n body quantum scattering theory in two hilbert spaces vii real energy limits a study is made of the real energy limits of approximate solutions of the chandler gibson equations as well as the real energy limits of the approximate equations themselves it is proved that 1 the approximate time independent transition operator t z and an auxiliary operator m z when restricted to finite energy intervals are trace class operators and have limits in trace norm for almost all values of the real energy 2 the basic dynamical equation that determines the operator m z when restricted to the space of trace class operators has a real energy limit in trace norm for almost all values of the real energy 3 the real energy limit of m z is a solution of the real energy limit equation 4 the diagonal on shell elements of the kernels of the real energy limit of t z and of all solutions of the real energy limit equation exactly equal the on shell transition operator implying that the real energy limit equation uniquely determines the physical transition amplitude and 5 a s

radiation conditions and scattering theory for n particle hamiltonians the correct form of the angular part of radiation conditions is found in scattering problem forn particle quantum systems the estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations

asymptotic completeness for n body stark hamiltonians we prove asymptotic completeness for short and long rangen body stark hamiltonians with local singularities of at most coulomb type our results include the usual models for atoms and molecules

nonautonomous impulsive systems with unbounded nonlinear terms in fenner and pinto 1999 23 proved that if the linear impulsive system satisfies the is condition see definition 2 1 with bounded nonlinear term f t x then the perturbed nonlinear impulsive system has a unique bounded solution see theorem a the method used to prove theorem a cannot be applied to the unbounded case in this paper we prove that if f t x e t m then the perturbed nonlinear impulsive system has a unique solution

structures of s matrices for three body schrodinger operators structures of the s matrix associated with the collision process from 2 clusters to 3 clusters are studied this s matrix is shown to have a continuous kernel except for 2 dimensional spheres on which 2 body subsystems have zero velocity on these spheres the s matrix has in general singularities whose existence arises from the zero eigenvalues and the zero resonances of the 2 body subsystems

the quantum n body problem this selective review is written as an introduction to the mathematical theory of the schrodinger equation for n particles characteristic for these systems are the cluster properties of the potential in configuration space which are expressed in a simple geometric language the methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results the central theme is the interplay between the spectral theory of n body hamiltonians and the space time and phase space analysis of bound states and scattering states

spline galerkin solution of integral equations for three body scattering above break up abstract we investigate a galerkin method for solving the integral equations for three body scattering at energies above the break up threshold the scattering equations that we consider are one dimensional integral equations that arise from a separable potential model cubic spline approximants with multiple knots are used to construct a non smooth solution function numerical results are obtained both for a system of spin 0 boson and spin 1 2 fermion particles interacting via separable two body potentials the results demonstrate that our numerical treatment of this problem is both robust and accurate with a small number of basis functions

spectral and scattering theory of spatially cut off p ϕ 2 hamiltonians we study spatially cut off p 2 hamiltonians we show the local finiteness of the pure point spectrum outside of thresholds the limiting absorption principle and asymptotic completeness of scattering for such hamiltonians our results imply the absence of singular continuous spectrum

the solution of faddeev integral equations for three body scattering by means of b splines abstract a method for the solution of fredholm integral equations of the second kind with sigularities both in the kernel and in the solution is developed based on the approximation of the solution by b splines the main problem of this method the distribution of knots is extensively investigated the faddeev equations describing the physical problem of pion deuteron elastic scattering are solved as an application of this method

dispersive charge transfer model with long range quantum interactions abstract we present a method of proving the asymptotic completeness of wave operators associated with the scattering of a quantum particle in a field of classical particles with long range interactions based on some propagation estimates of the type recently developed in the theory of n body systems the approach allows us to treat a class of dispersive systems where the free hamiltonian has the form h 0 p d covering for example the case of the relativistic hamiltonian p c 2 2 1 2

