the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

generalized point interactions in one dimensional quantum mechanics there is a four parameter family of point interactions in one dimensional quantum mechanics they represent all possible self adjoint extensions of the kinetic energy operator if time reversal invariance is imposed the number of parameters is reduced to three one of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known we present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so called interaction we emphasize that there is little resemblance between the interaction and what its name suggests

potential approximations to δ an inverse klauder phenomenon with norm resolvent convergence we show that there is a family schrodinger operators with scaled potentials which approximates the interaction hamiltonian in the norm resolvent sense this approximation based on a formal scheme proposed by cheon and shigehara has nontrivial convergence properties which are in several respects opposite to those of the klauder phenomenon

spectral asymptotics for schrodinger operators with periodic point interactions abstract spectrum of the second order differential operator with periodic point interactions in l 2 r is investigated classes of unitary equivalent operators of this type are described spectral asymptotics for the whole family of periodic operators are calculated it is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely it is proven that the spectrum of the operators with anomalous spectral asymptotics when the ratio between the lengths of the bands and gaps tends to zero at infinity can be approximated by standard periodic weighted operators with step wise density functions it is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal generalized resolvent of the periodic schrodinger operator with certain energy dependent boundary conditions the operator acting in an extended hilbert space such that its resolvent restricted to l 2 r coincides with the formal resolvent is constructed explicitly

a new class of solvable models in quantum mechanics describing point interactions on the line the authors provide a detailed analysis of properties of the schrodinger operator in l2 r which formally can be written as d2 dx2 sigma y in gamma nu y delta y where delta is the derivative of dirac s delta function and y contained in implied by r is discrete this model allows for an explicit calculation of spectral properties special emphasis is given to the periodic case y z nu nu j j in z where the spectrum and the density of states are explicitly computed also the spectrum for a half crystal is given the authors study in detail spectral consequences when various defects and impurities are added to the periodic case

distribution theory for discontinuous test functions and differential operators with generalized coefficients investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions such a distribution theory for test functions having discontinuity at one point is developed a four parameter family of schrodinger operators formed by the operators with singular potential singular metrics and singular gauge field is considered it is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point approximation by operators with smooth coefficients is discussed c 1996 academic press inc

point transfer matrices for the schrodinger equation the algebraic theory this paper deals with the theory of point interactions for the one dimensional schrodinger equation the familiar example of the potential v x g x x 0 for which the transfer matrix across the singularity point transfer matrix is given by is extended to cover cases in which the transfer matrix m z is dependent on the complex spectral parameter z and which can be obtained as limits of transfer matrices across finite intervals for sequences of approximating potentials v n the case of point transfer matrices polynomially dependent on z is treated in detail with a complete characterization of such matrices and a proof of their factorization as products of point transfer matrices linearly dependent on z the theory presented here has applications to the study of point interactions in quantum mechanics and provides new classes of point interactions which can be obtained as limiting cases of regular potentials

exclusion regions for systems of equations branch and bound methods for finding all zeros of a nonlinear system of equations in a box frequently have the difficulty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box this has the effect that near each zero many small boxes are created by repeated splitting whose processing may dominate the total work spent on the global search this paper discusses the reasons for the occurrence of this so called cluster effect and how to reduce the cluster effect by defining exclusion regions around each zero found that are guaranteed to contain no other zero and hence can safely be discarded such exclusion regions are traditionally constructed using uniqueness tests based on the krawczyk operator or the kantorovich theorem these results are reviewed moreover refinements are proved that significantly enlarge the size of the exclusion region existence and uniqueness tests are also given

weak convergence of symmetric diffusion processes in this paper we show the convergence of forms in the sense of mosco associated with the part form on relatively compact open set of dirichlet forms with locally uniform ellipticity and the locally uniform boundedness of ground states under regular dirichlet space setting we also get the same assertion under dirichlet space in infinite dimensional setting as a result of this we get the weak convergence under some conditions on initial distributions and the growth order of the volume of the balls defined by modified pseudo metric used in k th sturm

