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

comment on what bell did according to maudlin bell showed that it is the world which is non lcoal and not just some particular theories of it i argue that this conclusion is arrived at by taking for granted all assumptions of realism or classicality if these are taken into account the resulting conclusion that a classical world which allows for bell inequality violations must be non local is in good agreement with the mainstream perception of bell s theorem s this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

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

relative convergences of the wkb and swkb approximations the relative convergences of the wkb and swkb approximations are examined by calculating the eigenenergies for four potentials by one and two term wkb and swkb approximations exact eigenenergies for these four potentials are also calculated by a numerical integration of the schrodinger equation varied results are found for the four potentials it is found that in general the effect of the second term in the wkb and swkb approximations depends on the potential the parameters involved and the quantum number of the state no simple generalizations are possible

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

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

steering or maybe why einstein did not go all the way to bellʼs argument it is shown that a main source of conflict between einstein and the mainstream quantum physicists was his insistence that wave functions like classical probability distributions do not refer to individual particles and in particular do not describe individual systems completely the epr paper was written to argue for this position by aiming at showing that wave functions are unsuitable as local hidden variables the authors failed to see that a slight extension could have ruled out such local hidden variables in general as background for this analysis of the epr argument the notion of steering is described and a version of the bell argument is proved which emphasizes non local signalling aspects finally some background is given concerning a well known paper by the present author which is celebrating 25 years this year and in which the first non steering models were constructed this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

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

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

conformal invariance in a dirac oscillator the conformal invariance properties of a dirac oscillator are established a set of operators is constructed whose algebra shows that it can be considered as a conformal system the operators are then used to solve the problem using algebraic techniques the superconformal generalization of the algebra is also worked out and some consequences of these invariances for the properties of the model are mentioned

new class of function dependent deformations of creation and annihilation operators the author gives formulae for point spectra of hermitian operators built from creation and annihilation type operators satisfying general algebraic relation which he calls g f f equivalence by choosing the parameters in a specific way the description of the quantum q oscillator is given in a model built from the q derivative and dilatation operator he shows the explicit form of eigenfunctions of the model from l2 r then he constructs the transformation between his model and the q oscillator with arbitrary q real

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

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

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

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

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

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

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

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

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

can observed randomness be certified to be fully intrinsic in general any observed random process includes two qualitatively different forms of randomness apparent randomness which results both from ignorance or lack of control of degrees of freedom in the system and intrinsic randomness which is not ascribable to any such cause while classical systems only possess the first kind of randomness quantum systems may exhibit some intrinsic randomness in this letter we provide quantum processes in which all the observed randomness is fully intrinsic these results are derived under minimal assumptions the validity of the no signaling principle and an arbitrary but not absolute lack of freedom of choice our results prove that quantum predictions cannot be completed already in simple finite scenarios for instance of three parties performing two dichotomic measurements moreover the observed randomness tends to a perfect random bit when increasing the number of parties thus defining an explicit process attaining full randomness amplification

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

relevance vector machine for survival analysis an accelerated failure time aft model has been widely used for the analysis of censored survival or failure time data however the aft imposes the restrictive log linear relation between the survival time and the explanatory variables in this paper we introduce a relevance vector machine survival rvms model based on weibull aft model that enables the use of kernel framework to automatically learn the possible nonlinear effects of the input explanatory variables on target survival times we take advantage of the bayesian inference technique in order to estimate the model parameters we also introduce two approaches to accelerate the rvms training in the first approach an efficient smooth prior is employed that improves the degree of sparsity in the second approach a fast marginal likelihood maximization procedure is used for obtaining a sparse solution of survival analysis task by sequential addition and deletion of candidate basis functions these two approaches denoted by smooth rvms and fast rvms typically use fewer basis functions than rvms and improve the rvms training time however they cause a slight degradation in the rvms performance we compare the rvms and the two accelerated approaches with the previous sparse kernel survival analysis method on a synthetic data set as well as six real world data sets the proposed kernel survival analysis models have been discovered to be more accurate in prediction although they benefit from extra sparsity the main advantages of our proposed models are 1 extra sparsity that leads to a better generalization and avoids overfitting 2 automatic relevance sample determination based on data that provide more accuracy in particular for highly censored survival data and 3 flexibility to utilize arbitrary number and types of kernel functions e g non mercer kernels and multikernel learning

