quasi orthogonal decompositions of structured frames a decomposition of a hilbert space h into a quasi orthogonal family of closed subspaces is introduced we shall investigate conditions in order to derive bounded families of corresponding quasi projectors or resolutions of the identity operator given a local family of atoms or generalized stable basis for each subspace we show that the union of the local atoms can generate a global frame for the hilbert space corresponding duals can be calculated in a flexible way by means of systems of quasi projectors an application to gabor frames is presented as example of the use of this technique for calculation of duals and explicit estimates of lattice constants

carleman estimates and absence of embedded eigenvalues let l w be a schrodinger operator with a potential w in l frac n 1 2 mathbb r n n geq 2 we prove that there is no positive eigenvalue the main tool is an l p l p prime carleman type estimate which implies that eigenfunctions to positive eigenvalues must be compactly supported the carleman estimate builds on delicate dispersive estimates established in 7 we also consider extensions of the result to variable coefficient operators with long range and short range potentials and gradient potentials

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

stability of g frames g frames are natural generalizations of frames which cover many other recent generalizations of frames e g bounded quasi projectors frames of subspaces outer frames oblique frames pseudo frames and a class of time frequency localization operators moreover it is known that g frames are equivalent to stable space splittings in this paper we study the stability of g frames we first present some properties for g bessel sequences then we prove that g frames are stable under small perturbations we also study the stability of dual g frames

measure functions for frames this paper addresses the natural question how should frames be compared we answer this question by quantifying the overcompleteness of all frames with the same index set we introduce the concept of a frame measure function a function which maps each frame to a continuous function the comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set we define the ultrafilter measure function an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions we explore additional properties of frame measure functions showing that they are additive on a large class of supersets those that come from so called non expansive frames we apply our results to the gabor setting computing the frame measure function of gabor frames and establishing a new result about supersets of gabor frames 2007 published by elsevier inc

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

stationary scattering in the semi classical limit for a schrodinger operator in a gauge field abstract we study the classical limit of the stationary scattering theory for a schrodinger operator in a compactly supported gauge field we show that under suitable hypotheses on the associated classical flow the scattering amplitude has a complete asymptotic expansion in the semi classical parameter and we determine the main term of this expansion

on the duality of fusion frames the fusion frames were considered recently by p g casazza g kutyniok and s li in connection with distributed processing and are related to the construction of global frames from local frames in this paper we give new results on the duality of fusion frames in hilbert spaces

persistence of embedded eigenvalues we consider conditions under which an embedded eigenvalue of a self adjoint operator remains embedded under small perturbations in the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m infinity we show that in favorable situations the set of small perturbations of a suitable banach space which do not remove the eigenvalue form a smooth submanifold of codimension in we also have results regarding the cases when the eigenvalue is degenerate or when the multiplicity of the continuous spectrum is infinite

mourre s method and smoothing properties of dispersive equations abstract the purpose of this paper is to point out that mourre s method in spectral theory is useful and powerful in studying the smoothing effect of dispersive equations especially we will see that the smoothing effect is a quite general phenomenon even for operators with variable coefficients

eigenvalue bounds for schrodinger operators with complex potentials iii laptev and safronov conjectured that any non positive eigenvalue of a schrodinger operator v in l 2 r with complex potential has absolute value at most a constant times v 2 2 for 0 2 in dimension 2 we prove this conjecture for radial potentials if 0 2 and we almost disprove it for general potentials if 1 2 2 in addition we prove various bounds that hold in particular for positive eigenvalues

scattering matrices for dissipative quantum systems abstract we consider a quantum system s interacting with another system s and susceptible of being absorbed by s the effective dissipative dynamics of s is supposed to be generated by an abstract pseudo hamiltonian of the form h h 0 v i c c the generator of the free dynamics h 0 is self adjoint v is symmetric and c is bounded we study the scattering theory for the pair of operators h h 0 we establish a representation formula for the scattering matrices and identify a necessary and sufficient condition to their invertibility this condition rests on a suitable notion of spectral singularity our main application is the nuclear optical model where h is a dissipative schrodinger operator and spectral singularities correspond to real resonances

some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties abstract we examine several modified versions of the heteroskedasticity consistent covariance matrix estimator of hinkley 1977 and white 1980 on the basis of sampling experiments which compare the performance of quasi t statistics we find that one estimator based on the jackknife performs better in small samples than the rest we also examine the finite sample properties of using modified critical values based on edgeworth approximations as proposed by rothenberg 1984 in addition we compare the power of several tests for heteroskedasticity and find that it may be wise to employ the jackknife heteroskedasticity consistent covariance matrix even in the absence of detected heteroskedasticity