mosco convergence of dirichlet forms in infinite dimensions with changing reference measures let e be an infinite dimensional locally convex space let n be a weakly convergent sequence of probability measures on e and let en be a sequence of dirichlet forms on e such that en is defined on l2 n general sufficient conditions for mosco convergence of the gradient dirichlet forms are obtained applications to gibbs states on a lattice and to the gaussian case are given weak convergence of the associated processes is discussed

point interactions in one dimensional quantum mechanics with coupled channels we explore generalized point interactions in one dimensional quantum mechanics with two coupled channels they represent possible self adjoint extensions of the nonrelativistic kinetic energy operator assuming time reversal invariance we find a family of self adjoint extensions with seven parameters this can be compared with the one channel case in which the corresponding number of parameters is three

subdivision direction selection in interval methods for global optimization the role of the interval subdivision selection rule is investigated in branch and bound algorithms for global optimization the class of rules that allows convergence for the model algorithm is characterized and it is shown that the four rules investigated satisfy the conditions of convergence a numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required cpu time the number of function and derivative evaluations and space complexity and two rules can provide substantial improvements in efficiency

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

strong solutions of stochastic equations with singular time dependent drift we prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time we only assume local l q l p integrability of b in g with d p 2 q 1 we also prove strong feller properties in this case if b is the gradient in x of a nonnegative function blowing up as g x g we prove that the conditions 2d t k 2d t ke 0 2 imply that the explosion time is infinite and the distributions of the solution have sub gaussian tails

many body system with a four parameter family of point interactions in one dimension we consider a four parameter family of point interactions in one dimension this family is a generalization of the usual function potential we examine a system consisting of many particles of equal masses that are interacting pairwise through such a generalized point interaction we follow mcguire who obtained exact solutions for the system when the interaction is the function potential we find exact bound states with the four parameter family for the scattering problem however we have not been so successful this is because as we point out the condition of no diffraction that is crucial in mcguire s method is not satisfied except when the four parameter family is essentially reduced to the function potential

uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity abstract we study uniqueness and stability problems of slowly oscillating periodic solutions of delay equations with small parameters if the nonlinearity decays to a negative number at and blows up at or vice versa we show that for sufficiently small parameters the slowly oscillating periodic solutions are unique and asymptotically stable provided that the decay rate can dominate the growth rate in an appropriate sense this result particularly implies that wright s equation has a unique and asymptotically stable slowly oscillating periodic solution for large parameter

computational fixed point theory for differential delay equations with multiple time lags abstract we introduce a general computational fixed point method to prove existence of periodic solutions of differential delay equations with multiple time lags the idea of such a method is to compute numerical approximations of periodic solutions using newton s method applied on a finite dimensional projection to derive a set of analytic estimates to bound the truncation error term and finally to use this explicit information to verify computationally the hypotheses of a contraction mapping theorem in a given banach space the fixed point so obtained gives us the desired periodic solution we provide two applications the first one is a proof of coexistence of three periodic solutions for a given delay equation with two time lags and the second one provides rigorous computations of several nontrivial periodic solutions for a delay equation with three time lags

rigorous computation of the global dynamics of integrodifference equations with smooth nonlinearities topological tools such as conley index theory have inspired rigorous computational methods for studying dynamics these methods rely on the construction of an outer approximation a combinatorial representation of the system that incorporates small bounded error in this work we present an automated approach to constructing outer approximations for systems in a class of integrodifference operators with smooth nonlinearities chebyshev interpolants and galerkin projections form the basis for the construction while analysis and interval arithmetic are used to incorporate explicit error bounds this represents a significant advance to the approach given by day junge and mischaikow siam j appl dyn syst 3 2004 pp 117 160 extending the nonlinearities that may be studied from low degree polynomials to smooth functions and the studied portion of phase space from a simulated attracting region to the global maximal invariant set as a demonstration of the techniques a morse decomposition of the

