the renormalized electron mass in non relativistic quantum electrodynamics abstract this work addresses the problem of infrared mass renormalization for a non relativistic electron minimally coupled to the quantized electromagnetic field the standard translationally invariant system of an electron in non relativistic qed we assume that the interaction of the electron with the quantized electromagnetic field is subject to an ultraviolet regularization and an infrared regularization parametrized by 0 for the value p 0 of the conserved total momentum of electron and photon field bounds on the renormalized mass are established which are uniform in 0 and the existence of a ground state is proved for p 0 sufficiently small bounds on the renormalized mass are derived for any fixed 0 a key ingredient of our proofs is the operator theoretic renormalization group based on the isospectral smooth feshbach map it provides an explicit finite algorithm for determining the renormalized electron mass at p 0 to any given precision

lowest energy states in nonrelativistic qed atoms and ions in motion within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and n electrons coupled to the radiation field since the total momentum p is conserved the hamiltonian h admits a fiber decomposition with respect to p with fiber hamiltonian h p a stable atom resp ion means that the fiber hamiltonian h p has an eigenvalue at the bottom of its spectrum we establish the existence of a ground state for h p under i an explicit bound on p ii a binding condition and iii an energy inequality the binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons

a new perspective on static and low order anti windup synthesis by viewing the anti windup problem as a decoupled set of subsystems and relating this configuration to a general static anti windup set up lmi conditions are established which guarantee stability and performance of the resulting closed loop system the approach taken and the mapping used for the performance index are logical and intuitive and it is argued central to the true anti windup objective the approach enables one to construct static anti windup compensators in a systematic and numerically tractable manner the idea is extended to allow low order anti windup compensators to be synthesized which while being sub optimal can improve transient performance and possess several desired properties such as low computational overhead and sensible closed loop pole locations in addition low order anti windup synthesis is often feasible when the corresponding static synthesis is not

composite quadratic lyapunov functions for constrained control systems a lyapunov function based on a set of quadratic functions is introduced in this paper we call this lyapunov function a composite quadratic function some important properties of this lyapunov function are revealed we show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids these results are used to study the set invariance properties of continuous time linear systems with input and state constraints we show that for a system under a given saturated linear feedback the convex hull of a set of invariant ellipsoids is also invariant if each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators then their convex hull can also be made invariant by the same actuators for a set of ellipsoids each invariant under a separate saturated linear feedback we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant

analyticity of resonances and eigenvalues and spectral properties of the massless spin boson model abstract we extend the method of pizzo multiscale analysis for resonances introduced in 5 in order to infer analytic properties of resonances and eigenvalues and their eigenprojections as well as estimates for the localization of the spectrum of dilated hamiltonians and norm bounds for the corresponding resolvent operators in neighborhoods of resonances and eigenvalues we apply our method to the massless spin boson model assuming a slight infrared regularization we prove that the resonance and the ground state eigenvalue and their eigenprojections are analytic with respect to the dilation parameter and the coupling constant moreover we prove that the spectrum of the dilated spin boson hamiltonian in the neighborhood of the resonance and the ground state eigenvalue is localized in two cones in the complex plane with vertices at the location of the resonance and the ground state eigenvalue respectively additionally we provide norm estimates for the resolvent of the dilated spin boson hamiltonian near the resonance and the ground state eigenvalue the topic of analyticity of eigenvalues and resonances has let to several studies and advances in the past however to the best of our knowledge this is the first time that it is addressed from the perspective of pizzo multiscale analysis once the multiscale analysis is set up our method gives easy access to analyticity essentially it amounts to proving it for isolated eigenvalues only and use that uniform limits of analytic functions are analytic the type of spectral and resolvent estimates that we prove are needed to control the time evolution including the scattering regime the latter will be demonstrated in a forthcoming publication the introduced multiscale method to study spectral and resolvent estimates follows its own inductive scheme and is independent and different from the method we apply to construct resonances

correlation at low temperature ii asymptotics the present paper is a continuation of ref 4 where the truncated two point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature under a weak coupling assumption in this paper we compute the asymptotics of the correlation function as the temperature goes to zero this paper thus extends ref 3 in two directions the hamiltonian function is allowed to have several local minima other than a unique global minimum and we do not require translation invariance of the hamiltonian function we are in particular able to handle spin systems on a general lattice

dynamics of a family of chebyshev halley type methods in this paper the dynamics of the chebyshev halley family is studied on quadratic polynomials a singular set that we call cat set appears in the parameter space associated to the family this set has interesting similarities with the mandelbrot set the parameter space has allowed us to find different elements of the family which have bad convergence properties since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method

