g are related by the conditions 0 and g as a result of their investigations the authors conjecture the possible existence of an infinite number of solution pairs having and g connected by several different relations

lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus the inversive congruential method with modulus m 2w for the generation of uniform pseudorandom numbers has recently been introduced the discrepancy z l of a tuples of consecutive pseudorandom numbers generated by such a generator with maximal period length m 2 is the cru cial quantity for the analysis of the statistical independence properties of these pseudorandom numbers by means of the serial test it is proved that for a pos itive proportion of the inversive congruential generators with maximal period length the discrepancy ij2 is at least of the order of magnitude m xl2 for all k 2 this shows that the bound d 2 2 0 m 1 2 logw 2 established by the second author is essentially best possible

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

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

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

a rodrigues formula approach to determining closed form solutions to the schrodinger equation for symmetric anharmonic oscillators a natural generalization to the rodrigues formula for the hermite polynomials leads to the definition of sets of polynomials under certain constraints they are used to explain the existence of closed form solutions to the schrodinger equation for symmetric anharmonic potential wells and to obtain further closed form solutions

the serial test for congruential pseudorandom numbers generated by inversions two types of congruential pseudorandom number generators based on inversions were introduced recently we analyze the statistical independence properties of these pseudorandom numbers by means of the serial test the results show that these pseudorandom numbers perform satisfactorily under the serial test the methods of proof rely heavily on bounds for character sums such as the weil stepanov bound for character sums over finite fields

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

statistical independence of a new class of inversive congruential pseudorandom numbers linear congruential pseudorandom numbers show several undesirable regularities which can render them useless for certain stochastic simulations this was the motiviation for important recent developments in nonlinear congruential methods for generating uniform pseudorandom numbers it is particularly promising to achieve nonlinearity by employing the operation of multiplicative inversion with respect to a prime modulus in the present paper a new class of such inversive congruential generators is introduced and analyzed it is shown that they have excellent statistical independence properties and model true random numbers very closely the methods of proof rely heavily on weil stepanov bounds for rational exponential sums 39 refs

a perturbation technique for the prediction of the displacement of a line driven plate with discontinuities a novel technique is presented for obtaining approximate analytic expressions for an inhomogeneous line driven plate the equation of motion for the inhomogeneous plate is transformed and the transform of the total displacement is written as a sum of the solution for a homogeneous line driven plate plus a term due to the inhomogeneity the result is an integral equation for the transform of the inhomogeneous contribution this expression may in general be solved numerically however by introducing a small parameter into the problem it may be solved approximately using perturbation techniques this series may not be convergent but its convergence may be improved using pade approximation results are presented for the case of a single mass discontinuity and a distribution of mass discontinuities

the schrodinger equation for the interaction potential x2 λx2 1 gx2 and the first heun confluent equation the authors present systematically the solutions and the quasi polynomial solutions for the schrodinger equation with the x2 lambda x2 1 gx2 interaction potential with the help of the first heun confluent equation or spheroidal heun equation

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

on generalized inversive congruential pseudorandom numbers the inversive congruential method with prime modulus for generating uniform pseudorandom numbers has several very promising properties very recently a generalization for composite moduli has been introduced in the present paper it is shown that the generated sequences have very attractive statistical independence properties

on the simultaneous eigenproblem for the x2 λ x2 1 gx2 1 interaction extension of gallas results an ansatz for the eigenfunctions of the x2 x2 1 gx2 1 interaction has been recently introduced by blecher and leach using this ansatz gallas has obtained five solution pairs for 2g 3g 2 gallas results are generalised and extended here

spiked and symmetrized decadic potentials supporting elementary n plets of bound states we show that the potential wells v x x10 ax8 bx6 cx4 dx2 f x2 with a central spike possess arbitrary finite multiplets of elementary exact bound states their strong asymptotic growth implies an ambiguity in their symmetrically generalized quantization via complex boundary conditions but the three eligible methods coincide at our exceptional solutions

pade oscillators and a new formulation of perturbation theory being guided by the problem of bound states in potentials close to their pade approximants a new rayleigh schrodinger type perturbation theory is developed the unperturbed system is understood here in a broader sense its solutions are not needed but merely the related nondiagonal unperturbed propagator r in particular all the chain models h0 es0 h0 s0 band matrices with arbitrary perturbations are then perturbatively solvable with r constructed in terms of auxiliary matrix continued fraction fn alternatively a generalized unperturbed spectrum fn may be required as an input the algebraically constructed asymptotics of the fn s play this role in our pade examples due to s i the sturmians may also be constructed in the test evaluations of the binding energies and or couplings the simultaneous upper and lower bounds of high precision are shown to be numerically obtainable

a simplified fixed point perturbation theory i the method for a simple potential the author proposes a conversion of the standard differential radial schrodinger equation into an infinite dimensional matrix equation a phi 0 and describes its systematic non numerical solution the results may be characterised as infinite asymptotic expansions of the wavefunctions phi and or of the secular determinant an inverse green function det a the inverse model space dimension plays a role of the small perturbation parameter

statistical mechanics of kinks for quasi exactly solvable potentials we study the thermodynamics of kinks in the presence of quasi exactly solvable qes potentials using a langevin code implemented on a massively parallel computer specifically we study the potential v 2 1 2 for which certain exact solutions of the schrodinger equation are known provided the parameters and satisfy certain constraints for 1 this potential has two degenerate minima otherwise it has only one minimum thus a system with this potential is capable of describing a second order phase transition a la the 4 model however no exact solutions of the schrodinger equation for this potential exist we have obtained an exact solution for the kink in this model in addition we have calculated in a 1d model such quantities as the probability density function pdf field configuration and field field correlation functions both above and below the transition temperature tc for several temperatures these quantities help us understand the contribution to the specific heat from coherent structures such as domain walls kinks as opposed to the contribution from lattice vibrations we have calibrated our results against known exact solutions for limiting cases with very high accuracy

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

on computing eigenvalues of the schrodinger equation for symmetrical potentials the problem of the numerical determination of eigenvalues of the one dimensional schrodinger equation with symmetrical potential v x is considered the problem is reduced to the definition of an eigenvalue function g e well defined for the given potential v it is theoretically proved that the even parity eigenvalues are given by g e 0 and the odd parity eigenvalues are given by g e infinity the method is canonical in the sense that it is independent of the eigenfunction yet along with the eigenvalue it allows the determination of the eigenfunction initial values this method is applied to the potentials v x2 v x6 bx2 v x2 lambda x2 1 gx2 where exact eigenvalues are available the numerical results compared with the exact ones and with those of previous confirmed methods show that the present method is accurate and efficient

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

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