one dimensional schrodinger operators with δ interactions on cantor type sets we introduce a novel approach for defining a interaction on a subset of the real line of lebesgue measure zero which is based on sturm liouville differential expression with measure coefficients this enables us to establish basic spectral properties e g self adjointness lower semiboundedness and spectral asymptotics of hamiltonians with interactions concentrated on sets of complicated structures

almost globally convergent interval methods for discretizations of nonlinear elliptic partial differential equations we introduce interval methods for the numerical solution of nonlinear systems of equations for discretizations of partial differential equations similar methods have been discussed in 23 all these methods go back to an idea of krawczyk 10 and a similar method presented by alefeld platzoder 3 they converge under relatively weak conditions to the solution provided an initial inclusion is known since such an inclusion can be easily computed for problems of the type we discuss it can be said that the convergence is almost global the methods of the present paper converge without a particular numerical strategy or auxiliary methods in contrast to the methods in 21 24 also they can be applied to a larger number of problems than those in 21 24 convexity conditions are not needed for the convergence of our methods the present paper contains a comparison of the iterates produced by the original method of krawczyk and our methods

on the spectral theory of gesztesy seba realizations of 1 d dirac operators with point interactions on a discrete set abstract we investigate spectral properties of gesztesy seba realizations d x and d x of the 1 d dirac differential expression d with point interactions on a discrete set x x n n 1 r here n n 1 and n n 1 r the gesztesy seba realizations d x and d x are the relativistic counterparts of the corresponding schrodinger operators h x and h x with and interactions respectively we define the minimal operator d x as the direct sum of the minimal dirac operators on the intervals x n 1 x n then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator d x in the case d x inf x i x j i j 0 it turns out that the boundary operators b x and b x parameterizing the realizations d x and d x are jacobi matrices these matrices substantially differ from the ones appearing in spectral theory of schrodinger operators with point interactions we show that certain spectral properties of the operators d x and d x correlate with the corresponding spectral properties of the jacobi matrices b x and b x respectively using this connection we investigate spectral properties self adjointness discreteness absolutely continuous and singular spectra of gesztesy seba realizations moreover we investigate the non relativistic limit as the velocity of light c most of our results are new even in the case d x 0

using machine learning to predict catastrophes in dynamical systems nonlinear dynamical systems which include models of the earth s climate financial markets and complex ecosystems often undergo abrupt transitions that lead to radically different behavior the ability to predict such qualitative and potentially disruptive changes is an important problem with far reaching implications even with robust mathematical models predicting such critical transitions prior to their occurrence is extremely difficult in this work we propose a machine learning method to study the parameter space of a complex system where the dynamics is coarsely characterized using topological invariants we show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner we are able to predict with high accuracy the locations of critical transitions in parameter space

automatic differentiation for fourier series and the radii polynomial approach abstract in this work we develop a computer assisted technique for proving existence of periodic solutions of nonlinear differential equations with non polynomial nonlinearities we exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system we compute a numerical fourier expansion of the periodic orbit for the augmented system and prove the existence of a true solution nearby using an a posteriori validation scheme the radii polynomial approach the problems considered here are given in terms of locally analytic vector fields i e the field is analytic in a neighborhood of the periodic orbit hence the computer assisted proofs are formulated in a banach space of sequences satisfying a geometric decay condition in order to illustrate the use and utility of these ideas we implement a number of computer assisted existence proofs for periodic orbits of the planar circular restricted three body problem pcrtbp

connections between morse sets for delay differential equations on etudie la dynamique globale de l equation differentielle a retard scalaire x t f x t x t 1 avec des conditions appropriees sur f

conley index of poincare maps in isolating segments abstract we calculate the discrete time conley index of the poincare map of a time periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment as an application we present results on the existence of bounded solutions of some planar equations