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

spectral and scattering theory in perturbed stratified fluids ii transmission problems and exterior domains abstract the spectral and scattering theory of the acoustic propagator in perturbed stratified fluids is studied in the cases of transmission problems and of exterior domains

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

duality in reconstruction systems we consider the notion of finite dimensional reconstructions systems rs s which includes the fusion frames as projective rs s we study erasures some geometrical properties of these spaces the spectral picture of the set of all dual systems of a fixed rs the spectral picture of the set of rs operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting we give several examples

frames and bases of subspaces in hilbert spaces abstract in this paper we develop the frame theory of subspaces for separable hilbert spaces we will show that for every parseval frame of subspaces w i i i for a hilbert space h there exists a hilbert space k h and an orthonormal basis of subspaces n i i i for k such that w i p n i where p is the orthogonal projection of k onto h we introduce a new definition of atomic resolution of the identity in hilbert spaces in particular we define an atomic resolution operator for an atomic resolution of the identity which even yield a reconstruction 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

asymptotic completeness in dissipative scattering theory abstract we consider an abstract pseudo hamiltonian for the nuclear optical model given by a dissipative operator of the form h h v i c c where h v h 0 v is self adjoint and c is a bounded operator we study the wave operators associated to h and h 0 we prove that they are asymptotically complete if and only if h does not have spectral singularities on the real axis for schrodinger operators the spectral singularities correspond to real resonances

linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method abstract we study the large time behavior of solutions to two dimensional euler and navier stokes equations linearized about shear flows of the mixing layer type in the unbounded channel t r under a simple spectral stability assumption on a self adjoint operator we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity we also prove a local form of the enhanced viscous dissipation that takes place at times of order 1 3 being the small viscosity to prove these results we use a hamiltonian approach following the conjugate operator method developed in the study of schrodinger operators combined with a hypocoercivity argument to handle the viscous case

decay estimates and strichartz estimates of fourth order schrodinger operator abstract we study time decay estimates of the fourth order schrodinger operator h 2 v x in r d for d 3 and d 5 we analyze the low energy and high energy behaviour of resolvent r h z and then derive the jensen kato dispersion decay estimate and local decay estimate for e i t h p a c under suitable spectrum assumptions of h based on jensen kato type decay estimate and local decay estimate we obtain the l 1 l estimate of e i t h p a c in 3 dimension by ginibre argument and also establish the endpoint global strichartz estimates of e i t h p a c for d 5 furthermore using the local decay estimate and the georgescu larenas soffer conjugate operator method we prove the jensen kato type decay estimates for some functions of h

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

zero energy asymptotics of the resolvent for a class of slowly decaying potentials we prove a limiting absorption principle at zero energy for two body schrodinger operators with long range potentials having a positive virial at infinity more precisely we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter tends to zero in cones which are adjacent to the positive real axis the principal tools are absence of eigenvalue at zero singular mourre theory and microlocal estimates

excess of a class of g frames abstract in this paper we study the relationship between frames for the super hilbert space h h and g frames for h with respect to c 2 we show that a g frame associated with a frame for h h remains a g frame whenever any one of its elements is removed furthermore we show that the excess of such a g frame is at least dim h

fusion frames and g frames fusion frames and g frames were considered recently as generalizations of frames in hilbert spaces in this paper we generalize some of the known results in frame theory to fusion frames and g frames we obtain new g frames by considering g frames for its components we also obtain some results about alternate dual g frames excess of g frames and stability of g frames under small perturbations

the shape resonance for a class of schrodinger operatorsh 2 2m v onl 2 n with potentials having minima embedded in the continuum of the spectrum and non trapping tails we show the existence of shape resonances exponentially close to the real axis as 0 the resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling

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

spectral analysis of second order elliptic operators on noncompact manifolds on demontre l estimation de mourre et les bornes qui y sont reliees pour une classe d operateurs elliptiques qui inclut des laplaciens pour des quotients cofinis et coinfinis d espaces hyperboliques

moderation analysis using a two level regression model moderation analysis is widely used in social and behavioral research the most commonly used model for moderation analysis is moderated multiple regression mmr in which the explanatory variables of the regression model include product terms and the model is typically estimated by least squares ls this paper argues for a two level regression model in which the regression coefficients of a criterion variable on predictors are further regressed on moderator variables an algorithm for estimating the parameters of the two level model by normal distribution based maximum likelihood nml is developed formulas for the standard errors ses of the parameter estimates are provided and studied results indicate that when heteroscedasticity exists nml with the two level model gives more efficient and more accurate parameter estimates than the ls analysis of the mmr model when error variances are homoscedastic nml with the two level model leads to essentially the same results as ls with the mmr model most importantly the two level regression model permits estimating the percentage of variance of each regression coefficient that is due to moderator variables when applied to data from general social surveys 1991 nml with the two level model identified a significant moderation effect of race on the regression of job prestige on years of education while ls with the mmr model did not an r package is also developed and documented to facilitate the application of the two level model

jackknife bootstrap and other resampling methods in regression analysis motivated by a representation for the least squares estimator we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time they are unbiased for homoscedastic errors and a special case the delete one jackknife is almost unbiased for heteroscedastic errors the method is extended to cover nonlinear parameters regression m estimators nonlinear regression and generalized linear models interval estimators can be constructed from the jackknife histogram three bootstrap methods are considered two are shown to give biased variance estimators and one does not have the bias robustness property enjoyed by the weighted delete one jackknife a general method for resampling residuals is proposed it gives variance estimators that are bias robust several bias reducing estimators are proposed some simulation results are reported

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

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

time decay of semigroups generated by dissipative schrodinger operators abstract we establish a representation formula for semigroups of contractions in terms of a global limiting absorption principle as applications we prove long time asymptotics and dispersive estimate of the semigroup generated by i h where h is a dissipative schrodinger operator

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

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

high frequency limit of the helmholtz equation with variable refraction index abstract we study the high frequency limit of the helmholtz equation with variable refraction index and a source term concentrated near a p dimensional affine subspace under some conditions we first derive uniform estimates in besov spaces for the solutions then we prove that the semi classical measure associated with these solutions satisfies the stationary liouville equation with an explicit source term and has certain radiation property at infinity

spectral and scattering theory for differential and hankel operators abstract we consider a class of hankel operators h realized in the space l 2 r as integral operators with kernels h t s where h t p ln t t 1 and p x x n p n 1 x n 1 p 0 is an arbitrary real polynomial of degree n this class contains the classical carleman operator when n 0 we show that a hankel operator h in this class can be reduced by an explicit unitary transformation essentially by the mellin transform to a differential operator a v q d v in the space l 2 r here q x x n q n 1 x n 1 q 0 is a polynomial determined by p x and v 1 2 cosh 1 2 is the universal function then the operator a v q d v reduces by the generalized liouville transform to the standard differential operator b d n b n 1 x d n 1 b 0 x with the coefficients b m x m 0 n 1 decaying sufficiently rapidly as x this allows us to use the results of spectral theory of differential operators for the study of spectral properties of generalized carleman operators in particular we show that the absolutely continuous spectrum of h is simple and coincides with r if n is odd and it has multiplicity 2 and coincides with 0 if n 2 is even the singular continuous spectrum of h is empty and its eigenvalues may accumulate to the point 0 only as a by product of our considerations we develop spectral theory of a new class of degenerate differential operators a v q d v where q x is an arbitrary real polynomial and v is an almost arbitrary real function decaying at infinity

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

g frames and g riesz bases g frames are generalized frames which include ordinary frames bounded invertible linear operators as well as many recent generalizations of frames e g bounded quasi projectors and frames of subspaces g frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients we give characterizations of g frames and prove that g frames share many useful properties with frames we also give a generalized version of riesz bases and orthonormal bases as an application we get atomic resolutions for bounded linear operators

schrodinger operators with an electric field and random or deterministic potentials we prove that the schrodinger operatorh d2 dx2 v x f x has purely absolutely continuous spectrum for arbitrary constant external fieldf for a large class of potentials this result applies to many periodic almost periodic and random potentials and in particular to random wells of independent depth for which we prove that whenf 0 the spectrum is almost surely pure point with exponentially decaying eigenfunctions

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

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

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