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

on developing a stable and quadratic convergent method for solving absolute value equation abstract we modify the generalized newton method proposed by mangasarian 2007 for solving np complete absolute value equation so that it is numerically stable and has convergence order two moreover the convergence conditions are weaker than already iterative methods hence this method can be applied to a broad range of problems applicability of the proposed method is tested for various examples

technical communique an anti windup scheme with closed loop stability considerations this paper considers the so called model based approach to anti windup compensation and analyses the stability of the closed loop system for a class of input nonlinearities the degrees of freedom in the anti windup scheme are used to minimise the h norm of the transfer function around the nonlinearity and the small gain theorem is used to prove stability

smoothness and analyticity of perturbation expansions in qed abstract we consider the ground state of an atom in the framework of non relativistic qed we assume that the ultraviolet cutoff is of the order of the rydberg energy and that the atomic hamiltonian has a non degenerate ground state we show that the ground state energy and the ground state are k times continuously differentiable functions of the fine structure constant and respectively the square root of the fine structure constant on some nonempty interval 0 c k

positive commutators in non equilibrium quantum statistical mechanics the method of positive commutators developed for zero temperature problems over the last twenty years has been an essential tool in the spectral analysis of hamiltonians in quantum mechanics we extend this method to positive temperatures i e to non equilibrium quantum statistical mechanics we use the positive commutator technique to give an alternative proof of a fundamental property of a certain class of large quantum systems called return to equilibrium this property says that equilibrium states are asymptotically stable if a system is slightly perturbed from its equilibrium state then it converges back to that equilibrium state as time goes to infinity

one boson scattering processes in the massive spin boson model abstract the spin boson model describes a two level quantum system that interacts with a second quantized boson scalar field recently the relation between the integral kernel of the scattering matrix and the resonance in this model has been established in 18 for the case of massless bosons in the present work we treat the massive case on the one hand one might rightfully expect that the massive case is easier to handle since in contrast to the massless case the corresponding hamiltonian features a spectral gap on the other hand it turns out that the non zero boson mass introduces a new complication as the spectrum of the complex dilated free hamiltonian exhibits lines of spectrum attached to every multiple of the boson rest mass energy starting from the ground and excited state energies this leads to an absence of decay of the corresponding complex dilated resolvent close to the real line which in 18 was a crucial ingredient to control the time evolution in the scattering regime with the new strategy presented here we provide a proof of an analogous formula for the scattering kernel as compared to the massless case and use the opportunity to provide the required spectral information by a mourre theory argument combined with a suitable application of the feshbach schur map instead of complex dilation

a biparametric extension of king s fourth order methods and their dynamics a class of two point quartic order simple zero finders and their dynamics are investigated in this paper by extending king s fourth order family of methods with the introduction of an error corrector having a weight function dependent on a function to function ratio higher order convergence is obtained through a variety of test equations numerical experiments strongly support the theory developed in this paper in addition relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes

families of rational maps and iterative root finding algorithms in this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root finding algorithms we answer a question of smale s by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more we settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics and we give a classification of all such algorithms in the context of conformal dynamics our main result is the following a stable algebraic family of rational maps is either trivial all its members are conjugate by mobius transformations or affine its members are obtained as quotients of iterated addition on a family of complex tori our classification of generally convergent algorithms follows easily from this result as another consequence of rigidity we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices this implies that bounded analytic functions nearly separate points on the moduli space of a rational map

