a hill series approach to wavefunction nodes the hill series method is shown to have several advantages for treating the schrodinger equation when the wavefunction obeys the dirichlet boundary condition r 0 a novel combination of the virial theorem the hill series method and a finite difference method is used to find the node positions of the wavefunction for several excited states of perturbed oscillator and coulombic systems and to find critical radii for the hydrogen atom

equivalence of unstable anharmonic oscillators and double wells we give a direct and general proof of the equality of the energy levels of a pair of different quantum problems the double well and the unstable anharmonic oscillator defined by complex translation the resonances of the two problems are also equal and are the analytic continuations of the anharmonic oscillator energy levels

a new method for summation of divergent power series a new method for summation of divergent power series is developed it only requires the knowledge of the form of both the small and large power expansion being the perturbation parameter and few coefficients of one of them to yield excellent results convergence is proved for a simple two level model and reasonable arguments are given for more complex and interesting models the method is quite general and contains some resummation techniques reported previously as particular cases the anharmonic mean square displacement function the ground state eigenvalue of the quantum mechanical anharmonic oscillator and the ground state energy of the hydrogen atom in a magnetic field calculated in this way are shown to be of striking accuracy in the whole range of the perturbation parameter

nonlinear aspects of the renormalization group flows of dyson s hierarchical model we review recent results concerning the renormalization group rg transformation of dyson s hierarchical model hm this model can be seen as an approximation of a scalar field theory on a lattice we introduce the hm and show that its large group of symmetry simplifies drastically the blockspinning procedure several equivalent forms of the recursion formula are presented with unified notations rigourous and numerical results concerning the recursion formula are summarized it is pointed out that the recursion formula of the hm is inequivalent to both wilson s approximate recursion formula and polchinski s equation in the local potential approximation despite the very small difference with the exponents of the latter we draw a comparison between the rg of the hm and functional rg equations in the local potential approximation the construction of the linear and nonlinear scaling variables is discussed in an operational way we describe the calculation of non universal critical amplitudes in terms of the scaling variables of two fixed points this question appears as a problem of interpolation between these fixed points universal amplitude ratios are calculated we discuss the large n limit and the complex singularities of the critical potential calculable in this limit the interpolation between the hm and more conventional lattice models is presented as a symmetry breaking problem we briefly introduce models with an approximate supersymmetry one important goal of this review is to present a configuration space counterpart suitable for lattice formulations of functional rg equations formulated in momentum space often called exact rg equations and abbreviated erge

non borel summable φ4 theory in zero dimension a toy model for testing numerical and analytical methods the exact analytical solution for the generating functional of the zero dimensional 4 theory with degenerate minima is obtained in the whole complex coupling parameter plane for testing purposes the efficiency and precision of different computing tools proposed in non borel summable field theories to obtain approximate solutions in both perturbative and nonperturbative regimes are analyzed furthermore a new resummation approach is proposed in order to successfully deal with factorially divergent series it provides a representation of the generating function in terms of an unambiguously defined laplace borel integral on the other hand a recent approach called the generalized borel transform is shown to be an accurate and robust technique to capture non perturbative contributions in the coupling parameter an extension of this approach to path integrals is proposed

exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator is studied in a region of the coupling constant plane in which there is a sequence of exponentially small subseries beyond the standard rayleigh schrodinger perturbation theory rspt power series we give a simple algorithm for the calculation of these subseries to any desired order in terms of the rspt coefficients expressed as polynomials in the quantum number and illustrate our results with numerical borel pade summations of the expansion up to third exponentially small order

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

robust sequential detection in distributed sensor networks we consider the problem of sequential binary hypothesis testing with a distributed sensor network in a non gaussian noise environment to this end we present a general formulation of the consensus innovations sequential probability ratio test mathcal ci sprt furthermore we introduce two different concepts for robustifying the mathcal ci sprt and propose four different algorithms namely the least favorable density mathcal ci sprt the median mathcal ci sprt the m mathcal ci sprt and the myriad mathcal ci sprt subsequently we analyze their suitability for different binary hypothesis tests before verifying and evaluating their performance in a shift in mean and a shift in variance scenario for different network connectivities and amounts of noise contamination

on designing heteroclinic networks from graphs abstract robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems these networks may have a complicated structure determined to a large extent by the constraints and dimension of the system as these networks are of great interest as dynamical models of biological and cognitive processes it is useful to understand how particular directed graphs can be realised as attracting robust heteroclinic networks between states in phase space this paper presents two methods of realising arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by odes we say the odes realise the graphs as heteroclinic networks between equilibria that represent the vertices suppose we have a directed graph on n v vertices with n e edges the simplex realisation embeds the graph as an invariant set of a flow on an n v 1 simplex this method realises the graph as long as it is one and two cycle free the cylinder realisation embeds a graph as an invariant set of a flow on a n e 1 dimensional space this method realises the graph as long as it is one cycle free in both cases we realise the graph as an invariant set within an attractor and discuss some illustrative examples including the influence of noise and parameters on the dynamics in particular we show that the resulting heteroclinic network may or may not display memory of the vertices visited