n body quantum scattering theory in two hilbert spaces iv approximate equations a rigorous mathematical theory of approximations is developed for the time independent transition operators of n body multichannel nonrelativistic quantum scattering theory new basic dynamical equations are derived and shown to specify uniquely the approximate time independent transition operators these operator equations represent coupled integral equations with compact kernels but it is not assumed that the equations that determine the exact transition amplitudes have compact kernels convergence of sequences of these approximate time independent transition operators to the exact transition operator is established in appropriate limits stability of the basic dynamical equations is proved resolvent type equations and their relation to the limiting absorption principle are investigated the relation of this theory to the petryshyn theory of a proper operators and to the feshbach unified theory of nuclear reactions is discussed

numerical solution of hybrid systems of differential algebraic equations abstract we present a mathematical framework for general over and under determined hybrid switched systems of differential algebraic equations hdaes we give a systematic formulation of hdaes and discuss existence and uniqueness of solutions the treatment of the switch points and how to perform consistent initialization at switch points we show how numerical solution methods for daes can be adapted for hdaes and present a numerical results for these methods for the real world example of simulating an automatic gearbox

asymptotic completeness for long range many particle systems with stark effect ii we prove the existence and the asymptotic completeness of the dollard type modified wave operators for many particle stark hamiltonians with long range potentials

scattering matrices in many body scattering in this paper we show that generalized eigenfunctions of many body hamiltonians h with short range two body interactions have distributional asymptotics at non threshold channels the leading terms of the asymptotics can be used to define a scattering matrix and we show that this is the same up to normalization as that arising from the standard wave operator approach we also prove the existence of local distributional asymptotics for locally approximate generalized eigenfunctions in the more general setting of short range perturbations of a scattering metric defined by melrose in 13

cluster properties in relativistic quantum mechanics of n particle systems a general technique is presented for constructing a quantum theory of a finite number of interacting particles satisfying poincare invariance cluster separability and the spectral condition irreducible representations and clebsch gordan coefficients of the poincare group are the central elements of the construction a different realization of the dynamics is obtained for each basis of an irreducible representation of the poincare group unitary operators that relate the different realizations of the dynamis are constructed this technique is distinguished from other solutions s n sokolov dokl akad nauk ussr 233 575 1977 f coester and w n polyzou phys rev d 26 1348 1982 of this problem because it does not depend on the kinematic subgroups of dirac s forms p a m dirac rev mod phys 21 392 1949 of dynamics special basis choices lead to kinematic subgroups

a criterion for asymptotic completeness in local relativistic qft we formulate a generalized concept of asymptotic completeness and show that it holds in any haag kastler quantum field theory with an upper and lower mass gap it remains valid in the presence of pairs of oppositely charged particles in the vacuum sector which invalidate the conventional property of asymptotic completeness our result can be restated as a criterion characterizing a class of theories with complete particle interpretation in the conventional sense this criterion is formulated in terms of certain asymptotic observables araki haag detectors whose existence as strong limits of their approximating sequences is our main technical result it is proven with the help of a novel propagation estimate which is also relevant to scattering theory of quantum mechanical dispersive systems

a sharp version of ehrenfest s theorem for general self adjoint operators we prove the ehrenfest theorem of quantum mechanics under sharp assumptions on the operators involved

the combined effects of optimal control in cancer remission we investigate a mathematical model depicting the nonlinear dynamics of immunogenic tumors as envisioned by kuznetsov et al 1 to understand the dynamics under what circumstances the cancer cells can be eliminated we implement the theory of optimal control we design two types of external treatment strategies one is adoptive cellular immunotherapy and another is interleukin 2 our aim is to establish the treatment regimens that maximize the effector cell count and minimize the tumor cell burden and the deleterious effects of the total amount of drugs we derive the existence of an optimal control by using the boundedness of solutions we characterize the optimality system in which the state system is coupled with co states the uniqueness of an optimal control of our problem is also analyzed finally we demonstrate the numerical illustrations that the optimal regimens reduce the tumor burden under different scenarios

slowly changing potential problems in quantum mechanics adiabatic theorems ergodic theorems and scattering we employ the recently developed multi time scale averaging method to study the large time behavior of slowly changing in time hamiltonians we treat some known cases in a new way such as the zener problem and we give another proof of the adiabatic theorem in the gapless case we prove a new uniform ergodic theorem for slowly changing unitary operators this theorem is then used to derive the adiabatic theorem do the scattering theory for such hamiltonians and prove some classical propagation estimates and asymptotic completeness

