absence of absolutely continuous spectrum of floquet operators the spectrum of the floquet operator associated with time periodic perturbations of discrete hamiltonians is considered if the gap between successive eigenvalues j of the unperturbed hamiltonian grows as j j 1 j and the multiplicity of j grows asj with 0 asj tends to infinity then the corresponding floquet operator possesses no absolutely continuous spectrum provided the perturbation is smooth enough

stability of oscillators driven by ergodic processes the evolution of a quantum harmonic oscillator forced by a stationary ergodic process is considered under suitable conditions on the dynamics of this process it is shown that generically the spectrum of the associated quasienergy operator is continuous these results also apply to kicked harmonic oscillators where the amplitudes of the kicks are governed by a stationary ergodic process

perturbation of an eigenvalue from a dense point spectrum an example we study an example of a perturbed floquet hamiltonian depending on a coupling constant the spectrum is pure point and dense we pick up an eigenvalue namely and show the existence of a function defined on such that for all 0 is a point of density for the set i and the rayleigh schrodinger perturbation series represents an asymptotic series for the function all ideas are developed and demonstrated when treating the explicit example but some of them are expected to have an essentially wider range of application

a spectral multiplier theorem for a sublaplacian on su 2 we prove a hormander type spectral multiplier theorem for a sublaplacian on su 2 with critical index determined by the euclidean dimension of the group this result is the analogue for su 2 of the result for the heisenberg group obtained by d muller and e m stein and by w hebisch

complexity of a quasiperiodically driven spin system a quasiperiodically forced purely quantum spin system can exhibit rather irregular behaviour with a continuous power spectrum we have shown analytically and numerically that in these regimes both the topological entropy of a corresponding symbolic sequence and the maximal lyapunov exponent vanish

diffusive energy growth in classical and quantum driven oscillators we study the long time stability of oscillators driven by time dependent forces originating from dynamical systems with varying degrees of randomness the asymptotic energy growth is related to ergodic properties of the dynamical system when the autocorrelation of the force decays sufficiently fast one typically obtains linear diffusive growth of the energy for a system with good mixing properties we obtain a stronger result in the form of a central limit theorem if the autocorrelation decays slowly or does not decay the behavior can depend on subtle properties of the particular model we study this dependence in detail for a family of quasiperiodic forces the solution involves the analysis of a small denominator problem that can be treated by fairly elementary methods in the special case of a periodic force the quantum stability problem can be expressed in terms of spectral properties of the floquet operator in the presence of resonances the spectrum is absolutely continuous we find explicitly the eigenvalues and eigenfunctions for the nonresonant case

quasi periodic solutions for two level systems we consider the schrodinger equation for a class of two level atoms in a quasi periodic external field in the case in which the spacing 2 between the two unperturbed energy levels is small and we study the problem of finding quasi periodic solutions of a related generalized riccati equation we prove the existence of quasi periodic solutions of the latter equation for a cantor set of values of around the origin which is of positive lebesgue measure such solutions can be obtained from the formal power series by a suitable resummation procedure the set can be characterized by requesting infinitely many diophantine conditions of mel nikov type

renormalization of cocycles and linear ode with almost periodic coefficients in the current paper we address the problem of classification of cocycles over an irrational rotation we use the renormalization group approach

on quantum stability for systems under quasiperiodic perturbations we prove that the spectrum defined in terms of the autocorrelation function of a harmonic subject to a quasiperiodic perturbation is at resonance transient absolutely continuous covering the whole line in the nonresonant case and under some supplementary diophantine condition it is pure point coinciding with the spectrum of a special almost periodic function

floquet spectrum for two level systems in quasiperiodic time dependent fields we study the time evolution ofn level quantum systems under quasiperiodic time dependent perturbations the problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged hilbert space or equivalently of a generalized floquet operator we discuss criteria for the appearance of pure point as well as continuous spectrum corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior we discuss two types of mechanisms that lead to instability the first one is due to near resonances while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation we treat explicitly an example of this type the stability of the pure point spectrum under small perturbations is proven using kam techniques

dynamical localization for two level systems periodically driven here we consider a two level system driven by an external periodic field we show that the coherent destruction of tunnelling as proved by grossmann and co workers 1991 phys rev lett 67 516 1992 europhys lett 18 571 in the case of a monochromatic field also appears for any periodic driving field given by an even regular function with zero mean value and satisfying a technical condition on the zeros of this function