minimax robust decentralized detection decentralized detection problems are studied where the sensor distributions are not specified completely the sensor distributions are assumed to belong to known uncertainty classes it is shown for a broad class of such problems that a set of least favorable distributions exists for minimax robust testing between the hypotheses it is hence established that the corresponding minimax robust tests are solutions to simple decentralized detection problems for which the sensor distributions are specified to be the least favorable distributions

expansion around instantons in quantum mechanics we calculate numerically a few terms of the corrections to the large order behavior of the ground state energy of the o n anharmonic oscillator by analyzing the perturbation series we then generate 94 terms of the perturbative expansion of the difference between the energies of the two low lying states of the double well potential and analyze their large order behavior

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

controlled perturbation induced switching in pulse coupled oscillator networks pulse coupled systems such as spiking neural networks exhibit nontrivial invariant sets in the form of attracting yet unstable saddle periodic orbits where units are synchronized into groups heteroclinic connections between such orbits may in principle support switching processes in these networks and enable novel kinds of neural computations for small networks of coupled oscillators we here investigate under which conditions and how system symmetry enforces or forbids certain switching transitions that may be induced by perturbations for networks of five oscillators we derive explicit transition rules that for two cluster symmetries deviate from those known from oscillators coupled continuously in time a third symmetry yields heteroclinic networks that consist of sets of all unstable attractors with that symmetry and the connections between them our results indicate that pulse coupled systems can reliably generate well defined sets of complex spatiotemporal patterns that conform to specific transition rules we briefly discuss possible implications for computation with spiking neural systems

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

simple method to make asymptotic series of feynman diagrams converge we show that for two nontrivial lambda phi 4 problems the anharmonic oscillator and the landau ginzburg hierarchical model improved perturbative series can be obtained by cutting off the large field contributions the modified series converge to values exponentially close to the exact ones for lambda larger than some critical value the method outperforms pade s approximants and borel summations the method can also be used for series which are not borel summable such as the double well potential series we show that semiclassical methods can be used to calculate the modified feynman rules estimate the error and optimize the field cutoff

quantum anharmonic oscillators a new approach the determination of the eigenenergies of a quantum anharmonic oscillator consists merely in finding the zeros of a function of the energy namely the wronskian of two solutions of the schr dinger equation which are regular respectively at the origin and at infinity we show in this paper how to evaluate that wronskian exactly except for numerical rounding errors the procedure is illustrated by application to the gx2 x2n n a positive integer oscillator

encoding via conjugate symmetries of slow oscillations for globally coupled oscillators we study properties of the dynamics underlying slow cluster oscillations in two systems of five globally coupled oscillators these slow oscillations are due to the appearance of structurally stable heteroclinic connections between cluster states in the noise free dynamics in the presence of low levels of noise they give rise to long periods of residence near cluster states interspersed with sudden transitions between them moreover these transitions may occur between cluster states of the same symmetry or between cluster states with conjugate symmetries given by some rearrangement of the oscillators we consider the system of coupled phase oscillators studied by hansel et al phys rev e 48 3470 1993 in which one can observe slow noise driven oscillations that occur between two families of two cluster periodic states in the noise free case there is a robust attracting heteroclinic cycle connecting these families the two families consist of symmetric images of two inequivalent periodic orbits that have the same symmetry for n 5 oscillators one of the periodic orbits has one unstable direction and the other has two unstable directions examining the behavior on the unstable manifold for the two unstable directions we observe that the dimensionality of the manifold can give rise to switching between conjugate symmetry orbits by applying small perturbations to the system we can easily steer it between a number of different marginally stable attractors finally we show that similar behavior occurs in a system of phase energy oscillators that are a natural extension of the phase model to two dimensional oscillators we suggest that switching between conjugate symmetries is a very efficient method of encoding information into a globally coupled system of oscillators and may therefore be a good and simple model for the neural encoding of information