construction of coarse grained order parameters in nonequilibrium systems we develop a renormalization group rg procedure that includes important system specific features the key ingredient is to systematize the coarse graining procedure that generates the rg flow the coarse graining technology comes from the control and operator theoretic model reduction the resulting generalized rg is a consistent generalization of the wilsonian rg we apply the procedure to a deterministic nonlinear wave equation nlwe with probabilistic initial conditions we derive the form of the projection operator from the dynamics of the nlwe and then use it to generate the rg flow for the distribution of initial conditions the probability density of the initial conditions is chosen to be a boltzmann weight that is quartic in the field variables in our calculation we find that in contrast to conventional implementations of the rg naive power counting breaks down we also show that the resulting rg equations are different from those derived from the conventional rg

study of iterative methods through the cayley quadratic test many iterative methods for solving nonlinear equations have been developed recently the main advantage claimed by their authors is the improvement of the order of convergence in this work we compare their dynamical behavior on quadratic polynomials with the one of newton s scheme this comparison is defined in what we call cayley quadratic test cqt which can be used as a first test to check the efficiency of such methods moreover we make a brief insight in cubic polynomials

renormalization group analysis of spectral problems in quantum field theory abstract in this paper we present a self contained and detailed exposition of the new renormalization group technique proposed in 1 2 its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator the partition function or correlation functions we apply this renormalization transformation to a hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing

spectral theory of pauli fierz operators abstract we study spectral properties of pauli fierz operators which are commonly used to describe the interaction of a small quantum system with a bosonic free field we give precise estimates of the location and multiplicity of the singular spectrum of such operators applications of these estimates which will be discussed elsewhere concern spectral and ergodic theory of non relativistic qed our proof has two ingredients the feshbach method which is developed in an abstract framework and mourre theory applied to the operator restricted to the sector orthogonal to the vacuum

on the virial theorem in quantum mechanics we review the various assumptions under which abstract versions of the quantum mechanical virial theorem have been proved we point out a relationship between the virial theorem for a pair of operators h a and the regularity properties of the map we give an example showing that the statement of the virial theorem in cfks is incorrect

estimation of asymptotic stability regions via composite homogeneous polynomial lyapunov functions in this article we present a new method to estimate the asymptotic stability regions for a class of nonlinear systems via composite homogeneous polynomial lyapunov functions where these nonlinear systems are approximated as a convex hull of some linear systems since level set of the composite homogeneous polynomial lyapunov functions is a union set of several homogeneous polynomial functions the composite homogeneous polynomial lyapunov functions are nonconservative compared with quadratic or homogeneous polynomial lyapunov functions numerical examples are used to illustrate the effectiveness of our method

on the cauchy problem for gross pitaevskii hierarchies the purpose of this paper is to investigate the cauchy problem for the gross pitaevskii infinite linear hierarchy of equations on rn n 1 we prove local existence and uniqueness of solutions in certain sobolev type spaces h of sequences of marginal density operators with n 2 in particular we give a clear discussion of all cases n 2 which covers the local well posedness problem for the gross pitaevskii hierarchy in this situation

a construction of attracting periodic orbits for some classical third order iterative methods we use a family of root finding iterative methods for finding roots of nonlinear equations we present a procedure for constructing polynomials so that superattracting periodic orbits of any prescribed period occur when these methods are applied this family includes chebyshev s method halley s method the super halley method and the c methods as particular cases

basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations several optimal eighth order methods to obtain simple roots are analyzed the methods are based on two step fourth order optimal methods and a third step of modified newton the modification is performed by taking an interpolating polynomial to replace either f z n or f z n in six of the eight methods we have used a hermite interpolating polynomial the other two schemes use inverse interpolation we discovered that the eighth order methods based on jarratt s optimal fourth order methods perform well and those based on king s or kung traub s methods do not in all cases tested the replacement of f z by hermite interpolation is better than the replacement of the derivative f z

an antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation the article considers linear systems subject to actuator saturation an antiwindup technique is used to enlarge the domain of attraction of the closed loop system under an a priori designed linear dynamic feedback law the design of the antiwindup compensation gain is formulated and solved as an iterative optimization problem with lmi constraints numerical examples are used to demonstrate the effectiveness of the proposed design technique

