l induced norm analysis of sampled data systems via piecewise constant and linear approximations this paper deals with the l 1 analysis of linear sampled data systems by which we mean the computation of the l induced norm of linear sampled data systems two computation methods based on piecewise constant and piecewise linear approximations are provided through fast lifting by which the sampling interval 0 h is divided into m subintervals with an equal width even though the central part of the method with the former approximation essentially coincides with a conventional method via fast sample fast hold fsfh approximation after all we show that both methods successfully lead to the upper and lower bounds of the l induced norm whose gap converges to 0 at the rate of 1 m in the former approximation and 1 m 2 in the latter extended approximation such achievements are in sharp contrast with an existing result on the former i e fsfh approximation which only shows the convergence rate of the error in the resulting estimate of the l induced norm without providing any readily computable upper and lower bounds a numerical example is given to illustrate the effectiveness of these methods

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

computing the l 0 h induced norm of a compression operator via fast lifting abstract this paper studies computing the induced norm of a compression operator defined on the banach space l 0 h which is a difficult problem since it is an infinite rank operator two methods are provided for this problem each of which can compute an upper bound and a lower bound of the induced norm by using an idea of staircase or piecewise linear approximation staircase approximation and piecewise linear approximation are applied through fast lifting by which the interval 0 h is divided into m subintervals with equal width and the approximation errors in these methods are ensured to be reciprocally proportional to m or m 2 the effectiveness of the proposed methods is demonstrated through numerical examples

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

semiclassical resolvent estimates for n body schrodinger operators in this paper we prove for a generalized n body schrodinger operator that the non trapping condition on the classical hamiltonian and all classical subhamiltonians is both necessary and sufficient for obtaining good semiclassical bounds on the boundary values of the resolvent and their energy derivatives we accomplish this by generalizing gerard s geometrical construction of an escape function for three body problems to n body problems this work is motivated by a recent paper of ch gerard 3 in which he proved for three body problems that if the classical hamiltonian and all classical sub hamiltonians are non trapping at the energy e then the resolvent of the corresponding semiclassical schrodinger operator satisfies the estimates

geodesic connectedness of multiwarped spacetimes abstract a new technique for the study of geodesic connectedness in a class of lorentzian manifolds is introduced it is based on arguments of brouwer s topological degree for the solution of functional equations it is shown to be very useful for multiwarped spacetimes which include different types of relativistic spacetimes connectedness by causal geodesics is also proved

quantum mechanical resonance and limiting absorption the many body problem we introduce a notion of quantum mechanical resonance that does not rely on analytic continuation of resolvent or scattering matrix and relate it to slow temporal decay of certain distinguished resonant states we proceed to prove existence of resonances for the generalized many body schrodinger operator for a rather large class of potentials containing coulomb and yukawa but also nonsymmetric and nonanalytic potentials with coulomblike singularities at the origin and certain differentiability and decay properties

induced norms for sampled data systems abstract in this paper we consider a general linear interconnection of a continuous time plant and a discrete time controller via sample and hold devices when the closed loop sampled data feedback system is internally stable bounded inputs produce bounded outputs we present some explicit formulae for the induced norm of the closed loop system with l i e peak value and 1 i e integral absolute norms on the input and output signals

minimization of the l sup infinity induced norm for sampled data systems it is shown that given any degree of accuracy there exists a standard discrete time l sup 1 problem that can be determined a priori whose solution yields a controller that is almost optimal in terms of the hybrid l sup infinity induced norm this is accomplished by first converting the hybrid system into an equivalent infinite dimensional discrete time system using the lifting technique in continuous time and then approximating the infinite dimensional parts of the system which model the intersample dynamics a thorough analysis of the approximation procedure is presented and it is shown that it is convergent at the rate of 1 n explicit bounds that are independent of the controller are obtained to characterize the approximation it is also shown that the geometry of the induced norm for the sampled data problem is different from that of the standard l sup 1 norm and hence there might not exist a linear isometry that maps the sampled data problem exactly to a standard discrete time problem