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

high order harmonic generation and dynamic localization in a driven two level system a non perturbative solution using the floquet green formalism non perturbative exact analytical results are obtained for the harmonically driven two level system with the use of a floquet green operator formalism from this operator we find the quasi energies and the fourier components of the floquet eigenstates which we use to construct the time evolution operator of the system as an application of our formalism we study dynamic localization and high order harmonic generation in this system our results show the existence of an initial condition that leads to an emission spectrum with only hyper raman lines for any frequency and strength of the driving field we show the important role of the initial condition in determining the emission spectrum of this system in the non perturbative regime the emission spectrum is found to be in general very sensitive to the amplitude of the driving field we also derive an expression for the coherent emission spectrum which shows its dependence on the degree of localization of the system

explicit formulas for laplace transforms of certain functionals of some time inhomogeneous diffusions we consider a process xt t 0 t given by the sde dxt b t xt dt t dbt t 0 t with initial condition x0 0 where t 0 r bt t 0 t is a standard wiener process b 0 t r 0 and 0 t 0 are continuously differentiable functions assuming ddt b t t 2 2kb t 2 t 2 t 0 t with some k r we derive an explicit formula for the joint laplace transform of 0tb s 2 s 2 xs 2ds and xt 2 for all t 0 t and for all r our motivation is that the maximum likelihood estimator mle t of can be expressed in terms of these random variables as an application we show that in case of k k 0 ik t t k l sign k 2 01wsdws 01 ws 2ds t 0 t where ik t denotes the fisher information for contained in the observation xs k s 0 t ws s 0 1 is a standard wiener process and l denotes equality in distribution we also prove asymptotic normality of the mle t of as t t for sign k sign k k 0 as an example for all r and t 0 we study the process xt t 0 t given by the sde dxt t txt dt dbt t 0 t with initial condition x0 0 in case of 0 this process is known as an wiener bridge and in case of 1 this is the usual wiener bridge

models of two level atoms in quasiperiodic external fields we prove that the spectrum of the generalized quasienergy operator of a plane polarized two level atom in a strong external quasiperiodic electromagnetic field with nonzero constant fourier component is pure point under diophantine conditions on the frequency ratio and excluding a small subset of resonant values the widespread belief that there may be only pure point spectrum in such models is briefly discussed in section 2 and the circularly polarized case a well known soluble model is revisited from the point of view of the quasienergy operator

recurrent versus diffusive dynamics for a kicked quantum system we study the dynamics of a two level quantum system subject to a time dependent kicking perturbation modulated along the thue morse sequence for a nontrivial set of the parameters the quantum autocorrelation function is explicitly calculated and splits into a purely recurrent and a purely diffusive part furthermore the diffusive part is directly related to the singular continuous correlation measure of the thue morse sequence

differential structure on the dual of a compact lie group abstract in this paper we define difference operators and homogeneous sobolev type spaces on the dual of a compact lie group as an application and to show that this defines a relevant differential structure we state and prove multiplier theorems of hormander mihlin and marcinkiewicz types together with the sharpness in the sobolev exponent for the result of hormander type

spectra of quasienergies abstract the spectra of the floquet operator for the standard kicked rotor and the quasiperiodic rabi oscillator are generically purely continuous for the kicked rotor the spectrum is purely singular continuous for a generic set of kick potentials with bounded variation

proof of dynamical localization for perturbations of discrete 1d schrodinger operators with uniform electric fields we present a proof of discrete spectrum and dynamical localization for small perturbations of discrete one dimensional schrodinger operators with uniform electric fields the proof of dynamical localization is based on the kam technique

hyperbolic low dimensional invariant tori and summations of divergent series we consider a class of a priori stable quasi integrable analytic hamiltonian systems and study the regularity of low dimensional hyperbolic invariant tori as functions of the perturbation parameter we show that under natural nonresonance conditions such tori exist and can be identified through the maxima or minima of a suitable potential they are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin the invariant tori admit an asymptotic series at the origin with taylor coefficients that grow at most as a power of a factorial and a remainder that to any order n is bounded by the n 1 st power of the argument times a power of n we show the existence of a summation criterion of the generically divergent series in powers of the perturbation size that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory

lp lq multipliers on locally compact groups abstract in this paper we discuss the l p l q boundedness of both spectral and fourier multipliers on general locally compact separable unimodular groups g for the range 1 p 2 q as a consequence of the established fourier multiplier theorem we also derive a spectral multiplier theorem on general locally compact separable unimodular groups we then apply it to obtain embedding theorems as well as time asymptotics for the l p l q norms of the heat kernels for general positive unbounded invariant operators on g we illustrate the obtained results for sub laplacians on compact lie groups and on the heisenberg group as well as for higher order operators we show that our results imply the known results for l p l q multipliers such as hormander s fourier multiplier theorem on r n or known results for fourier multipliers on compact lie groups the new approach developed in this paper relies on advancing the analysis in the group von neumann algebra and its application to the derivation of the desired multiplier theorems

global hypoellipticity for strongly invariant operators abstract in this note by analyzing the behavior at infinity of the matrix symbol of an invariant operator p with respect to a fixed elliptic operator we obtain a necessary and sufficient condition to guarantee that p is globally hypoelliptic as an application we obtain the characterization of global hypoellipticity on compact lie groups and examples on the sphere and the torus we also investigate relations between the global hypoellipticity of p and global subelliptic estimates

quantum averaging for driven systems with resonances abstract we discuss the effects of resonances in driven quantum systems within the context of quantum averaging techniques in the floquet representation we consider in particular iterative methods of kam type and the extensions needed to take into account resonances the approach consists in separating the coupling terms into resonant and nonresonant components at a given scale of time and intensity the nonresonant part can be treated with perturbative techniques which we formulate in terms of kam type unitary transformations that are close to the identity these can be interpreted as averaging procedures with respect to the dynamics defined by effective uncoupled hamiltonians the resonant parts are treated with a different kind of unitary transfomations that are not close to the identity and are adapted to the structure of the resonances they can be interpreted as a renormalization of the uncoupled hamiltonian that yields an effective dressed hamiltonian around which a perturbation expansion can be developed the combination of these two ingredients provides a strongly improved approximation technique

floquet hamiltonians with pure point spectrum we consider floquet hamiltonians of the type k f i partial t h 0 beta v omega t whereh0 a selfadjoint operator acting in a hilbert space has simple discrete spectrume1 0 for a given 0 t v t is 2 periodic andr times strongly continuously differentiable as a bounded operator on and are real parameters and the periodic boundary condition is imposed in time we show roughly that providedr is large enough small enough and non resonant then the spectrum ofkf is pure point the method we use relies on a successive application of the adiabatic treatment due to howland and the kam type iteration settled by bellissard and extended by combescure both tools are revisited adjusted and at some points slightly simplified

floquet green function formalism for harmonically driven hamiltonians a method is proposed for the calculation of the floquet green function of a general hamiltonian with harmonic time dependence we use matrix continued fractions to derive an expression for the dynamical effective potential that can be used to calculate the floquet green function of the system we demonstrate the formalism for the simple case of a space periodic in the tight binding approximation hamiltonian with a defect whose on site energy changes harmonically with time we study the local density of states for this system and the behaviour of the localized states as a function of the different parameters that characterize the system

sharp garding inequality on compact lie groups abstract we establish the sharp garding inequality on compact lie groups the positivity condition is expressed in the non commutative phase space in terms of the full matrix symbol which is defined using the representations of the group applications are given to the l 2 and sobolev boundedness of pseudo differential operators

non stochasticity of time dependent quadratic hamiltonians and the spectra of canonical transformations the authors study the quantum mechanical evolution generated by hamiltonians which are quadratic polynomials in q and p with time dependent coefficients this evolution is determined by a unitary implementation of the phase flow of the corresponding classical hamiltonian in the case of quadratic hamiltonians which are periodic in time the floquet operator is shown to have either a pure point spectrum or a purely transient absolutely continuous spectrum thus the motion is non stochastic in a simple model of a quadratic hamiltonian with random time dependence the quantum mechanical motion is shown to be non stochastic almost surely

hardy littlewood hausdorff young paley inequalities and lp lq fourier multipliers on compact homogeneous manifolds abstract in this paper we prove new inequalities describing the relationship between the size of a function on a compact homogeneous manifold and the size of its fourier coefficients these inequalities can be viewed as noncommutative versions of the hardy littlewood inequalities obtained by hardy and littlewood 17 on the circle for the example case of the group su 2 we show that the obtained hardy littlewood inequalities are sharp yielding a criterion for a function to be in l p su 2 in terms of its fourier coefficients we also establish paley and hausdorff young paley inequalities on general compact homogeneous manifolds the latter is applied to obtain conditions for the l p l q boundedness of fourier multipliers for 1 p 2 q on compact homogeneous manifolds as well as the l p l q boundedness of general non invariant operators on compact lie groups we also record an abstract version of the marcinkiewicz interpolation theorem on totally ordered discrete sets to be used in the proofs with different plancherel measures on the unitary duals