infrared renormalization in non relativistic qed and scaling criticality abstract we consider a spin 1 2 electron in the framework of non relativistic quantum electrodynamics qed let h p denote the fiber hamiltonian corresponding to the conserved total momentum p r 3 of the electron and the photon field regularized by a fixed ultraviolet cutoff in the interaction term and an infrared regularization parametrized by 0 1 ultimately our goal is to remove the latter by taking 0 we prove that there exists a constant 0 0 1 independent of 0 such that for all p 1 3 and all values of the finestructure constant 0 0 there exists a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum moreover we prove that the renormalized electron mass satisfies 1 m ren p 1 c uniformly in 0 in units where the bare mass has the value 1 and we prove the existence of the renormalized mass in the limit 0 our analysis uses the isospectral renormalization group method of bach frohlich sigal introduced in v bach j frohlich i m sigal quantum electrodynamics of confined non relativistic particles adv math 137 2 1998 299 395 v bach j frohlich i m sigal renormalization group analysis of spectral problems in quantum field theory adv math 137 1998 205 298 and further developed in v bach t chen j frohlich i m sigal smooth feshbach map and operator theoretic renormalization group methods j funct anal 203 1 2003 44 92 v bach t chen j frohlich i m sigal the renormalized electron mass in non relativistic qed j funct anal 243 2 2007 426 535 the limit 0 determines a scaling critical or endpoint type renormalization group problem in which the interaction is strictly marginal of scale independent size a main result of this paper is the development of a method that provides rigorous control of the renormalization of a strictly marginal quantum field theory characterized by a non trivial scaling limit the key ingredients entering this analysis include a hierarchy of exact algebraic cancelation identities exploiting the spatial and gauge symmetries of the model and a combination of the isospectral renormalization group method with the strong induction principle

one boson scattering processes in the massless spin boson model a non perturbative formula abstract in scattering experiments physicists observe so called resonances as peaks at certain energy values in the measured scattering cross sections per solid angle these peaks are usually associate with certain scattering processes e g emission absorption or excitation of certain particles and systems on the other hand mathematicians define resonances as poles of an analytic continuation of the resolvent operator through complex dilations a major challenge is to relate these scattering and resonance theoretical notions e g to prove that the poles of the resolvent operator induce the above mentioned peaks in the scattering matrix in the case of quantum mechanics this problem was addressed in numerous works that culminated in simon s seminal paper 33 in which a general solution was presented for a large class of pair potentials however in quantum field theory the analogous problem has been open for several decades despite the fact that scattering and resonance theories have been well developed for many models in certain regimes these models describe very fundamental phenomena such as emission and absorption of photons by atoms from which quantum mechanics originated in this work we present a first non perturbative formula that relates the scattering matrix to the resolvent operator in the massless spin boson model this result can be seen as a major progress compared to our previous works 14 and 12 in which we only managed to derive a perturbative formula

resolvent smoothness and local decay at low energies for the standard model of non relativistic qed we consider an atom interacting with the quantized electromagnetic field in the standard model of non relativistic qed the nucleus is supposed to be fixed we prove smoothness of the resolvent and local decay of the photon dynamics for quantum states in a spectral interval i just above the ground state energy our results are uniform with respect to i their proofs are based on abstract mourre s theory a mourre inequality established by frohlich griesemer and sigal see frohlich et al 2008 14 hardy type estimates in fock space and a low energy dyadic decomposition

smooth feshbach map and operator theoretic renormalization group methods abstract a new variant of the isospectral feshbach map defined on operators in hilbert space is presented it is constructed with the help of a smooth partition of unity instead of projections and is therefore called smooth feshbach map it is an effective tool in spectral and singular perturbation theory as an illustration of its power a novel operator theoretic renormalization group method is described and applied to analyze a general class of hamiltonians on fock space the main advantage of the new renormalization group method over its predecessors is its technical simplicity which it owes to the use of the smooth feshbach map

design of a nonlinear anti windup gain by using a composite quadratic lyapunov function this technical note revisits the problem of designing a static anti windup gain for enlarging the domain of attraction of the resulting closed loop system by utilizing a composite quadratic lyapunov function which was originally proposed to study the stabilization problem for linear systems under actuator saturation an existing lmi based design algorithm is enhanced to result in a nonlinear possibly continuous anti windup gain this nonlinear anti windup gain enables us to obtain an estimate of the domain of attraction in the form of the convex hull of a group of ellipsoids instead of a single ellipsoid that would result from a single lyapunov function based design simulation results demonstrate the features of the proposed design the composite quadratic lyapunov function brings about