a proof of wright s conjecture abstract wright s conjecture states that the origin is the global attractor for the delay differential equation y t y t 1 1 y t for all 0 2 when y t 1 this has been proven to be true for a subset of parameter values we extend the result to the full parameter range 0 2 and thus prove wright s conjecture to be true our approach relies on a careful investigation of the neighborhood of the hopf bifurcation occurring at 2 this analysis fills the gap left by complementary work on wright s conjecture which covers parameter values further away from the bifurcation point furthermore we show that the branch of slowly oscillating periodic orbits originating from this hopf bifurcation does not have any subsequent bifurcations and in particular no folds for 2 2 6 830 10 3 when combined with other results this proves that the branch of slowly oscillating solutions that originates from the hopf bifurcation at 2 is globally parametrized by 2

weak convergence of diffusion processes on wiener space let be a gaussian measure on a suslin space x h be the corresponding cameron martin space and e i h be an orthonormal basis of h suppose that n n is a sequence of probability measures which converges weakly to a probability measure consider a sequence of dirichlet forms varepsilon n where varepsilon n f int x nabla h f 2 h rho n rm d gamma and sqrt rho n in w 2 1 gamma we prove some sufficient conditions for mosco convergence varepsilon n to varepsilon where varepsilon f int x nabla h f 2 h rho rm d gamma in particular if x is a hilbert space sup n sqrt rho n w 2 1 gamma infty and partial e i rho n rho n can be uniformly approximated by finite dimensional conditional expectations rm ie mathcal f n mu n big partial e i rho n rho n big for every fixed e i then under broad assumptions varepsilon n to varepsilon mosco and the distributions of the associated stochastic processes converge weakly

comments on the schrodinger equation with delta interaction in one dimension the boundary conditions at the singular point for the schrodinger equation with delta interaction in one dimension are proposed it is also shown that the boundary conditions adopted by albeverio et al 1988 are irrelevant to the delta interaction

recent advances about the uniqueness of the slowly oscillating periodic solutions of wright s equation abstract an old conjecture in delay equations states that wright s equation y t y t 1 1 y t r has a unique slowly oscillating periodic solution sops for every parameter value 2 we reformulate this conjecture and we use a method called validated continuation to rigorously compute a global continuous branch of sops of wright s equation using this method we show that a part of this branch does not have any fold point partially answering the new reformulated conjecture

symbolic dynamics for geodesic floes the subject of symbolic dynamics is of central importance in the modern theory of dynamical systems the use of symbolic sequences to study dynamical properties of geodesies originates in the work of koebe 21 22 and morse 24 25 and is already foreshadowed in hadamard 15 and jordan 19 the method of koebe and of morse is to code a geodesic on a surface m of negative curvature by recording the order in which it traverses a given set of labelled curves on m the treatment of morse allows variable curvature but assumes at least two boundary components whereas koebe assumes constant curvature but allows infinite connectivity and nonorientability and treats the more difficult case of a closed surface this last case is handled by recording crossings of a fixed pants decomposition anticipating the thurston parameterization of simple curves as described in 11 both koebe and morse used their codings to demonstrate the existence of countably many closed geodesies and of everywhere dense transitive geodesies morse further constructed the first nonsynthetic example of a recurrent nonperiodic discontinuous motion in modern terminology a minimal nowhere dense set later 26 morse treated the special closed surfaces of genus g associated to tesselations of the disc d by regular 4 gons each edge of a region in such a tesselation may be labelled by the isometry which glues it to another side of the same region in forming the quotient sin face m the set of isometries appearing as labels generate tti m thus any geodesic is coded as a doubly infinite sequence of generators of iri m the same method applies quite generally to tesselations associated to any fuchsian group we call the sequences thus obtained cutting sequences and refer to generating sets of this kind as geometric the difficulty of course is to determine precisely the class of sequences which occur for a surface with boundary one obtains exactly reduced sequences in the generators in generalthe problem4s gompligated1 henge the difficulties encountered in 26 see also theorem 3 below there is another method of coding geodesies using certain boundary expansions for points at infinity in the universal cover of m for the modular surface h sl 2 2 the appropriate expansions are continued fractions and for the symmetrical genus g surfaces above they are the nielsen boundary expansions of 27