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

propagation properties of quantum scattering states the change of localization of a scattering state under the free and interacting time evolutions for quantum mechanical potential scattering is controlled if a particle of mass m is localized in a ball of radius r and has energy below mv22 then at time t the component of the state in the forbidden region x r vt decays integrably in time various sharper estimates for the free time evolution are collected in section ii the momentum support of a state in terms of its full energy support in regions where the potential is weak is controlled the potential may be fairly general of short or long range

phase space analysis on some black hole manifolds abstract the schwarzschild and reissner nordstrom solutions to einstein s equations describe space times which contain spherically symmetric black holes we consider solutions to the linear wave equation in the exterior of a fixed black hole space time of this type we show that for solutions with initial data which decay at infinity and at the bifurcation sphere a weighted l 6 norm in space decays like t 1 3 this weight vanishes at the event horizon but not at infinity to obtain this control we require only an loss of angular derivatives

detecting and controlling unstable periodic orbits that are not part of a chaotic attractor a method for controlling unstable periodic orbits upos that have not been controllable before is presented the method is based on detecting upos that are situated outside the skeleton of a chaotic attractor the main idea is to exploit flexible parts of the attractor which under weak external perturbations allow variable excursions of the trajectory away from its originally determined path after the perturbation the trajectory of the autonomous system seeks its path back to the chaotic attractor and reveals additional upos that are otherwise not used by the system it is shown that these upos can be controlled as easily as the upos that form the basic chaotic attractor the effectiveness of the proposed method is demonstrated on two different chaotic systems with very distinct response abilities to external perturbations additionally some applications of the method in the fields of laser technology information encoding and biomedical engineering are discussed

asymptotic properties of solutions to 3 particle schrodinger equations we construct a generalized fourier transformation f associated with the 3 body schrodinger operator h a v a x a and characterize all solutions of h u 0 in the agmon hormander space b as the image of f these stationary solutions admit asymptotic expansions in b in terms of spherical waves associated with scattering channels

a new k matrix approach to n body scattering the chandler gibson theory of n body scattering is used to define a new k matrix for n body quantum scattering systems the half on shell k matrix has the proper channel thresholds and may be computed using a k matrix form of the cg equations the on shell k matrix is an hermitian matrix for all energies and is related to a unitary scattering matrix via a cayley transform a b spline solution method is developed and applied to a two body and a three body test problem the three body numerical calculations are within 0 5 of the exact solution both below and above the breakup threshold

on inverse scattering for then body schrodinger equation abstract we consider the generalized n body schrodinger operator h q x q x a a v a x a x r d x a x a r d x a x a with short range regular interactions we obtain in particular the following results 1 we give new formulas and proofs relating high energy asymptotics in a weak sense of the scattering operators with the same cluster decomposition a for and and the x ray transform pi defined on the set of all lines in x a with nonsingular directions of the effective potentials i x a x a x a these results significantly clarify some of those given in the literature 2 we describe completely ker p and give a method for reconstruction of i mod ker p from pi 3 we prove pointwise high energy asymptotics for the two cluster two cluster scattering amplitudes f with the fourier transform i in the leading term for the case when the cluster decomposition for and is the same 4 we give several additional results for the case of perturbed stratified medium when q x v a x a v b x b x a x 1 x b x 1 x d x and each v c x c rapidly decreases as x c c a b our results including proofs in some cases significantly simplify methods of high energy inverse scattering for the n body schrodinger operator given earlier by wang and by enss and weder