weyl asymptotics for the laplacian on asymptotically euclidean spaces we prove a trace type formula for the laplacian on an asymptotically euclidean space and use this to obtain weyl asymptotics of the scattering phase other results include an explicit calculation of the leading order singularity of the scattering matrix and results on the behavior of the resolvent and scattering matrix near 0

fr operator approach to the h sub 2 analysis and synthesis of sampled data systems recently a frequency domain operator called frequency response fr operator was defined and shown to represent the transfer characteristics of a stable sampled data system using this novel frequency domain notion and introducing its extended notion called hybrid fr operator we define an h sub 2 norm for sampled data systems in this paper then sampled data h sub 2 control problems are formulated and solved whereby the usefulness of these frequency domain notions is demonstrated both in the analysis and synthesis aspects of sampled data systems for the case of sampled data systems with hybrid i e both continuous time and discrete time input and output signals the h sub 2 norm defined by a hybrid fr operator turns out to be slightly different from that defined in previous studies the source of the discrepancy is also identified

geodesics on non static lorentz manifolds of reissner nordstrom type a geodesic y is called time like null or space like if er is less equal or greater than zero respectively the problem of the existence and multiplicity of geodesics in non geodesically complete static lorentz manifolds has been treated in 3 under assumptions satisfied by the schwartzschild spacetime here we consider the non static case under assumptions satisfied by the non static region of the reissner nordstr6m spacetime the reissner nordstrsm spacetime see 6 is the 4 dimensional lorentz manifold outside a spherically symmetric charged body whose curvature is due to

scattering metrics and geodesic flow at infinity any compact manifold with boundary admits a riemann metric on its interior taking the form x 4 dx 2 x 2 h near the boundary where x is a boundary defining function and h is a smooth symmetric 2 cotensor restricting to be positive definite and hence a metric h on the boundary the scattering theory associated to the laplacian for such a scattering metric was discussed by the first author and here it is shown as conjectured that the scattering matrix is a fourier integral operator which quantizes the geodesic flow on the boundary for the metric h at time to prove this the poisson operator of the associated generalized boundary problem is constructed as a fourier integral operator associated to a singular legendre manifold

on the spectral theory of dispersive n body hamiltonians in this work we describe a general class of dispersive n body hamiltonians for which we prove the hunziker van winter and zislin hwz theorem and a mourre estimate outside a closed and countable set of energies called thresholds as a consequence of the mourre estimate we prove a strong form of the limiting absorption principle which implies the absence of a singular continuous spectrum and gives criteria of local smoothness

fundamental solutions and eigenfunction expansions for schrodinger operators ii eigenfunction expansions where we assume that v x satisfies assumptions i and ii of part i namely v x is a real valued function of xern in studying eigenfunction expansions for h we further assume the existence and intertwining property of the modified channel wave operator w or w 9 whose definition will be given in a general form in sect 1 under these assumptions we first construct eigendistributions f for if which expand the range of w and prove a structure theorem for them using the fourier integral operator expression of the fundamental solution e ttb obtained in part i this structure theorem is then used in sect 2 to prove a regularity result for x fj indicating the variables in question the pairing i 9 py g rm pe rm m is taken to satisfy f for pesf rm where is the z inner product for hilbert spaces jf j b j jf denotes the space of bounded operators from 2tf into tf and b j b j j y

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

h 2 and hinf design of sampled data systems using lifting part ii properties of systems in the lifted domain an important class of sampled data systems becomes time invariant when the systems are lifted into an appropriate domain the purpose of the present paper is to study some basic properties of systems represented by state space equations in the lifted domain hidden modes invariant zeros and riccati equations are investigated and the connection between representations in continuous and lifted domains are clarified the material treated here is important for solving optimal control problems of generalized sampled data systems

smoothing effects and dispersion of singularities for the schrodinger evolution group we describe smoothing effects and dispersion of singularities for the schrodinger evolution group in the weighted sobolev spaces under a fairly general assumption on the potential it is shown that all singularities in the wavefunction vanish instantly whenever the initial state has sufficient decay we measure the regularity gained by the wavefunction by the decay property of the initial state no assumptions on the regularity of the initial state are imposed throughout the paper