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

polynomial type eigenfunctions the differential equation for the harmonic oscillator is generalised to include an interaction potential containing a positive definite quadratic denominator conditions are developed under which certain eigenfunctions take the form of an exponential function multiplied by a polynomial the problem reduces to that of finding the eigenvalues of a certain matrix that is tridiagonal in form properties of the eigenvalues of this matrix are investigated since they are functions of a parameter occurring in the positive definite quadratic form the asymptotic forms of these eigenvalues are developed together with computed results expressed as curves showing the variation of the eigenvalues with respect to this parameter

quantum equilibrium and the origin of absolute uncertainty what is randomness probability certainty knowledge these are old and difficult questions and we shall not focus on them here nonetheless we shall obtain sharp striking conclusions concerning the relationship between these concepts

l2 series solutions of the dirac equation for power law potentials at rest mass energy we obtain solutions of the three dimensional dirac equation for radial power law potentials at rest mass energy as an infinite series of square integrable functions these are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the dirac operator is tridiagonal the wave equation results in a three term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials these are modified versions of the meixner pollaczek polynomials and of the continuous dual hahn polynomials the choice depends on the values of the angular momentum and the power of the potential

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

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

the evidence framework applied to classification networks three bayesian ideas are presented for supervised adaptive classifiers first it is argued that the output of a classifier should be obtained by marginalizing over the posterior distribution of the parameters a simple approximation to this integral is proposed and demonstrated this involves a moderation of the most probable classifier s outputs and yields improved performance second it is demonstrated that the bayesian framework for model comparison described for regression models in mackay 1992a b can also be applied to classification problems this framework successfully chooses the magnitude of weight decay terms and ranks solutions found using different numbers of hidden units third an information based data selection criterion is derived and demonstrated within this framework

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

efficient svm training using low rank kernel representations svm training is a convex optimization problem which scales with the training set size rather than the feature space dimension while this is usually considered to be a desired quality in large scale problems it may cause training to be impractical the common techniques to handle this difficulty basically build a solution by solving a sequence of small scale subproblems our current effort is concentrated on the rank of the kernel matrix as a source for further enhancement of the training procedure we first show that for a low rank kernel matrix it is possible to design a better interior point method ipm in terms of storage requirements as well as computational complexity we then suggest an efficient use of a known factorization technique to approximate a given kernel matrix by a low rank matrix which in turn will be used to feed the optimizer finally we derive an upper bound on the change in the objective function value based on the approximation error and the number of active constraints support vectors this bound is general in the sense that it holds regardless of the approximation method

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

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

gaussian kernel width optimization for sparse bayesian learning sparse kernel methods have been widely used in regression and classification applications the performance and the sparsity of these methods are dependent on the appropriate choice of the corresponding kernel functions and their parameters typically the kernel parameters are selected using a cross validation approach in this paper a learning method that is an extension of the relevance vector machine rvm is presented the proposed method can find the optimal values of the kernel parameters during the training procedure this algorithm uses an expectation maximization approach for updating kernel parameters as well as other model parameters therefore the speed of convergence and computational complexity of the proposed method are the same as the standard rvm to control the convergence of this fully parameterized model the optimization with respect to the kernel parameters is performed using a constraint on these parameters the proposed method is compared with the typical rvm and other competing methods to analyze the performance the experimental results on the commonly used synthetic data as well as benchmark data sets demonstrate the effectiveness of the proposed method in reducing the performance dependency on the initial choice of the kernel parameters

new class of conditionally exactly solvable potentials in quantum mechanics motivated by an idea of dutra 1993 we obtain a new class of one dimensional conditionally exactly solvable potentials for which the entire spectra can be obtained in an algebraic manner provided one of the potential parameters is assigned a fixed negative value it is shown that using shape invariant potentials as input one may generate different classes of such potentials even in more than one dimension we also illustrate that wkb and supersymmetry inspired wkb methods provide very good approximations for these potentials with the latter doing comparatively better

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

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

the two bell s theorems of john bell many of the heated arguments about the meaning of bell s theorem arise because this phrase can refer to two different theorems that john bell proved the first in 1964 and the second in 1976 his 1964 theorem is the incompatibility of quantum phenomena with the dual assumptions of locality and determinism his 1976 theorem is the incompatibility of quantum phenomena with the unitary property of local causality this is contrary to bell s own later assertions that his 1964 theorem began with that single and indivisible assumption of local causality even if not by that name while there are other forms of bell s theorems which i present to explain the relation between jarrett completeness fragile locality and epr completeness i maintain that bell s two versions are the essential ones although the two bell s theorems are logically equivalent their assumptions are not and the different versions of the theorem suggest quite different conclusions which are embraced by different communities for realists the notion of local causality ruled out by bell s 1976 theorem is motivated implicitly by reichenbach s principle of common cause and explicitly by the principle of relativistic causality and it is the latter which must be forgone operationalists pay no heed to reichenbach s principle but wish to keep the principle of relativistic causality which bolstered by an implicit principle of agent causation implies their notion of locality thus for operationalists bell s theorem is the 1964 one and implies that it is determinism that must be forgone i discuss why the two camps are drawn to these different conclusions and what can be done to increase mutual understanding