a new proof of the propagation theorem for n body quantum systems a new proof of i sigal s and a soffer s propagation theorem is given this theorem describes a large class of operators which are kato smooth with respect to ann body schrodinger operator

n body quantum scattering theory in two hilbert spaces v computation strategy in this paper the development of our previously published theory of approximations for the chandler gibson cg equations is continued in particular our approximation theory is rigorously brought to the point where n particle scattering calculations can begin this is accomplished by mapping the cg operator equations into a function equation form where the unknowns belong to a new third computational hilbert space l this mapping is facilitated by rescaling the jacobi momentum variables for the relative free motion of the asymptotic clusters so that surfaces of constant kinetic energy are hyperspheres the input terms to the resulting equations are expanded in a basis on the surface of the kinetic energy hypersphere this leads to a system of infinitely many coupled one dimensional integral equations with the kinetic energy as the continuous variable the half on shell variant of these equations is then transformed to a k matrix form our approximations result from truncating this system to a finite

scattering theory for riemannian laplacians we introduce a notion of scattering theory for the laplace beltrami operator on non compact connected and complete riemannian manifolds a principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity another condition is certain bounds of derivatives up to order one of the trace of this quantity these conditions are shown to be optimal for existence and completeness of a wave operator our theory does not involve prescribed asymptotic behavior of the metric at infinity like asymptotic euclidean or hyperbolic metrics studied previously in the literature a consequence of the theory is spectral theory for the laplace beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum

asymptotic completeness for n body short range quantum systems a new proof we give an alternative geometrical proof of asymptotic completeness for an arbitrary number of quantum particles interacting through shortrange pair potentials it relies on an estimate showing that the intercluster motion concentrates asymptotically on classical trajectories

propagation estimates for n body schroedinger operators we prove propagation estimates of strong type for long rangen body hamiltonians emphasis is on phase space analysis in the free channel region

the mourre theory for analytically fibered operators abstract we develop in this paper the mourre theory for an abstract class of fibered self adjoint operators which we call analytically fibered operators we construct a conjugate operator for which we prove that a mourre estimate holds examples of analytically fibered operators are given and finally perturbations of such operators are considered

the n particle scattering problem asymptotic completeness for short range systems on demontre la completude asymptotique pour des systemes de mecanique quantique a courte portee constitues d un nombre arbitraire de particules

the wave equation on asymptotically de sitter like spaces in this paper we obtain the asymptotic behavior of solutions of the klein gordon equation on lorentzian manifolds x g which are de sitter like at infinity such manifolds are lorentzian analogues of the so called riemannian conformally compact or asymptotically hyperbolic spaces under global assumptions on the null bicharacteristic flow namely that the boundary of the compactification x is a union of two disjoint manifolds y and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to and to the other manifold as the parameter goes to we also define the scattering operator and show that it is a fourier integral operator associated to the bicharacteristic flow from y to y

optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems based on the lyapunov stability and adaptive synchronization theory optimization design of adaptive controllers and parameter observers with controllable gain coefficient are investigated in detail the linear errors of corresponding variables and parameters are used to construct different appropriate positive lyapunov functions v and the parameter observers and adaptive controllers are approached analytically by simplifying the differential inequality dv dt 0 particularly an optional gain coefficient is selected in the parameter observers and positive lyapunov function which decides the transient period to identify the unknown parameters and reach synchronization the scheme is used to study the synchronization of two non identical hyperchaotic rossler systems the theoretical and numerical results confirm that the four unknown parameters in the drive system are estimated exactly and the two hyperchaotic systems reach complete synchronization when the controllers and parameter observers work on the driven system to confirm the model independence of this scheme an alternative hyperchaotic system is investigated whereby the results confirm that the five unknown parameters are identified rapidly and exactly and that the two hyperchaotic systems reach complete synchronization as well