positive commutators at the bottom of the spectrum abstract bony and hafner have recently obtained positive commutator estimates on the laplacian in the low energy limit on asymptotically euclidean spaces these estimates can be used to prove local energy decay estimates if the metric is non trapping we simplify the proof of the estimates of bony hafner and generalize them to the setting of scattering manifolds i e manifolds with large conic ends by applying a sharp poincare inequality our main result is the positive commutator estimate i h 2 g i 2 h 2 g a i h 2 g c i h 2 g 2 where h is a large parameter i is a compact interval in 0 and i its indicator function and where a is a differential operator supported outside a compact set and equal to 1 2 r d r r d r near infinity the laplacian can also be modified by the addition of a positive potential of sufficiently rapid decay the same estimate then holds for the resulting schrodinger operator

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

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

symbolic functional calculus and n body resolvent estimates abstract we construct a functional calculus for symbolic functions of scattering pseudodifferential operators on manifolds with boundary this is applied to obtain mapping results on the scattering wavefront set of the b operator used by gerard et al 1994 commutator algebra and resolvent estimates advanced studies in pure mathematics vol 23 math soc japan tokyo to obtain n body resolvent estimates as an application we construct the plane waves for an n body short range schrodinger operator and obtain a scattering wavefront set estimate

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

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

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

graded c algebras in the n body problem an algebraic formalism is presented which simplifies and makes natural various arguments in the theories where some notion of connected operator appears as a first example in this paper the case of n body systems is considered

h 2 and h infty design of sampled data systems using lifting part i general framework and solutions this paper presents a complete solution to the h2 and h infty problems for sampled data systems as opposed to previous works in the area it is assumed here that all or some of the sampling function the discrete time controller and the hold function are available for design the solution is obtained by transforming the problem to discrete time using the well known lifting technique it is then shown that the desired components of the sampled data controller can be peeled off from the inherently infinite dimensional description in the lifted domain the procedure for doing this last step is central to the approach in this paper both new and revised solutions are presented in this paper the solution to the h2 problems is completely new the solution to the h infty problems is presented in a unifying framework and is more transparent than the previous existing solutions in the literature transparency pays in the form of clearer results in particular a separation structure is established between the design of sub optimal hold and sampler

l_ 1 discretization for sampled data controller synthesis via piecewise linear approximation this paper develops a new discretization method with piecewise linear approximation for the l 1 optimal controller synthesis problem of sampled data systems which is the problem of minimizing the l infty induced norm of sampled data systems we apply fast lifting on the top of the lifting technique by which the sampling interval 0 h is divided into m subintervals with an equal width the signals on each subinterval are then approximated by linear functions by introducing two types of linearizing operators for input and output which leads to piecewise linear approximation of sampled data systems by using the arguments of preadjoint operators we provide an important inequality that forms a theoretical basis for tackling the l 1 optimal controller synthesis problem of sampled data systems more efficiently than the conventional method more precisely a mathematical basis for the piecewise linear approximation method associated with the convergence rate is shown through this inequality and this suggests that the piecewise linear approximation method may drastically outperform the conventional method in the l 1 optimal controller synthesis problem of sampled data systems we then provide a discretization procedure of sampled data systems by which the l 1 optimal controller synthesis problem is converted to the discrete time l 1 optimal controller synthesis problem finally effectiveness of the proposed method is demonstrated through a numerical example

on p convex sets and geodesics on etudie les geodesiques sur une variete contrainte m a bord irregulier on considere une classe d ensembles irreguliers m qui permet d obtenir des resultats d existence et de multiplicite pour les geodesiques

