an algorithm to generate canonical transformations application to normal forms we describe a new method to generate symplectic diffeomorphisms this method is based on primitive functions and it was introduced in nonlinearity 13 2000 1483 we apply the algorithm to the construction of normal forms of kam tori a concrete application developed in detail and implemented numerically is normal forms near invariant circles

the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

erratic behavior of invariant circles in standard like mappings abstract we study mappings like the standard map but with the sine replaced by another function by varying a parameter in this function it changes smoothly from the sine to a piecewise linear pwl function the function is always analytic except when it is pwl taking the function close enough to the pwl function it is found that kam tori can reappear after they disappeared when the nonlinearity is increased when plotting the boundary around the area where a kam torus with noble winding number exists in the plane of the non linearity versus the distance of the function to the pwl function self similarity is observed

renormalization and transition to chaos in area preserving nontwist maps abstract the problem of transition to chaos i e the destruction of invariant circles or kam kolmogorov arnold moser curves in area preserving nontwist maps is studied within the renormalization group framework nontwist maps are maps for which the twist condition is violated along a curve known as the shearless curve in renormalization language this problem is that of finding and studying the fixed points of the renormalization group operator r that acts on the space of maps a simple period two fixed point of r whose basin of attraction contains the nontwist maps for which the shearless curve exists is found also a critical period 12 fixed point of r with two unstable eigenvalues is found the basin of attraction of this critical fixed point contains the nontwist maps for which the shearless curve is at the threshold of destruction this basin defines a new universality class for the transition to chaos in area preserving maps

nontrivial solutions for a class of quasilinear problems with jumping nonlinearities via a cohomological local splitting abstract we establish the existence of a nontrivial solution for a class of quasilinear elliptic boundary value problems with jumping nonlinearities by means of variational arguments and a cohomological local splitting

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

generalization of the birman schwinger method for the number of bound states we generalize the birman schwinger method and derive a general upper bound on the number of bound states in the s wave for a spherically symmetric potential this general bound includes of course the bargmann bound but also leads for increasing negative potentials to a calogero cohn type bound finally we show that for a large class among these potentials one can obtain further improvements

the three dimensional fourier grid hamiltonian method a method to compute the bound state eigenvalues and eigenfunctions of a schrodinger equation or a spinless salpeter equation with central interaction is presented this method is the generalization to the three dimensional case of the fourier grid hamiltonian method for a one dimensional schrodinger equation it requires only the evaluation of the potential at equally spaced grid points and yields the radial part of the eigenfunctions at the same grid points it can be easily extended to the case of coupled channel equations and to the case of nonlocal inter actions

necessary and sufficient conditions for existence of bound states in a central potential we obtain using the birman schwinger method a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics these conditions yield a monotonic series of lower limits on the critical value of the strength of the potential for which a first bound state appears which converges to the exact critical strength we also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential

generalised meshes for quantum mechanical problems a new method to discretise schrodinger equations on a mesh is described this method is based on an accurate approximation of a variational calculation the regularly spaced mesh and meshes based on the zeros of orthogonal polynomials are studied in detail it is shown that with each type of mesh is associated a particular kinetic energy operator and an optimal formula for its discretized form the applications to some simple potential problems show that the method is very accurate as well as very simple applications to many body problems indicate that the accuracy of the results is improved by an order of magnitude with respect to conventional mesh calculations

multiparameter critical situations universality and scaling in two dimensional period doubling maps we review critical situations linked with period doubling transition to chaos which require using at least two dimensional maps as models representing the universality classes each of them corresponds to a saddle solution of the two dimensional generalization of feigenbaum cvitanovic equation and is characterized by a set of distinct universal constants analogous to feigenbaum s and one type of criticality designated h was discovered by several authors in 80 th in the context of period doubling in conservative dynamics but occurs as well in dissipative dynamics as a phenomenon of codimension 2 second is bicritical behavior which takes place in systems allowing decomposition onto two dissipative period doubling subsystems each of which is brought by parameter tuning onto a threshold of chaos types of criticality designated as fq and c occur in non invertible two dimensional maps we present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems in particular we indicate a possibility for realization of the h type criticality without vanishing dissipation but with its compensation in a self oscillatory system next we present a number of examples coupled henon like maps coupled driven oscillators coupled chaotic self oscillators which manifest bicritical behavior for fq type we indicate possibility to arrange it in non symmetric systems of coupled period doubling subsystems e g in henon like maps and in chua s circuits for c type we present examples of its appearance in a driven rossler oscillator at the period doubling accumulation on the edge of syncronization tongue and in a model map with the neimark sacker bifurcation