the l2 anti windup problem for discrete time linear systems definition and solutions abstract the anti windup problem for discrete time linear systems is formalized in terms of the l 2 norm of the deviation of the actual response of the system with saturation and anti windup compensation from the ideal unconstrained response we show that paralleling continuous time results a r teel n kapoor the l 2 anti windup problem its definition and solution in proceedings of the 4th ecc brussels belgium july 1997 the problem is globally solvable if and only if the plant is non exponentially unstable and it is robustly globally solvable if and only if the plant is exponentially stable we provide a constructive solution whenever the problem is solvable also offered is a high performance global solution for exponentially stable plants based on receding horizon control illustrative simulations are included

a unified framework for the study of anti windup designs we present a unified framework for the study of linear time invariant lti systems subject to control input nonlinearities the framework is based on the following two step design paradigm design the linear controller ignoring control input nonlinearities and then add anti windup bumpless transfer awbt compensation to minimize the adverse eflects of any control input nonlinearities on closed loop performance the resulting awbt compensation is applicable to multivariable controllers of arbitrary structure and order all known lti anti windup and or bumpless transfer compensation schemes are shown to be special cases of this framework it is shown how this framework can handle standard issues such as the analysis of stability and performance with or without uncertainties in the plant model the actual analysis of stability and performance and robustness issues are problems in their own right and hence not detailed here the main result is the unification of existing schemes for awbt compensation under a general framework

ground states and spectrum of quantum electrodynamics of nonrelativistic particles a system consisting of finitely many nonrelativistic particles bound on an external potential and minimally coupled to a massless quantized radiation field without the dipole approximation is considered an ultraviolet cut off is imposed on the quantized radiation field the hamiltonian of the system is defined as a self adjoint operator in a hilbert space the existence of the ground states of the hamiltonian is established it is shown that there exist asymptotic annihilation and creation operators hence the location of the absolutely continuous spectrum of the hamiltonian is specified

brief paper anti windup synthesis for linear control systems with input saturation achieving regional nonlinear performance in this paper we present lmi based synthesis tools for regional stability and performance of linear anti windup compensators for linear control systems we consider both static and dynamic compensators algorithms are developed that minimize the upper bound on the regional l 2 gain for exogenous inputs with l 2 norm bounded by a given value and that minimize this upper bound with a guaranteed reachable set or domain of attraction based on the structure of the optimization problems it is shown that for systems whose plants have poles in the closed left half plane plant order dynamic anti windup can achieve semiglobal exponential stability and finite l 2 gain for exogenous inputs with l 2 norm bounded by any finite value the problems are studied in a general setting where the only requirement on the linear control system is well posedness and internal stability the effectiveness of the proposed techniques is illustrated with an example

an anti windup design for linear systems with input saturation abstract an anti windup design problem is posed and it is shown that all observer based anti windup modifications solve this problem at least locally sufficient conditions for an observer based anti windup modification to solve the global version of this problem are presented a novel observer based anti windup design is then proposed such that the anti windup problem can be interpreted in terms of a reduced order system in particular the proposed anti windup design induces an invariant subspace for the dynamic behavior of the mismatch between the constrained and unconstrained closed loops the dynamics in this invariant subspace are identical to the behavior of the plant with input saturation starting at the origin stabilized by linear state feedback and driven by the mismatch between the unconstrained input and this input passed through a saturation function the second part of the paper shows how the original dynamic compensator can be modified while retaining those dynamic features that produce a desirable closed loop steady state response to ensure that the requisite invariant subspace exists and it is reasonably tuned for input saturation two case studies are carried out on systems that have been investigated in the literature

brief multiplier theory for stability analysis of anti windup control systems we apply the passivity theorem with appropriate choice of multipliers to develop a general framework for analyzing stability of anti windup bumpless transfer awbt schemes for appropriate choices of the multipliers we reduce the resulting sufficient conditions for stability to equivalent linear matrix inequalities lmis we show that a number of previously reported attempts to analyze awbt stability using seemingly diverse techniques such as the popov circle and off axis circle criteria the optimally scaled small gain theorem generalized m upper bound and describing functions can all be generalized within the framework of this paper