propagation of singularities in many body scattering in the presence of bound states abstract in this paper we describe the propagation of singularities of tempered distributional solutions u s of h u 0 where h is a many body hamiltonian h v 0 v a v a and is not a threshold of h under the assumption that the inter particle e g two body interactions v a are real valued polyhomogeneous symbols of order 1 e g coulomb type with the singularity at the origin removed here the term singularity refers to a microlocal description of the lack of decay at infinity thus we prove that the set of singularities of u is a union of maximally extended broken bicharacteristics of h these are curves in the characteristic variety of h which can be quite complicated due to the existence of bound states we use this result to describe the wave front relation of the s matrices we also analyze lagrangian properties of this relation which shows that the relation is not too large in terms of its dimension

the critical value of a nonlinear bvp and the symmetrization of the data conjectures and computer examples on considere le probleme aux valeurs propres non lineaires lu x u x x f u x pour x 1 1 u 1 0 ou est un poids positif f une fonction de classe c 2 positive croissante et convexe r

parametrices for a class of schrodinger equations the article describes the construction of a parametrix global in space in the initial value problem for the schrodinger equation i u t u q x u v x u with q a real quadratic form and v a real potential in the hypothesis on v is that v x const x and v x const x 1 as x by conjugating the equation with the unitary group exp ith h q x the problem is reduced to a hyperbolic pseudodifferential operator to which a global version of the geometric optics method applies the symbolic calculus used by the construction is based on weights that reflect the role of the propagation set

n body quantum scattering theory in two hilbert spaces iii theory of approximations abstract a rigorous mathematical theory of approximations is developed for abstract nonrelativistic quantum scattering systems within the two hilbert space framework an approximate space of asymptotic states and an approximate asymptotic hamiltonian must be specified initially an approximate n particle hamiltonian is then constructed and proved to be self adjoint approximate wave operators are shown to exist and in certain interesting cases to be asymptotically complete certain sequences of the approximate wave operators are proved to converge to the exact wave operators in an appropriate limit thus approximate scattering operators are unitary and converge to the exact scattering operator

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

scattering for schrodinger operators in a class of domains with non compact boundaries abstract schrodinger operators in a class of domains with asymptotic cones are considered a generalized fourier transform representing the absolutely continuous part of the schrodinger operator as multiplication by 2 in the asymptotic cone is constructed wave operators relating the free laplacian to schrodinger operators are computed using the generalized fourier transform the wave operators relating schrodinger operators acting in domains with the same asymptotic cone are computed and shown to be complete

continuous spectrum of automorphism groups and the infraparticle problem this paper presents a general framework for a refined spectral analysis of a group of isometries acting on a banach space which extends the spectral theory of arveson the concept of a continuous arveson spectrum is introduced and the corresponding spectral subspace is defined the absolutely continuous and singular continuous parts of this spectrum are specified conditions are given in terms of the transposed action of the group of isometries which guarantee that the pure point and continuous subspaces span the entire banach space in the case of a unitarily implemented group of automorphisms acting on a c algebra relations between the continuous spectrum of the automorphisms and the spectrum of the implementing group of unitaries are found the group of spacetime translation automorphisms in quantum field theory is analyzed in detail in particular it is shown that the structure of its continuous spectrum is relevant to the problem of existence of infra particles in a given theory

asymptotic completeness for the massless spin boson model abstract we consider generalized versions of the massless spin boson model building on the recent work in 3 and 4 we prove asymptotic completeness

n body quantum scattering theory in two hilbert spaces vi compactness conditions it is shown how to implement in a practical way the approximation theory previously developed j funct anal 52 80 1983 for nonrelativistic n body quantum systems of particles interacting via pair potentials belonging to a certain general class this is done by constructing the projection operators which generate the approximations and by proving that certain operators j j i are hilbert schmidt and that certain other operators v e are trace class for all finite real intervals two types of projections are considered the results for the first type generalize previous results of combes and simon for asymptotic channels with only two clusters the results for the second type provide an alternative approach to n body scattering and spectral problems which is both practical and theoretically correct the compactness results are used to prove that the approximate theories are exact theories for approximate hamiltonians that the approximate wave operators are asymptotically complete and sat