manifold reconnection and diffusion in strong chaos abstract we analyse diffusion across a reconnecting zone in a regime of strong chaos numerical estimates from the angular correlation functions averaged over a suitable set of phase space initial conditions are compared to a local diffusion coefficient obtained using fick s law it emerges that the diffusion coefficient is enhanced from small to larger values as successive reconnection like processes or more usual bifurcations take place this feature is illustrated using a quadratic nontwist map

fractal boundary for the existence of invariant circles for area preserving maps observations and renormalisation explanation abstract breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters the critical line in parameter space turns out to have a cantor set of cusps which can be organized in a self similar tree a theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator the results of some other researchers on similar systems are shown to fit this explanation as well

the defocusing mechanism of isochronous resonances abstract we present a numerical study concerning the defocusing mechanism of isochronous resonance island chains in the presence of two permanent robust tori the process is initialized and concluded through bifurcations of fixed points located on the robust tori our approach is based on a hamiltonian system derived from the resonant normal form choosing a convenient parameter in this system we are able to depict a comprehensive analysis of the dynamics of the problem

renormalization group operators for maps and universal scaling of universal scaling exponents renormalization group rg methods provide a unifying framework for understanding critical behaviour such as transition to chaos in area preserving maps and other dynamical systems which have associated with them universal scaling exponents recently de la llave et al 2007 10 have formulated the principle of approximate combination of scaling exponents pacse for short which relates exponents for different criticalities via their combinatorial properties the main objective of this paper is to show that certain integrable fixed points of rg operators for area preserving maps do indeed follow the pacse

combination laws for scaling exponents and relation to the geometry of renormalization operators the principle of approximate combination of scaling exponents renormalization group has become a standard tool for describing universal properties of different routes to chaos period doubling in unimodal maps quasiperiodic transitions in circle maps dynamics on the boundaries of siegel disks destruction of invariant circles of area preserving twist maps and others the universal scaling exponents for each route are related to the properties of the corresponding renormalization operators

a sufficient condition for the existence of bound states for scalar spherically symmetric potentials we consider the class of three dimensional attractive finite range or similar potentials depending on a strength constant beyond a critical value the potential has at least one bound state for the s wave we propose simple bounds for obtained by formally solving the schrodinger equation for the zero energy bound state the various bounds are compared with the exact value for a set of usual potentials they are also compared with bounds derived in earlier works by glaser et al and by calogero we show that the case is solved by equations very similar to ones

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

local approximation to the critical parameters of quantum wells we calculate the critical parameters for some simple quantum wells by means of the riccati pade method the original approach converges reasonably well for nonzero angular momentum quantum number l but rather too slowly for the s states we therefore propose a simple modification that yields remarkably accurate results for the latter case the rate of convergence of both methods increases with l and decreases with the radial quantum number n we compare rpm results with wkb ones for sufficiently large values of l as illustrative examples we choose the one dimensional and central field gaussian wells as well as the yukawa potential the application of perturbation theory by means of the rpm to a class of rational potentials yields interesting and baffling unphysical results

the levinson theorem the levinson theorem is a fundamental theorem in quantum scattering theory which shows the relation between the number of bound states and the phase shift at zero momentum for the schrodinger equation the levinson theorem was established and developed mainly with the jost function with the green function and with the sturm liouville theorem in this review we compare three methods of proof study the conditions of the potential for the levinson theorem and generalize it to the dirac equation the method with the sturm liouville theorem is explained in some detail references to development and application of the levinson theorem are introduced

on the birman schwinger principle applied to δ m2 m the condition for e 0 to be an eigenvalue of the operator m2 m v is obtained through the use of the birman schwinger principle theorem 3 2 by setting e 2 and using the analyticity of the corresponding birman schwinger kernel theorem 3 1 the series development of 1 is obtained up to second order on theorem 4 1

critical behaviour of weakly bound systems we consider the class of three dimensional finite range or similar potentials depending on a strength constant we study the behaviour of the eigenvalue e as a function of where is the critical value at the transition from 0 1 bound state for the case we find whereas the relationship is linear for treating as a continuous parameter in the radial schrodinger equation we give the evolution of the power law between and besides spherically symmetric scalar potentials we also discuss the case of a repulsive scalar potential combined with a spin orbit component of the thomas form

upper and lower limits on the number of bound states in a central potential in a recent paper new upper and lower limits were given in the context of the schrodinger or klein gordon equations for the number n0 of s wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity in this paper these results are extended to the number nl of bound states for the lth partial wave and results are also obtained for potentials that are not monotonic and even somewhere positive new results are also obtained for the case treated previously including the remarkably neat lower limit nl 2l 1 1 2 with 2 max0 r r v r 1 2 valid in the schrodinger case for a class of potentials that includes the monotonically nondecreasing ones entailing the following lower limit for the total number n of bound states possessed by a monotonically nondecreasing central potential vanishing at infinity n 1 2 3 2 2 here the double braces denote the integer part