quantum electrodynamics of confined nonrelativistic particles abstract we consider a system of finitely many nonrelativistic quantum mechanical electrons bound to static nuclei the electrons are minimally coupled to the quantized electromagnetic field but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives and the interactions between the radiation field and electrons localized very far from the nuclei are turned off for a class of hamiltonians we prove exponential localization of bound states establish the existence of a ground state and derive sufficient conditions for its uniqueness furthermore we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field to this end we develop a novel renormalization transformation which acts directly on the space of hamiltonians

continuous renormalization group analysis of spectral problems in quantum field theory the isospectral renormalization group is a powerful method to analyze the spectrum of operators in quantum field theory it was introduced in 1995 see 2 4 and since then it has been used to prove several results for non relativistic quantum electrodynamics after the introduction of the method there have been many works in which extensions simplifications or clarifications are presented see 7 11 13 in this paper we present a new approach in which we construct a flow of operators parametrized by a continuous variable in the positive real axis while this is in contrast to the discrete iteration used before this is more in spirit of the original formulation of the renormalization group introduced in theoretical physics in 1974 22 the renormalization flow that we construct can be expressed in a simple way it can be viewed as a single application of the feshbach schur map with a clever selection of the spectral parameter another advantage of the method is that there exists a flow function for which the renormalization group that we present is the orbit under this flow of an initial hamiltonian this opens the possibility to study the problem using different techniques coming from the theory of evolution equations

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

ground states in non relativistic quantum electrodynamics the excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay but such a particle should have a true ground state one that minimizes the energy and satisfies the schrodinger equation we prove quite generally that this state exists for all values of the fine structure constant and the ultraviolet cutoff we also show the same thing for a many particle system under physically natural conditions

anti windup control design for exponentially unstable lti systems with actuator saturation in this paper a new saturation control technique in the framework of anti windup compensation is developed for exponentially unstable linear time invariant systems subject to input nonlinearities the proposed control algorithm guarantees regional stability in the existence of input saturation and provides less conservative performance than most existing anti windup schemes moreover an explicit form of anti windup controller with its order no more than the order of the plant is derived an inverted pendulum example is used to demonstrate the advantages of the newly proposed anti windup control technique

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

a stable family with high order of convergence for solving nonlinear equations recently li et al 2014 have published a new family of iterative methods without memory with order of convergence five or six which are not optimal in the sense of kung and traub s conjecture therefore we attempt to modify this suggested family in such a way that it becomes optimal to this end we consider the same two first steps of the mentioned family and furthermore we introduce a better approximation for f z in the third step based on interpolation idea as opposed to the taylor s series used in the work of li et al theoretical dynamical and numerical aspects of the new family are described and investigated in details

markov chain approximations to nonsymmetric diffusions with bounded coefficients we consider a certain class of nonsymmetric markov chains and obtain heat kernel bounds and parabolic harnack inequalities using the heat kernel estimates we establish a sufficient condition for the family of markov chains to converge to nonsymmetric diffusions as an application we approximate nonsymmetric diffusions in divergence form with bounded coefficients by nonsymmetric markov chains this extends the results by stroock and zheng to the nonsymmetric divergence forms 2012 wiley periodicals inc

existence of resonances for the spin boson model with critical coupling function abstract a two level atom coupled to the quantized radiation field is studied in the physical relevant situation the coupling function modeling the interaction between the two components behaves like k 1 2 as the photon momentum tends to zero this behavior is referred to as critical as it constitutes a borderline case previous results on non existence state that in the general case neither a ground state nor a resonance exists hasler and herbst 2011 have shown however that a ground state does exist if the absence of self interactions is assumed bach ballesteros konenberg and menrath 2017 have then explicitly constructed the ground state this specific case using the multiscale analysis known as pizzo s method pizzo 2003 in the present paper a resonance eigenvalue of the complex deformed hamiltonian is constructed by applying an inductive multiscale approach developed by bach ballesteros and pizzo 2013 2017 thus showing the existence of resonances in the critical case although the proof does not mainly rely on spectral renormalization some techniques enter in the form of a suitable feshbach schur map to control the exponential decay of the terms in the neumann series expansions overall the interaction terms obtained only involve second powers of the creation and annihilation operators and thus allow for a simpler proof