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

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 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

riesz transform and l p cohomology for manifolds with euclidean ends let m be a smooth riemannian manifold which is the union of a compact part and a finite number of euclidean ends rr n setminus b 0 r for some r 0 each of which carries the standard metric our main result is that the riesz transform on m is bounded from l p m to l p m t m for 1 n the result is new for 2 2 for a more general class of manifolds assume that m is a n dimensional complete manifold satisfying the nash inequality and with an o r n upper bound on the volume growth of geodesic balls we show that boundedness of the riesz transform on l p for some p 2 implies a hodge de rham interpretation of the l p cohomology in degree 1 and that the map from l 2 to l p cohomology in this degree is injective

a topological method for geodesic connectedness of space times outer kerr space time the geodesic connectedness of outer kerr space time is proven by using a topological method the proof is based on arguments about brouwer s degree for the solutions of functional equations the applicability of these topological arguments for dealing with geodesics in space times is stressed

frequency response of sampled data systems this paper develops a frequency domain theory that provides a method to analyze and design sampled data control systems including their intersample behaviors the key idea is to consider the signal space x x t x t n x xn exp j t jn st n x xn 2 where s is the sampling angular frequency it is shown that a stable sampled data system equipped with a strictly proper pre filter before the sampler maps x into x l2 in the steady state that mapping is denoted by q j and is referred to as an fr operator it is proved that the norm of the sampled data system as an operator from l2 to l2 is given by max q j 2 2 where 12 s 12 s a set of equations relating outputs of a closed loop system to its inputs in the frequency domain is derived and their solution is given in an explicit form based on that solution the sensitivity fr operator y j and the complementary sensitivity fr operator t j are defined for feedback control systems and it is shown that y j gives the improvement of the sensitivity of the transfer characteristics from the reference to the controlled output and also represents the ability of rejecting disturbances and that t j represents the degree of robust stability and at the same time gives the effect of detection noises it is also shown that y j t j i and thus a frequency domain paradigm for the design of sampled data control systems which is exactly parallel to the continuous time case is established

technical communique approximation of frequency response for sampled data control systems this paper proves that the frequency response gains of fast sample fast hold approximations of a sampled data system converge to that of the original system as the sampling rate gets faster while this may appear to hold trivially there is a serious technical difficulty and the proof is indeed nontrivial it is also guaranteed that this convergence is uniform on the total frequency range the latter property is necessary to guarantee that a single approximant can be used for frequency response computation for the overall frequency range

quantum scattering at low energies for a class of negative slowly decaying potentials including v x x with 0 2 we study the quantum mechanical scattering theory in the low energy regime using appropriate modifiers of the isozaki kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the hamiltonian including the energy 0 we show that the modified scattering matrices s are welldefined and strongly continuous down to the zero energy threshold similarly we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces these results are used to derive oscillatory asymptotics of the standard short range and dollard type s matrices for the subclasses of potentials where both kinds of s matrices are defined for potentials whose leading part is x we show that the location of singularities of the kernel of s experiences an abrupt change from passing from positive energies to the limiting energy 0 this change corresponds to the behaviour of the classical orbits under stronger conditions one can extract the leading term of the asymptotics of the kernel of s at its singularities

further results on the l 1 analysis of sampled data systems via kernel approximation approach abstractthis paper gives two methods for the l1 analysis of sampled data systems by which we mean computing the l induced norm of sampled data systems this is achieved by developing what we call the kernel approximation approach in the setting of sampled data systems we first consider the lifting treatment of sampled data systems and give an operator theoretic representation of their input output relation we further apply the fast lifting technique by which the sampling interval 0 h is divided into m subintervals with an equal width and provide methods for computing the l induced norm in contrast to a similar approach developed earlier called the input approximation approach we use an idea of kernel approximation in which the kernel function of an input operator and the hold function of an output operator are approximated by piecewise constant or piecewise linear functions furthermore it is shown that the approximation errors in the piecewise constant approximation or piecewise linear appro

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

modified fast sample fast hold approximation and γ independent h discretisation for general sampled data systems by fast lifting this article is concerned with the fast lifting approach to h analysis and design of sampled data systems and extends our preceding study on modified fast sample fast hold fsfh approximation in which the direct feedthrough matrix d 11 from the disturbance w to the controlled output z was assumed to be zero more precisely this article removes this assumption and shows that a independent h discretisation is still possible in a nontrivial fashion by applying what we call quasi finite rank approximation of an infinite rank operator and then the loop shifting technique as in the case of d 11 0 the modified fsfh approach retains the feature that both the upper and lower bounds of the h norm or the frequency response gain can be computed where the gap between the upper and lower bounds can be bounded with the approximation parameter n and is independent of the discrete time controller this feature is significant in applying the new method especially to control system design and this study in