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

local asymptotic mixed normality for semimartingale experiments we give conditions for local asymptotic mixed normality of experiments when the observed process is a semimartingale and the observation time increases to infinity as a consequence we obtain asymptotic efficiency of various estimators several special models for counting process s diffusion processes and diffusions with jumps are studied

a proof of the dynamical version of the bombieri taylor conjecture an adaptation of the bombieri taylor conjecture on diffraction has been applied to some time dependent quantum systems here the problem is mathematically stated and a proof is presented simple examples give an idea as to what extent the conjecture is valid

correlation dimension and spectrum for the almost periodic bouncing ball abstract we analyze the dependence of the autocorrelation functions and correlation dimensions on the degree of almost periodicity of the classical bouncing ball model

pure point spectrum for two level systems in a strong quasi periodic field we consider two level atoms in a strong external quasi periodic field with diophantine frequency vector we show that if the field is an analytic function with zero average then for a large set of values of its frequency vector characterized by imposing infinitely many diophantine conditions the spectrum of the quasi energy operator is pure point as in the case of nonzero average which was already known in literature

the structure of the resolvent of elliptic pseudodifferential operators abstract we show that the resolvent kernel of an elliptic b pseudodifferential operator on a compact manifold with corners of arbitrary codimension is a polyhomogeneous or classical function on a certain manifold with corners the singularities of the resolvent kernel are shown to localize near the diagonal as the resolvent parameter goes to infinity explicit descriptions of the expansions including logarithmic terms are given in particular the asymptotics of the resolvent restricted to the diagonal follows as a corollary applications to the asymptotic behavior of the heat kernel and to the analysis of the poles of complex powers are also given

dynamical localization criterion for driven two level systems in this paper we consider a two level system under the effect of an external time dependent field we give a precise criterion for dynamical localization we then apply our result to the cases of external ac dc and bichromatic fields

perturbative analysis of dynamical localization in this paper we present a mathematical investigation of the phenomenon of dynamical localization in a class of quasi periodically and periodically time dependent two level systems our results are based on a sort of renormalization procedure which is developed here in a systematic way in order to adapt it to the case of dynamical localization in the quasi periodic case this procedure leads to a formal perturbative expansion free of secular terms in the periodic case a convergent perturbative solution is obtained and in particular a convergent perturbative expansion for the secular frequency is presented the case of ac dc fields is discussed in some detail leading to the conclusion that the phenomenon of dynamical localization is not exact in that situation

l p fourier multipliers on compact lie groups in this paper we prove l p fourier multiplier theorems for invariant and also non invariantoperatorsoncompactliegroupsinthespiritofthewell knownhormander mikhlin theorem on r n and its variants on tori t n we also give applications to a priori estimates for non hypoelliptic operators already in the case of tori we get an interesting refinement of the classical multiplier theorem

two level systems driven by a stochastic perturbation here we consider a two level system driven by an external harmonic field whose amplitude is perturbed by a white noise term in the limit of small splitting dynamical localization i e coherent destruction of tunneling is proved for times of the order of 1 e where e is the two level splitting the same type of localization is proved if the driving field is simply the white noise

spectrum for quantum duffing oscillator and small divisor equation with large variable coefficient by establishing a new estimate of the solutions for the small divisor equation with large variable coefficients and using the kam technique we prove a reduction theorem that entails the pure point nature of the floquet spectrum of the quantum duffing oscillator with a small perturbation temporal quasi periodic with nonresonant frequencies 2010 wiley periodicals inc

global functional calculus for operators on compact lie groups abstract in this paper we develop the functional calculus for elliptic operators on compact lie groups without the assumption that the operator is a classical pseudo differential operator consequently we provide a symbolic descriptions of complex powers of such operators as an application we give a constructive symbolic proof of the garding inequality for operators in classes in the setting of compact lie groups