maximal velocity of photons in non relativistic qed abstract we consider the problem of propagation of photons in the quantum theory of non relativistic matter coupled to electromagnetic radiation which is presently the only consistent quantum theory of matter and radiation assuming that the matter system is in a localized state i e for energies below the ionization threshold we show that the probability to find photons at time t at the distance greater than c t where c is the speed of light vanishes as t as an inverse power of t

convergent expansions in non relativistic qed analyticity of the ground state we consider the ground state of an atom in the framework of non relativistic qed we show that the ground state as well as the ground state energy are analytic functions of the coupling constant which couples to the vector potential under the assumption that the atomic hamiltonian has a non degenerate ground state moreover we show that the corresponding expansion coefficients are precisely the coefficients of the associated raleigh schrodinger series as a corollary we obtain that in a scaling limit where the ultraviolet cutoff is of the order of the rydberg energy the ground state and the ground state energy have convergent power series expansions in the fine structure constant with dependent coefficients which are finite for 0

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

on the smooth feshbach schur map a new variant of the feshbach map called smooth feshbach map has been introduced recently by bach et al in connection with the renormalization analysis of non relativistic quantum electrodynamics we analyze and clarify its algebraic and analytic properties and we generalize it to non selfadjoint partition operators and

renormalization analysis for degenerate ground states abstract we consider a hamilton operator which describes a finite dimensional quantum mechanical system with degenerate eigenvalues coupled to a field of relativistic bosons we show that the ground state projection and the ground state energy are analytic functions of the coupling constant in a cone with apex at the origin provided a mild infrared assumption holds to show the result operator theoretic renormalization is used and extended to degenerate situations

existence of ground state eigenvalues for the spin boson model with critical infrared divergence and multiscale analysis abstract a two level atom coupled to the radiation field is studied first principles in physics suggest that the coupling function representing the interaction between the atom and the radiation field behaves like k 1 2 as the photon momentum k tends to zero previous results on non existence of ground state eigenvalues suggest that in the most general case binding does not occur in the spin boson model i e the minimal energy of the atom photon system is not an eigenvalue of the energy operator hasler and herbst have shown 12 however that under the additional hypothesis that the coupling function be off diagonal which is customary to assume binding does indeed occur in this paper an alternative proof of binding in case of off diagonal coupling is given i e it is proven that if the coupling function is off diagonal the ground state energy of the spin boson model is an eigenvalue of the hamiltonian we develop a multiscale method that can be applied in the situation we study with the help of a key symmetry operator which we use to demonstrate that the most singular terms appearing in the multiscale analysis vanish

on a homogenization problem we study the average green s function of stochastic uniformly elliptic operators of divergence form on zd mathbb z d when the randomness is independent and has small variance we prove regularity of the fourier transform of the self energy the proof relies on the schur complement formula and the analysis of singular integral operators combined with a steinhaus system

antiwindup design with guaranteed regions of stability an lmi based approach this note addresses the design of antiwindup gains for obtaining larger regions of stability for linear systems with saturating inputs considering that a linear dynamic output feedback has been designed to stabilize the linear system without saturation a method is proposed for designing an antiwindup gain that maximizes an estimate of the basin of attraction of the closed loop system it is shown that the closed loop system obtained from the controller plus the antiwindup gain can be modeled by a linear system with a deadzone nonlinearity a modified sector condition is then used to obtain stability conditions based on quadratic lyapunov functions differently from previous works these conditions are directly in linear matrix inequality form some numerical examples illustrate the effectiveness of the proposed design technique when compared with the previous ones

an improvement of chebyshev halley methods free from second derivative abstract in this paper a family of modified chebyshev halley s methods free from second derivative is presented per iteration the new methods require three evaluations of the function and one of its first derivatives a detailed convergence analysis of the new methods shows that the new methods are at least fifth order convergent and especially the modified super halley s method is sixth order convergent numerical examples are given to illustrate the efficiency and performance of the new methods