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

spectral theory of time periodic many body systems we study the spectrum of the monodromy operator for an n body quantum system in a time periodic external field with time mean equal to zero this includes ac stark and circularly polarized fields and pair potentials with a local singularity up to and including the coulomb singularity in the framework of floquet theory we prove a local commutator estimate and use it to prove a limiting absorption principle for the floquet hamiltonian as well as exponential decay estimates on non threshold eigenfunctions these two results are then used to obtain a second order perturbation theory for embedded eigenvalues the principal tool is a new extended mourre theory

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

local time decay of high energy scattering states for the schrodinger equation on considere l operateur de schrodinger h h 0 v s v l dans l espace de hilbert 7b h l 2 r n n n ou h 0 c 0 r n et v s et v l sont des fonctions a valeurs reelles satisfaisant certaines conditions

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

minimization of the l induced norm for sampled data systems the following problem is addressed given a continuous time plant with continuous time performance objectives expressed in terms of the l induced norm design a digital controller that delivers or optimizes this performance this problem differs from the standard discrete time methods in that it takes into consideration the inter sample behavior of the closed loop system the resulting closed loop system dynamics consist of both continuous time and discrete time dynamics and thus such systems are known as hybrid systems it is shown that given any degree of accuracy there exists a standard discrete time l 1 problem which can be determined a priori whose solution yields a controller that is almost optimal in terms of the hybrid l induced norm

exponential bounds and absence of positive eigenvalues for n body schrodinger operators for a large class ofn body potentialsv we prove that if is an eigenfunction of v with eigenvaluee then sup 2 e 0 exp x l2 is either a threshold or consequences of this result are the absence of positive eigenvalues and optimal l2 exponential lower bounds

asymptotic behavior of generalized eigenfunctions inn body scattering abstract in this paper an asymptotic expansion is proved for locally at infinity outgoing functions on asymptotically euclidian spaces this is applied to n body scattering where the two body interactions are one step polyhomogeneous symbols of order 1 or 2 hence long range and short range respectively the asymptotic behavior of the n body resolvent applied to schwartz functions is thereby deduced away from the singular set where some of the potentials do not decay at infinity

l sub 1 analysis and design of sampled data systems the focus of this work is l sub 1 optimal control of sampled data systems a converging approximation procedure is derived to compute the l sub infinity induced norm of closed loop finite dimensional linear time invariant lti sampled data systems an approximation method is developed to synthesize l sub 1 optimal sampled data regulators finally an example is provided that illustrates the l sub 1 analysis and design techniques presented

upper lower bounds of generalized h2 norms in sampled data systems with convergence rate analysis and discretization viewpoint abstract this paper considers linear time invariant lti sampled data systems and studies their generalized h 2 norms they are defined as the induced norms from l 2 to l in which two types of the l norm of the output are considered as the temporal supremum magnitude under the spatial norm and 2 norm the input output relation of sampled data systems is first formulated under their lifting based treatment we then develop a method for computing the generalized h 2 norms with operator theoretic gridding approximation this method leads to readily computable upper bounds as well as lower bounds of the generalized h 2 norms whose gaps tend to 0 at the rate of 1 n with the gridding approximation parameter n an approximately equivalent discretization method of the generalized plant is further provided as a fundamental step to addressing the controller synthesis problem of minimizing the generalized h 2 norms of sampled data systems finally a numerical example is given to show the effectiveness of the computation method

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

perturbation of embedded eigenvalues in the generalized n body problem we discuss the perturbation of continuum eigenvalues without analyticity assumptions among our results we show that generally a small perturbation removes these eigenvalues in accordance with fermi s golden rule thus generically in a baire category sense the schrodinger operator has no embedded non threshold eigenvalues

asymptotic properties of generalized eigenfunctions for three body schrodinger operators we study the spatial asymptotics of generalized eigenfunctions of three body schrodinger operators and derive all the s matrices with initial state of 2 clusters