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

asymptotic completeness of long range n body quantum systems where x is a euclidean space and lra a e a is a finite family of projections onto certain subspaces xa of x we will always assume that the family xa a e a is closed with respect to the algebraic sum and contains xamin 0 the class of operators of the form 0 2 is a generalization of 0 1 up to a linear change of coordinates in the configuration space x they usually go under the name of generalized n body schrbdinger operators and were first considered in a in what follows we will drop the word generalized we refer the reader to rsi vol iii for a general introduction to n body schr6dinger operators defined as in equation 0 1 we will consistently use the definition 0 2 and assume that the reader is familiar with the physical meaning of various objects under study

asymptotic completeness for n body quantum systems with long range interactions in a time periodic electric field we show the asymptotic completeness for n body quantum systems with long range interactions in a time periodic electric field whose mean in time is non zero where n 2 one of the main ingredients of this paper is to give some propagation estimates for physical propagators generated by time periodic hamiltonians which govern the systems under consideration

asymptotic completeness for 3 particle long range systems we prove asymptotic completeness for 3 particle systems with pair potentials satisfying the virial condition and decreasing at infinity as for 1 2

an inequality for the critical value of nonlinear eigenvalue problems soit un domaine borne de r n de classe c 2 pour 0 1 soit l un operateur differentiel d ordre 2 fortement uniformement elliptique on considere le probleme aux valeurs propres non lineaire lu pf u dans u 0 ou est un poids positif et f c 2 positive croissante et convexe on demontre une inegalite sur la valeur propre critique

long range scattering for higher order schrodinger operators abstract this paper is concerned with the scattering problem for the higher order schrodinger operator m v x where v x is a long range potential at this moment the classical wave operators do not exist so we provide one kind of modified wave operators and show the existence and the asymptotic completeness of such wave operators as well as the expression of the corresponding scattering operator

synchronization transition in degenerate optical parametric oscillators induced by nonlinear coupling in this paper the synchronization of certain degenerate optical parametric oscillators is investigated in detail complete and or partial synchronization can be reached when linear controller constructed by the real part or imaginary part of the subharmonic mode is imposed on the chaotic degenerate optical parametric oscillators with appropriate coupling intensity the lyapunov exponents under different coupling coefficients are calculated to find the critical condition for complete synchronization transition from complete synchronization to partial synchronization is observed when nonlinear coupling is introduced into the two chaotic oscillators it is found that synchronization of chaotic oscillators keeps robust when the intensity of the nonlinear coupling is less than the intensity of the linear coupling the complete synchronization state is destructed and transient period for partial synchronization is in certain delay when the intensity of the nonlinear coupling is beyond the intensity of the linear coupling

scattering theory for a class of fermionic pauli fierz models abstract the scattering theory for a class of fermionic pauli fierz models is considered we give a proof of the asymptotic completeness of the dynamics in the case of massive fermions the result applied to the hamiltonian of a quantized spin 1 2 dirac particle interacting with an external field through a cutoff yukawa interaction and to the hamiltonian of a system of finitely many confined particles coupled to a fermionic field with a quadratic interaction

the scattering amplitude for the schrodinger equation with a long range potential we consider the schrodinger operator with a long range potential v x in the space our goal is to study spectral properties of the corresponding scattering matrix and a diagonal singularity of its kernel the scattering amplitude it turns out that in contrast to the short range case the dirac function singularity of at the diagonal disappears and the spectrum of the scattering matrix covers the whole unit circle for an asymptotically homogeneous function v x of order we show that typically where the module w and the phase are asymptotically homogeneous functions as of orders and respectively leading terms of asymptotics of w and at are calculated in the case 1 our results generalize in the limit the well known formula of gordon and mott