morse decompositions for delay differential equations abstract the attractor for the scalar delay differential equation x t f x t x t 1 is shown to possess a morse decomposition under the hypothesis of a negative feedback condition in the delay term using this decomposition we prove results on the asymptotic rate of oscillation of solutions of the initial value problem existence of both slowly and rapidly oscillating periodic solutions is also proved

rigorous numerics for nonlinear differential equations using chebyshev series a computational method based on chebyshev series to rigorously compute solutions of initial and boundary value problems of analytic nonlinear vector fields is proposed the idea is to recast solutions as fixed points of an operator defined on a banach space of rapidly decaying chebyshev coefficients and to use the so called radii polynomials to show the existence of a unique fixed point near an approximate solution as applications solutions of initial value problems in the lorenz equations and symmetric connecting orbits in the gray scott equation are rigorously computed the symmetric connecting orbits are obtained by solving a boundary value problem with one of the boundary values in the stable manifold

on energy gaps in a new type of analytically solvable model in quantum mechanics abstract a new type of analytically solvable one dimensional model in quantum mechanics formally given by d2 dx2 j z j x yj j yj r is analyzed besides calculating the density of states of various periodic arrangements of such and models we also prove a generalization of the saxon hutner conjecture concerning the existence of gaps in the energy spectrum of certain substitutional alloys of or interactions in addition the spectrum of the corresponding random hamiltonians i e j j z ergodic stochastic processes is investigated and explicitly determined in certain cases

stability and uniqueness of slowly oscillating periodic solutions to wright s equation abstract in this paper we prove that wright s equation y t y t 1 1 y t has a unique slowly oscillating periodic solution sops for all parameter values 1 9 6 0 up to time translation our proof is based on the same strategy employed earlier by xie 27 show that every sops is asymptotically stable we first introduce a branch and bound algorithm to control all sops using bounding functions at all parameter values 1 9 6 0 once the bounding functions are constructed we then control the floquet multipliers of all possible sops by solving rigorously an eigenvalue problem again using a formulation introduced by xie using these two main steps we prove that all sops of wright s equation are asymptotically stable for 1 9 6 0 and the proof follows this result is a step toward the proof of the jones conjecture formulated in 1962

infinite series in cohomology attractors and conley index in this paper we study the cohomological conley index of arbitrary isolated invariant continua for continuous maps f u subseteq mathbb r d rightarrow mathbb r d by analyzing the topological structure of their unstable manifold we provide a simple dynamical interpretation for the first cohomological conley index describing it completely and relate it to the cohomological conley index in higher degrees a number of consequences are derived including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3 our approach exploits a certain attractor repeller decomposition of the unstable manifold reducing the study of the cohomological conley index to the relation between the cohomology of an attractor and its basin of attraction this is a classical problem that in the present case is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated to address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology if z is a metric space and k subseteq z is a compact global attractor for a continuous map g z rightarrow z we show how to interpret series of the form sum j ge 0 a j g j as endomorphisms of the cohomology group of the pair z k

chaos near a resonant inclination flip abstract horseshoes play a central role in dynamical systems and are observed in many chaotic systems however most points in a neighborhood of the horseshoe escape after finitely many iterations in this work we construct a new model by re injecting the points that escape the horseshoe we show that this model can be realized within an attractor of a flow arising from a three dimensional vector field after perturbation of an inclination flip homoclinic orbit with a resonance the dynamics of this model without considering the re injection often contains a cuspidal horseshoe with positive entropy and we show that for a computational example the dynamics with re injection can have more complexity than the cuspidal horseshoe alone

indices of the iterates of r 3 homeomorphisms at lyapunov stable fixed points abstract given any positive sequence c n n n we construct orientation preserving homeomorphisms f r 3 r 3 such that fix f per f 0 0 is lyapunov stable and lim sup i f m 0 c m we will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms

the fermi pseudo potential in one dimension wu and yu recently examined point interactions in one dimension in the form of the fermi pseudo potential on the other hand there are point interactions in the form of self adjoint extensions saes of the kinetic energy operator we examine the relationship between the point interactions in these two forms in the one channel and two channel cases in the one channel case the pseudo potential leads to the standard three parameter family of saes in the two channel case the pseudo potential furnishes a ten parameter family of saes

capacity and quantum mechanical tunneling we connect the notion of capacity of sets in the theory of symmetric markov process and dirichlet forms with the notion of tunneling through the boundary of sets in quantum mechanics in particular we show that for diffusion processes the notion appropriate to a boundary without tunneling is more refined than simply capacity zero we also discuss several examples in d

on the connection between schrodinger and dirichlet forms relations between schrodinger forms associated with schrodinger operators in l2 dnx rn open n 1 and the corresponding dirichlet forms are investigated various concrete examples are presented

conley index for random dynamical systems conley index theory is a very powerful tool in the study of dynamical systems in this paper we generalize conley index theory to discrete random dynamical systems our constructions are basically the random version of franks and richeson in j franks d richeson shift equivalence and the conley index trans amer math soc 352 2000 3305 3322 for maps and the relations of isolated invariant sets between time continuous random dynamical systems and corresponding time h maps are discussed two examples are presented to illustrate results in this paper

distribution theory for schrodinger s integral equation much of the literature on point interactions in quantum mechanics has focused on the differential form of schrodinger s equation this paper in contrast investigates the integral form of schrodinger s equation while both forms are known to be equivalent for smooth potentials this is not true for distributional potentials here we assume that the potential is given by a distribution defined on the space of discontinuous test functions first by using schrodinger s integral equation we confirm a seminal result by kurasov which was originally obtained in the context of schrodinger s differential equation this hints at a possible deeper connection between both forms of the equation we also sketch a generalisation of kurasov s j math anal appl 201 1 297 323 1996 result to hypersurfaces second we derive a new closed form solution to schrodinger s integral equation with a delta prime potential this potential has attracted considerable attention including some controversy interestingly the derived propagator satisfies boundary conditions that were previously derived using schrodinger s differential equation third we derive boundary conditions for super singular potentials given by higher order derivatives of the delta potential these boundary conditions cannot be incorporated into the normal framework of self adjoint extensions we show that the boundary conditions depend on the energy of the solution and that probability is conserved this paper thereby confirms several seminal results and derives some new ones in sum it shows that schrodinger s integral equation is a viable tool for studying singular interactions in quantum mechanics

isolating blocks and epsilon chains for maps abstract the relationship between the outputs and computer simulations of dynamical systems and their true orbit structures is explored epsilon chains are used as a mathematical representation of computer generated pseudo orbits the question when is a pseudo orbit shadowed by a true orbit is discussed isolated invariant sets are defined and are shown to be contained in special neighborhoods called isolating blocks the conley index is extended and applied to maps

boundary conditions at the derivative of a delta function the author derives the joining conditions on psi at a point where the potential is the nth derivative of a dirac delta function

chaos in the lorenz equators a computer assisted proof part ii details details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described the technique combines abstract existence results based on the conley index theory with rigorous computer assisted computations as an application of these methods it is proven that for some explicit parameter values the lorenz equations exhibit chaotic dynamics

on the existence of resonances in the transmission probability for interactions arising from derivatives of dirac s delta function the scattering properties of regularizing finite range potentials constructed in the form of squeezed rectangles which approximate the first and second derivatives of the dirac delta function x are studied in the zero range limit particularly for a countable set of interaction strength values a non zero transmission through the point potential x defined as the weak limit in the standard sense of distributions of a special dipole like sequence of rectangles is shown to exist when the rectangles are squeezed to zero width a tripole sequence of rectangles which gives in the weak limit the distribution x is demonstrated to exhibit the total transmission for a countable sequence of the rectangle s width that tends to zero however this tripole sequence does not admit a well defined point interaction in the zero range limit making sense only for a finite range of the regularizing rectangular like potentials