critical strength of attractive central potentials we obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive purely negative central potentials these conditions yield several sequences of upper and lower limits on the critical value g of the coupling constant strength g of the potential v r gv r for which a first wave bound state appears which converges to the exact critical value

a general mechanism of diffusion in hamiltonian systems qualitative results we present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable hamiltonian systems our approach relies on the scattering map outer dynamics and on the recurrence property of the inner dynamics restricted to a normally hyperbolic invariant manifold we apply topological methods to find trajectories that follow these two dynamics this method differs in several crucial aspects from earlier works there are virtually no assumptions on the inner dynamics as the method does not use at all the invariant objects for the inner dynamics e g primary and secondary tori lower dimensional hyperbolic tori and their stable unstable manifolds aubry mather sets the method applies when the unperturbed hamiltonian is not necessarily convex and of arbitrary degrees of freedom in addition this mechanism is easy to verify analytically or numerically in concrete examples as well as to establish diffusion in generic systems we include several applications such as bridging large gaps in a priori unstable models in any dimension and establishing diffusion in cases when the inner dynamics in a non twist map

upper and lower limits for the number of s wave bound states in an attractive potential new upper and lower limits are given for the number of s wave bound states yielded by an attractive monotonic potential in the context of the schrodinger or klein gordon equation

another look at the saddle centre bifurcation vanishing twist abstract in the saddle centre bifurcation a pair of periodic orbits is created out of nothing in a hamiltonian system with two degrees of freedom it is the generic bifurcation with multiplier one we show that out of nothing should be replaced by out of a twistless torus more precisely we show that invariant tori of the normal form have vanishing twist right before the appearance of the new orbits vanishing twist means that the derivative of the rotation number with respect to the action for constant energy vanishes we explicitly derive the position of the twistless torus in phase and in parameter space near the saddle centre bifurcation the theory is applied to the area preserving henon map

recurrence of kolmogorov arnold moser tori in nonanalytic twist maps we study a class of twist maps where the functiong 1 2 z 1 is nonanalytic c1 and endowed with a varying degree of inflectionz whenz 3 reappearance of a kam torus after its breakup has been observed we introduce an inverse residue criterion to determine the reappearance point scaling behavior at the transition points is also studied for 2 z 3 the scaling exponents are found to vary withz whereas forz 3 they are independent ofz in this sensez 3 plays a role quite similar to that of the upper critical dimension in phase transitions

generalized comparison theorems in quantum mechanics this paper is concerned with the discrete spectra of schrodinger operators h v where v r is an attractive potential in n spatial dimensions two principal results are reported for the bottom of the spectrum of h in each angular momentum subspace l i an optimized lower bound when the potential is a sum of terms v r v 1 r v 2 r and the bottoms of the spectra of v 1 r and v 2 r in l are known and ii a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials v 1 r and v 2 r intersect in a controlled way pure power law potentials are studied and an application of the results to the coulomb plus linear potential v r a r br is presented in detail for this problem an earlier formula for energy bounds is sharpened and generalized to n dimensions

on the number of bound states for the one dimensional schrodinger equation the number of bound states of the one dimensional schrodinger equation is analyzed in terms of the number of bound states corresponding to fragments of the potential when the potential is integrable and has a finite first moment the sharp inequalities 1 p j 1p nj n j 1p nj are proved where p is the number of fragments n is the total number of bound states and nj is the number of bound states for the jth fragment when p 2 the question of whether n n1 n2 or n n1 n2 1 is investigated in detail an illustrative example is also provided

sufficient conditions for the existence of bound states in a central potential we show how a large class of sufficient conditions for the existence of bound states in non positive central potentials can be constructed these sufficient conditions yield upper limits on the critical value g l c of the coupling constant strength g and of the potential v r gv r for which a first l wave bound state appears these upper limits are significantly more stringent than hitherto known results

analytic continuation of taylor series and the boundary value problems of some nonlinear ordinary differential equations abstract we compare and discuss the respective efficiency of three methods with two variants for each of them based respectively on taylor maclaurin series pade approximants and conformal mappings for solving quasi analytically a two point boundary value problem of a nonlinear ordinary differential equation ode six configurations of ode and boundary conditions are successively considered according to the increasing difficulties that they present after having indicated that the taylor series method almost always requires the recourse to analytical continuation procedures to be efficient we use the complementarity of the two remaining methods pade and conformal mapping to illustrate their respective advantages and limitations we emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods particularly for the so called pade hankel method we show that this latter method is equivalent to pushing a movable pole to infinity for each configuration we determine the singularity distribution in the complex plane of the independent variable of the solution sought and show how this distribution controls the efficiency of the two methods in general the method based on pade approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required

upper bounds on the number of eigenvalues of stationary schrodinger operators in this article we provide several upper bounds on the number of eigenvalues respectively for schrodinger operators of one dimensional case central potential case and the case with point interactions compared with bargman s and calogero s results the new bounds can give finite estimates when the potential function has a heavy tail besides a generalization of calogero s bound is given which can be applied to bell shaped potential functions rather than merely monotonous ones with respect to schrodinger operators containing dirac functions some bounds are also established