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

low energy asymptotics for schrodinger operators with slowly decreasing potentials low energy behavior of schrodinger operators with potentials which decay slowly at infinity is studied it is shown that if the potential is positive then the zero energy is very regular and the resolvent is smooth near 0 this implies rapid local decay for the solutions of the schrodinger equation on the other hand if the potential is negative then the resolvent has discontinuity at zero energy thus one cannot expect local decay faster than ordert 1 ast

the h 2 problem for sampled data systems m for sampled data systems in this paper we pose and give a complete solution to an analog of the h2 problem for sampled data system we motivate and develop a natural generalization of the h2 cost to periodic systems which is then applied to the continuous time closed loop mapping in a sampled data control system it is shown that the cost criterion developed is actually a norm in an space of hilbert schmidt operator valued functions we give state space solutions to the optimal and suboptimal controllers synthesis problems in this new norm by establishing an equivalent standard h2 problem

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

h sub 2 optimal sampled data control the h sub 2 optimal control of continuous time linear time invariant systems by sampled data controllers is discussed two different solutions state space and operator theoretic are given in both cases the h sub 2 sampled data problem is shown to be equivalent to a certain discrete time h sub 2 problem other topics discussed include input output stability of sampled data systems performance recovery in digital implementation of analog controllers and sampled data control of systems with the possibility of multiple time delays

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

absolute continuity of the floquet spectrum for a nonlinearly forced harmonic oscillator we prove that the floquet spectrum of the time periodic schrodinger equation corresponding to a mildly nonlinear resonant forcing is purely absolutely continuous for suitably small

h optimal sampled data control in continuous time systems we consider the design of h optimal discrete time digital controllers in continuous time systems an apparent difficulty especially in utilizing modern transform domain analysis in this context stems from the absence of an appropriate transfer function model for the hybrid time discrete and continuous closed loop system this difficulty is overcome through the introduction of an equivalent difference equation model for the continuous time system with distributed inputs and outputs equivalence being in the sense that the continuous and discrete time inputs and outputs are essentially identical using the interplay between the discrete and the continuous time models solutions of the well known purely continuous time and purely discrete time standard problems extend to solutions of the problem at hand they comprise riccati equation characterizations of feasible combinations of sampling rates and bounds on the closed loop induced input output norm and parameterization of compensators we consid

spectral asymptotics for sub riemannian laplacians i quantum ergodicity and quantum limits in the 3d contact case this is the first paper of a series in which we plan to study spectral asymptotics for sub riemannian laplacians and to extend results that are classical in the riemannian case concerning weyl measures quantum limits quantum ergodicity quasi modes trace formulae even if hypoelliptic operators have been well studied from the point of view of pde s global geometrical and dynamical aspects have not been the subject of much attention as we will see already in the simplest case the statements of the results in the sub riemannian setting are quite different from those in the riemannian one let us consider a sub riemannian sr metric on a closed three dimensional manifold with an oriented contact distribution there exists a privileged choice of the contact form with an associated reeb vector field and a canonical volume form that coincides with the popp measure we establish a quantum ergodicity qe theorem for the eigenfunctions of any associated sr laplacian under the assumption that the reeb flow is ergodic the limit measure is given by the normalized popp measure this is the first time that such a result is established for a hypoelliptic operator whereas the usual shnirelman theorem yields qe for the laplace beltrami operator on a closed riemannian manifold with ergodic geodesic flow to prove our theorem we first establish a microlocal weyl law which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sr laplacian then we derive a birkhoff normal form along this characteristic manifold thus showing that in some sense all 3d contact structures are microlocally equivalent the quantum version of this normal form provides a useful microlocal factorization of the sr laplacian using the normal form the factorization and the ergodicity assumption we finally establish a variance estimate from which qe follows we also obtain a second result which is valid without any ergodicity assumption every quantum limit ql can be interpreted as the sum of two mutually singular measures the first measure is supported on the unit cotangent bundle and is invariant under the sr geodesic flow and the second measure is supported on the characteristic manifold of the sr laplacian and is invariant under the lift of the reeb flow moreover we prove that the first measure is zero for most ql s