on the global dynamics of attractors for scalar delay equations a semi conjugacy from the dynamics of the global attractors for a family of scalar delay differential equations with negative feedback onto the dynamics of a simple system of ordinary differential equations is constructed the construction and proof are done in an abstract setting and hence are valid for a variety of dynamical systems which need not arise from delay equations the proofs are based on the conley index theory institute for dynamics university of cincinnati cincinnati ohio 45221 0025 e mail address chris mccorduc edu center for dynamical systems and nonlinear studies georgia institute of tech nology atlanta georgia 30332 e mail address mischaikfmath gatech edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 05 19 39 utc all use subject to http about jstor org terms

comments on a recent note on the schrodinger equation with a delta interaction for original paper see bao heng zhao ibid vol 25 p l617 1992 it is argued that a recent letter by bao heng zhao 1992 on the one dimensional schrodinger equation with a delta interaction is seriously flawed and hence arrives at wrong conclusions

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

realization of all dold s congruences with stability the main goal of this paper is to prove that for each n 2 every sequence of integers satisfying dold s congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving r n homeomorphism at an isolated stable fixed point

a proof of jones conjecture abstract in this paper we prove that wright s equation y t y t 1 1 y t has a unique slowly oscillating periodic solution for parameter values 2 1 9 up to time translation this result proves jones conjecture formulated in 1962 that there is a unique slowly oscillating periodic orbit for all 2 furthermore there are no isolas of periodic solutions to wright s equation all periodic orbits arise from hopf bifurcations

rigorous numerics for ill posed pdes periodic orbits in the boussinesq equation in this paper we develop computer assisted techniques for the analysis of periodic orbits of ill posed partial differential equations as a case study our proposed method is applied to the boussinesq equation which has been investigated extensively because of its role in the theory of shallow water waves the idea is to use the symmetry of the solutions and a newton kantorovich type argument the radii polynomial approach to obtain rigorous proofs of existence of the periodic orbits in a weighted l1 banach space of space time fourier coefficients with exponential decay we present several computer assisted proofs of the existence of periodic orbits at different parameter values

index filtrations and the homology index braid for partially ordered morse decompositions abstract on a morse decomposition of an invariant set in a flow there are partial orderings defined by the flow these are called admissable orderings of the morse decomposition the index filtrations for a total ordering of a morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissable partial orderings of a morse decomposition it is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration the homology index braid of the corresponding admissable ordering of the morse decomposition is obtained by passing to homology in the chain complex braid keywords morse decomposition conley index index filtration

spectral theory for systems of ordinary differential equations with distributional coefficients abstract we study the spectral theory for the first order system j u q u w f of differential equations on the real interval a b when j is a constant invertible skew hermitian matrix and q and w are matrices whose entries are distributions of order zero with q hermitian and w non negative also we do not pose the definiteness condition often required for the coefficients of the equation specifically we construct minimal and maximal relations and study self adjoint restrictions of the maximal relation for these we determine green s function and prove the existence of a spectral or generalized fourier transformation we have a closer look at the special cases when the endpoints of the interval a b are regular as well as the case of a 2 2 system two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations

convergence of dirichlet forms and associated schrodinger operators abstract we study the convergence of local dirichlet forms and associated schrodinger operators conditions are given for n in order that the schrodinger operators n n associated with dirichlet forms u x 2 n 2 x dx generate contraction semigroups which converge strongly in l2 r d moreover conditions are given which permit one to identify the limit contraction semigroups and schrodinger operators

symbolic dynamics for the modular surface and beyond in this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development a geometric method stems from a 1898 work of j hadamard and was developed by m morse in the 1920s it consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated fuchsian groups another method of arithmetic nature uses continued fraction expansions of the end points of the geodesic at infinity and is even older it comes from the gauss reduction theory introduced to dynamics by e artin in a 1924 paper this method was used to exhibit dense geodesics on the modular surface for 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics geometry and combinatorial group theory we present some of the ideas results old and recent and interpretations that illustrate the multiple facets of the subject

validated continuation for equilibria of pdes one of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor corrector continuation techniques in the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output we introduce a new technique that combines the information obtained from the predictor corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation using the cahn hilliard and swift hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone

1 d schrodinger operators with local point interactions on a discrete set abstract spectral properties of 1 d schrodinger operators h x d 2 d x 2 x n x n x x n with local point interactions on a discrete set x x n n 1 are well studied when d inf n k n x n x k 0 our paper is devoted to the case d 0 we consider h x in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding weyl functions we show that the spectral properties of h x like self adjointness discreteness and lower semiboundedness correlate with the corresponding spectral properties of certain classes of jacobi matrices based on this connection we obtain necessary and sufficient conditions for the operators h x to be self adjoint lower semibounded and discrete in the case d 0 the operators with type interactions are investigated too the obtained results demonstrate that in the case d 0 as distinguished from the case d 0 the spectral properties of the operators with and type interactions are substantially different

resonant tunnelling through short range singular potentials a three parameter family of point interactions constructed from sequences of symmetric barrier well barrier and well barrier well rectangles is studied in the limit when the rectangles are squeezed to zero width but the barrier height and the well depth become infinite the zero range limit the limiting generalized potentials are referred to as the second derivative of dirac s delta function x with a renormalized coupling constant 0 or simply as like point interactions as a result a whole family of self adjoint extensions of the one dimensional schrodinger operator is shown to exist which results in full and partial resonant tunnelling through this class of singular potentials the resonant tunnelling occurs for countable sets of interaction strength values in the space which are the roots of several transcendental equations the comparison with the previous results for like point interactions is also discussed

δ interaction as a limit of a thin neumann waveguide with transversal window abstract we consider a waveguide like domain consisting of two thin straight tubular domains connected through a tiny window the perpendicular size of this waveguide is of order e under the assumption that the window is appropriately scaled we prove that the neumann laplacian on this domain converges in a kind of norm resolvent sense as e 0 to a one dimensional schrodinger operator corresponding to a interaction of a non negative strength we estimate the rate of this convergence also we prove the convergence of spectra

detection of renewal system factors via the conley index let n be an isolating neighborhood for a map f if we can decompose n into the disjoint union of compact sets n 1 and n 2 then we can relate the dynamics on the maximal invariant set inv n to the shift on two symbols by noting which component of n each iterate of a point x inv n lies in we examine a method based on work by mischaikow szymczak et al for using the discrete conley index to detect explicit subshifts of the shift associated to n in essence we measure the difference between the conley index of inv n and the sum of the indices of inv n 1 and inv n 2

a unified approach to eigenvalues and resonances of schrodinger operators using fredholm determinants we study in detail the threshold behaviour of eigenvalues and resonances of the schrodinger operator h v in l2 r3 as they are absorbed at the bottom of the continuous spectrum when 0 0 a so called coupling constant threshold using fredholm determinants we give their analytic structure in terms of 0 and compute the first order terms explicitly

one dimensional schrodinger operators with interactions singular on a discrete set on etudie le caractere auto adjoint des operateurs de schrodinger sur y u j i x j y j ou on suppose y j x j 1

the essential self adjointness of generalized schrodinger operators a necessary and sufficient condition is given for the generalized schrodinger operator a 12 i 1n di di to be essentially self adjoint in l2 dx under general assumptions on and for arbitrary domains in rn in particular if is strictly positive and locally lipschitz continuous on rn then a is essentially self adjoint examples of non essential self adjointness and a complete discussion of the one dimensional case are also given these results have applications to the problem of the essential self adjointness of quantum hamiltonians and to the uniqueness problem of markov processes