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

bound states in one and two spatial dimensions in this article we study the number of bound states for potentials in one and two spatial dimensions we first show that in addition to the well known fact that an arbitrarily weak attractive potential has a bound state it is easy to construct examples where weak potentials have an infinite number of bound states these examples have potentials which decrease at infinity faster than expected using somewhat stronger conditions we derive explicit bounds on the number of bound states in one dimension using known results for the three dimensional zero angular momentum a change of variables which allows us to go from the one dimensional case to that of two dimensions results in a bound for the zero angular momentum case finally we obtain a bound on the total number of bound states in two dimensions first for the radial case and then under stronger conditions for the noncentral case

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

renormalization for breakup of invariant tori we present renormalization group operators for the breakup of invariant tori with winding numbers that are quadratic irrationals we find the simple fixed points of these operators and interpret the map pairs with critical invariant tori as critical fixed points coordinate transformations on the space of maps relate these fixed points and also induce conjugacies between the corresponding operators

on a sufficient condition of the existence of n particle bound states the authors use a birman schwinger type analysis to derive a sufficient condition for the existence of bound states in a certain class of n body systems this condition is given in terms of the underlying subsystem with the threshold energy sigma inf sigma ess h

generalization of the calogero cohn bound on the number of bound states it is shown that for the calogero cohn type upper bounds on the number of bound states of a negative spherically symmetric potential v r in each angular momentum state that is bounds containing only the integral 0 v r 1 2 dr the condition v r 0 is not necessary and can be replaced by the less stringent condition d dr r1 2p v 1 p 0 1 2 p 1 which allows oscillations in the potential the constants in the bounds are accordingly modified depend on p and l and tend to the standard value for p 1 2

the fucik spectrum of the p laplacian and jumping nonlinear problems abstract we prove the existence of a nontrivial curve in the fucik spectrum of the p laplacian using the sequence of minimax eigenvalues constructed by cohomological index furthermore we study the solvability of jumping nonlinear problems p u u p 1 u p 1 f x u 0 x

bertlmann martin inequality for a weakly bound particle the bertlmann martin bm inequality for the rms radius is checked against the exact value in the case of a particle in a weak central potential having a single bound state a number of usual potentials are considered in one and three dimensions the bm inequality is poorly saturated but the correction factor needed to recover the exact rms radius is weakly depending on the shape of the potential except for the hulth n potential for potentials having a hard core component we found that which depends on the bound state energy tends to a universal curve as the hard core radius increases

labyrinthic standard non twist map we introduce the labyrinthic standard non twist map which is a modified non twist standard map with an additional perturbation which is a period multiple of the original perturbation period in phase space this map can present several sets of invariant meandering curves due to the reconnection process of isochrones resonances inside each meander set we can find a non twist invariant curve called the shearless curve with an extreme rotation number which generates a wide stickiness effect in phase space when it is broken the birth break of these shearless curves as well as their effects on the dynamical transport are studied showing the formation of many transport barriers in the system

some remarks on the fucik spectrum of the p laplacian and critical groups let be a bounded domain in n n 1 for 1 p the fucik spectrum of the p laplacian on w 1 p 0 is defined as the set p of those points a b 2 such that pu a u p 1 b u p 1 u w 1 p 0 1 1 has a nontrivial solution here pu div u p 2 u and u max u 0 it is known that the first eigenvalue 1 of p on w 1 p 0 is positive simple and admits a positive eigenfunction 1 w 1 p 0 c1 see lindqvist 8 so p contains the two lines 1 and 1 a first nontrivial curve c2 in p through 2 2 where 2 is the second eigenvalue of p was recently constructed and variationally characterized by a

semirelativistic lagrange mesh calculations the lagrange mesh method is a very powerful procedure to compute eigenvalues and eigenfunctions of nonrelativistic hamiltonians the trial eigenstates are developed in a basis of well chosen functions and the computation of hamiltonian matrix elements requires only the evaluation of the potential at grid points it is shown that this method can be used to solve semirelativistic two body eigenvalue equations as in the nonrelativistic case it is very accurate fast and very simple to implement