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

factorization of quasiclassical integrable systems abstract using asymptotic methods one calculates the first temperature correction to the free energy of non linear kink bearing scalar fields the method allows to make a distinct separation between the contribution of the anharmonic optical phonon field and that of the dilute gas of kinks the same technique can be used to calculate the multi kink contributions to the free energy

the confined system approximation for solving non separable potentials in three dimensions the hilbert space to which the wavefunction of the three dimensional schrodinger equation belongs has been replaced by where is a bounded region the energy spectrum of the usual unbounded system is then determined by showing that the dirichlet and neumann problems in generate upper and lower bounds respectively to the eigenvalues required highly accurate numerical results for the quartic and sextic oscillators are presented for a wide range of the coupling constants

a note on the non perturbative zero dimensional λϕ4 model abstract we exhibit the partition function of the zero dimensional 4 model as a simple exact expression in terms of the macdonald function for re 0 then by analytic continuation we obtain an expression defined in the complex coupling constant plane for arg consequently the partition function understood as an analytic continuation is defined for all values of except for a branch cut along the real negative axis this shows that at least in zero dimension the partition function can be defined for negative coupling constant where the integral is formally divergent provided it has a non vanishing imaginary part we also evaluate the partition function on perturbative grounds using the borel summation technique and we found that in the common domain of validity for re 0 it coincides precisely with the exact expression furthermore a connection between the non perturbative zero dimensional solution and the ultralocal 4 model in arbitrary dimension d is presented

higher order corrections to instantons the energy levels of the double well potential receive beyond perturbation theory contributions which are non analytic in the coupling strength these are related to instanton effects for example the separation between the energies of odd and even parity states is given at leading order by the one instanton contribution however to determine the energies more accurately multi instanton configurations have also to be taken into account we investigate here the two instanton contributions first we calculate analytically higher order corrections to multi instanton effects we then verify that the difference betweeen numerically determined energy eigenvalues and the generalized borel sum of the perturbation series can be described to very high accuracy by two instanton contributions we also calculate higher order corrections to the leading factorial growth of the perturbative coefficients and show that these are consistent with analytic results for the two instanton effect and with exact data for the first 200 perturbative coefficients

instantons in quantum mechanics numerical evidence for a conjecture in previous articles we have given a proper definition of multi instanton contributions in quantum mechanics and calculated these contributions to leading order we have also presented a conjecture about the form of the expansion of these multi instanton contributions to all orders in powers of the coupling constant we give here some numerical results which support this conjecture

high precision numerical determination of eigenvalues for a double well potential related to the zinn justin conjecture a numerical method of high precision is used to calculate the energy eigenvalues and eigenfunctions for a symmetric double well potential the method is based on enclosing the system within two infinite walls with a large but finite separation and developing a power series solution for the schrodinger equation the obtained numerical results are compared with those obtained on the basis of the zinn justin conjecture and found to be in excellent agreement

variational perturbation theory for summing divergent tunnelling amplitudes we present a method for evaluating divergent series with factorially growing coefficients of equal sign the method is based on an analytic continuation of variational perturbation theory from the regime of alternating signs we demonstrate its power first by applying it to the exactly known partition function of the anharmonic oscillator in zero space time dimensions the simple integral then we consider the quantum mechanical case of one space time dimension and derive the imaginary part of the ground state energy of the anharmonic oscillator for all negative values of the coupling constant g including the non analytic tunnelling regime at small g as a highlight of the theory we extract from the divergent perturbation expansion the action of the critical bubble and the contribution of the higher loop fluctuations around the bubble some figures in this article are in colour only in the electronic version

large order perturbation theory for the o 2 anharmonic oscillator with negative anharmonicity and for the double well potential for the o 2 anharmonic oscillator with negative anharmonicity and one dimensional double well potential u x x2 r2 2 8r2 r is a large parameter for both problems conventional rayleigh schrodinger expansions in power series of 1 r2 for the energy eigenvalues agree the authors studied asymptotic expansions for the eigenvalues which contain in addition to the perturbation series exponentially small terms in r2 due to the tunnelling through corresponding potential barriers numerical calculations are also performed and comparisons with the asymptotic formulae are given

robust hypothesis testing for bounded classes of probability densities corresp least favorable pairs of probability density functions and robust tests are obtained for classes of density functions specified as bands with upper and lower bounds the results are shown to extend huber s solution for the epsilon contaminated classes which is contained as a special case a minimum distance property of the least favorable density pair is also made explicit

virtual turning points and bifurcation of stokes curves for higher order ordinary differential equations for a higher order linear ordinary differential operator p its stokes curve bifurcates in general when it hits another turning point of p this phenomenon is most neatly understandable by taking into account stokes curves emanating from virtual turning points together with those from ordinary turning points this understanding of the bifurcation of a stokes curve plays an important role in resolving a paradox recently found in a system of linear differential equations associated with the fourth painleve equation

conformal mappings versus other power series methods for solving ordinary differential equations illustration on anharmonic oscillators the simplicity and the efficiency of a quasi analytical method for solving nonlinear ordinary differential equations ode is illustrated on the study of anharmonic oscillators ao with a potential v x x2 x2m m 0 the method bervillier 2008 nucl phys b801 296 applies a priori to any ode with two point boundaries one being located at infinity the solution of which has fixed singularities in the complex plane of the independent variable x a conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed taylor series of the generic solution so that the boundary condition at infinity can be easily imposed in principle this constraint when applied on the logarithmic derivative of the wavefunction determines the eigenvalues to an arbitrary level of accuracy in practice for 0 or slightly negative the accuracy of the results obtained is astonishingly large with regard to the modest computing power used various aspects of the method and comparisons with some seemingly similar methods based also on expressing the solution as a taylor series are shortly reviewed presented and discussed

frequency locking and complex dynamics near a periodically forced robust heteroclinic cycle received 18 july 2006 published 6 november 2006robust heteroclinic cycles occur naturally in many classes of nonlinear differential equations with invarianthyperplanes in particular they occur frequently in models for ecological dynamics and uid mechanicalinstabilities we consider the effect of small amplitude time periodic forcing and describe how to reduce thedynamics to a two dimensional map in the limit where the heteroclinic cycle loses asymptotic stability intervals of frequency locking appear in the opposite limit where the heteroclinic cycle becomes stronglystable the dynamics remains chaotic and no frequency locking is observed doi 10 1103 physreve 74 055201 pacs number s 05 45 a 82 40 bj

minimax robust hypothesis testing minimax robust hypothesis testing is studied for the cases where the collected data samples are corrupted by outliers and are mismodeled due to modeling errors for the former case huber s clipped likelihood ratio test is introduced and analyzed for the latter case first a robust hypothesis testing scheme based on the kullback leibler divergence is designed this approach generalizes a previous work by levy second dabak and johnson s asymptotically robust test is introduced and other possible designs based on f divergences are investigated all proposed and analyzed robust tests are extended to fixed sample size and sequential probability ratio tests simulations are provided to exemplify and evaluate the theoretical derivations

arbitrarily accurate eigenvalues for one dimensional polynomial potentials we show that the riccati form of the one dimensional schrodinger equation can be reformulated in terms of two linear equations depending on an arbitrary function g when g and the potential as for anharmonic oscillators are polynomials the solutions of these two equations are entire functions l and k and the zeros of k are identical to those of the wavefunction requiring such a zero at a large but finite value of the argument yields low energy eigenstates with exponentially small errors approximate formulae for these errors are provided we explain how to choose g in order to dramatically improve the numerical treatment the method yields many significant digits with modest computer means we discuss the extension of this method in the case of several variables

exact quantization condition for anharmonic oscillators in one dimension an exact quantization condition is given for the one dimensional schrodinger operator with a homogeneous anharmonic potential q2m it has the form of an explicit mapping from level sequences to level sequences involving a bohr sommerfeld like quantization step and having the exact spectrum as fixed point numerical tests and an approximate linear theory both suggest at least for the few lowest m that the mapping has a contractive region when an initial level sequence is only asymptotically correct to lowest order its iterates are seen to converge term by term towards the exact eigenvalues this type of approach ought to extend to general polynomial potentials

exact semiclassical expansions for one dimensional quantum oscillators a set of rules is given for dealing with wkb expansions in the one dimensional analytic case whereby such expansions are not considered as approximations but as exact encodings of wave functions thus allowing for analytic continuation with respect to whichever parameters the potential function depends on with an exact control of small exponential effects these rules which include also the case when there are double turning points are illustrated on various examples and applied to the study of bound state or resonance spectra in the case of simple oscillators it is thus shown that the rayleigh schrodinger series is borel resummable yielding the exact energy levels in the case of the symmetrical anharmonic oscillator one gets a simple and rigorous justification of the zinn justin quantization condition and of its solution in terms of multi instanton expansions

langer cherry derivation of the multi instanton expansion for the symmetric double well the multi instanton expansion for the eigenvalues of the symmetric double well is derived using a langer cherry uniform asymptotic expansion of the solution of the corresponding schrodinger equation the langer cherry expansion is anchored to either one of the minima of the potential and by construction has the correct asymptotic behavior at large distance while the quantization condition amounts to imposing the even or odd parity of the wave function this method leads to an efficient algorithm for the calculation to virtually any desired order of all the exponentially small series of the multi instanton expansion and with trivial modifications can also be used for nonsymmetric double wells

a tractable example of perturbation theory with a field cutoff the anharmonic oscillator for 4 models the introduction of a large field cutoff improves significantly the accuracy that can be reached with perturbative series but the calculation of the modified coefficients remains a challenging problem we show that this problem can be solved numerically and analytically in the limits of large and small field cutoffs for the ground state energy of the anharmonic oscillator for the two lowest orders in the approximate formulae obtained in the large field cutoff limit extend unexpectedly far in the low field cutoff region and there is a significant overlap with the small field cutoff approximation for the higher orders this is not the case however the shape of the transition between the small field cutoff regime and the large field cutoff regime is approximately order independent