spectral theory of massless pauli fierz models we study the spectral theory of massless pauli fierz models using an extension of the mourre method we prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant in addition we show that the expectation value of the number operator is finite on all eigenvectors

exponential decay and ionization thresholds in non relativistic quantum electrodynamics abstract spatial localization of the electrons of an atom or molecule is studied in models of non relativistic matter coupled to quantized radiation we give two definitions of the ionization threshold one in terms of spectral data of cluster hamiltonians and one in terms of minimal energies of non localized states we show that these two definitions agree and that the electrons described by a state with energy below the ionization threshold are localized in a small neighborhood of the nuclei with a probability that approaches 1 exponentially fast with increasing radius of the neighborhood the latter result is derived from a new general result on exponential decay tailor made for our problem but applicable to many non relativistic quantum systems outside quantum electrodynamics as well

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

three step iterative methods with eighth order convergence for solving nonlinear equations a family of eighth order iterative methods for the solution of nonlinear equations is presented the new family of eighth order methods is based on king s fourth order methods and the family of sixth order iteration methods developed by chun et al per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has the efficiency index which equals 1 682 kung and traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n 1 thus we provide a new example which agrees with the conjecture of kung traub for n 4 numerical comparisons are made to show the performance of the presented methods

basin attractors for various methods abstract there are many methods for the solution of a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several methods of various orders and present the basin of attraction for several examples it can be seen that not all higher order methods were created equal newton s halley s murakami s and neta johnson s methods are consistently better than the others in two of the examples neta s 16th order scheme was also as good

different anomalies in a jarratt family of iterative root finding methods abstract in this paper the behavior of a jarratt family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family be means of studying the dynamical behavior of this fourth order family of methods parameter spaces are shown and the study of the stability of all the fixed points is presented dynamical planes for members with good and bad dynamical behavior are also provided

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

analyticity of the self energy in total momentum of an atom coupled to the quantized radiation field abstract we study a neutral atom with a non vanishing electric dipole moment coupled to the quantized electromagnetic field for a sufficiently small dipole moment and small momentum p the one particle self energy of an atom is proven to be a real analytic function of its momentum the main ingredient of our proof is a suitable form of the feshbach schur spectral renormalization group

existence and construction of resonances for atoms coupled to the quantized radiation field abstract the processes of emission and absorption of photons by atoms can be rigorously understood in the low energy limit if we neglect the creation and annihilation of electrons they are related to eigenvalues of the atomic hamiltonian that are embedded in the continuous spectrum of the free hamiltonian of the electromagnetic field the mathematical analysis of the perturbation of these eigenvalues due to the electromagnetic interaction relies on a complex deformation technique relating the original hamiltonian to a non selfadjoint operator we develop a new technique to analyze the spectrum of operators used in non relativistic quantum electrodynamics our method can be applied to prove most of the results that previously required an involved renormalization group construction we use analytic perturbation theory of operators in hilbert spaces instead more precisely we extend the multi scale analysis introduced by one of us in 2003 15 which was used so far only for the study of selfadjoint operators to non selfadjoint operators compared to the selfadjoint case see for example 3 the analysis of these non selfadjoint operators is more difficult because we cannot make use of the functional calculus spectral theorem and the min max principle in some crucial estimates

functional integral representations of the pauli fierz model with spin 1 2 a feynman kac type formula for a levy and an infinite dimensional gaussian random process associated with a quantized radiation field is derived in particular a functional integral representation of e thpf generated by the pauli fierz hamiltonian with spin 1 2 in non relativistic quantum electrodynamics is constructed when no external potential is applied hpf turns translation invariant and it is decomposed as a direct integral hpf r3 hpf p dp the functional integral representation of e thpf p is also given although all these hamiltonians include spin nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions as an application of the functional integral representations energy comparison inequalities are derived

on optimal fourth order iterative methods free from second derivative and their dynamics in this paper new fourth order optimal root finding methods for solving nonlinear equations are proposed the classical jarratt s family of fourth order methods are obtained as special cases we then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three the basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance