spectral theory for 1 body stark operators abstract we investigate spectral theory for a one body stark hamiltonian under minimum regularity and decay conditions on the potential actually allowing sub linear growth at infinity our results include rellich s theorem the limiting absorption principle radiation condition bounds and sommerfeld s uniqueness and most of these are stated and proved in sharp form employing besov type spaces for the proofs we adopt a commutator scheme by ito skibsted 13 a feature of the paper is a special choice of an escape function related to parabolic coordinates which conforms well with classical mechanics for the stark hamiltonian the whole setting of the paper such as the conjugate operator and the besov type spaces is generated by this single escape function we apply our results in the sequel paper 5

the scattering matrix for the schrodinger operator with a long range electromagnetic potential we consider the schrodinger operator h i a 2 v in the space l2 rd with long range electrostatic v x and magnetic a x potentials using the scheme of smooth perturbations we give an elementary proof of the existence and completeness of modified wave operators for the pair h0 h our main goal is to study spectral properties of the corresponding scattering matrix s we obtain its stationary representation and show that its singular part up to compact terms is a pseudodifferential operator with an oscillating amplitude which is an explicit function of v and a finally we deduce from this result that in general for each 0 the spectrum of s covers the whole unit circle

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

isometries fock spaces and spectral analysis of schrodinger operators on trees abstract we construct conjugate operators for the real part of a completely non unitary isometry and we give applications to the spectral and scattering theory of a class of operators on complete fock spaces natural generalizations of the schrodinger operators on trees we consider c algebras generated by such hamiltonians with certain types of anisotropy at infinity we compute their quotient with respect to the ideal of compact operators and give formulas for the essential spectrum of these hamiltonians

estimations de la resolvante pour une molecule diatomique dans l approximation de born oppenheimer making use of an adiabatic operator that takes several electronic states into account we derive a born oppenheimer approximation of the resolvent for a diatomic molecule this is an improvement of a result in kmw1 such a resolvent approximation is useful to obtain an adiabatic approximation of total cross sections see jec2 the strategy we use based on mourre s commutator method and on a new kind of global escape function may be carried over to control the resolvent of some matricial schrodinger operators in the same way we obtain a semiclassical estimate for the resolvent of the semiclassical dirac operator with scalar electric potential extending a result of ce

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

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

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

spiraling attractors and quantum dynamics for a class of long range magnetic fields abstract we consider the long time behavior of a quantum particle in a 2d magnetic field which is homogeneous of degree 1 if the field never vanishes above a certain energy the associated classical dynamical system has a globally attracting periodic orbit in a reduced phase space for that energy regime we construct a simple approximate evolution based on this attractor and prove that it completely describes the quantum dynamics of our system

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

index theory for quasi crystals i computation of the gap label group abstract in this paper we give a complete solution to the gap labelling conjecture for quasi crystals the method adopted relies on the index theory for laminations and the main tools are the connes skandalis longitudinal k theory index morphism together with the measured index formula

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

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

on the mathematical theory of the aharonov bohm effect we consider the schr dinger operator h i a 2 in the space l2 2 with a magnetic potential a x a x2 x1 x 2 where a is an arbitrary function on the unit circle our goal is to study spectral properties of the corresponding scattering matrix s 0 we obtain its stationary representation and show that its singular part up to compact terms is a pseudodifferential operator of zero order whose symbol is an explicit function of a we deduce from this result that the essential spectrum of s does not depend on and consists of two complex conjugated and perhaps overlapping closed intervals of the unit circle finally we calculate the diagonal singularity of the scattering amplitude kernel of s considered as an integral operator in particular we show that for all these properties only the behaviour of a potential at infinity is essential the preceding papers on this subject treated the case a const and used the separation of variables in the schr dinger equation in the polar coordinates this technique does not of course work for arbitrary a from an analytical point of view our paper relies on some modern tools of scattering theory and well known properties of pseudodifferential operators

gap labelling conjecture with nonzero magnetic field abstract given a constant magnetic field on euclidean space r p determined by a skew symmetric p p matrix and a z p invariant probability measure on the disorder set which is by hypothesis a cantor set where the action is assumed to be minimal the corresponding integrated density of states of any self adjoint operator affiliated to the twisted crossed product algebra c z p where is the multiplier on z p associated to takes on values on spectral gaps in the magnetic gap labelling group the magnetic frequency group is defined as an explicit countable subgroup of r involving pfaffians of and its sub matrices we conjecture that the magnetic gap labelling group is a subgroup of the magnetic frequency group we give evidence for the validity of our conjecture in 2d 3d the jordan block diagonal case and the periodic case in all dimensions

on three body scattering cross sections using a hilbert space version of the faddeev method the authors prove finiteness and continuity as a function of the energy of the total scattering cross section for three quantum mechanical particles with two body initial states it is assumed that the pair potentials are locally in lq r3 for some q 3 2 and decay as mod x mod 2 delta at infinity

am fm energy detection and separation in noise using multiband energy operators this paper develops a multiband or wavelet approach for capturing the am fm components of modulated signals immersed in noise the technique utilizes the recently popularized nonlinear energy operator psi s s sup 2 ss to isolate the am fm energy and an energy separation algorithm esa to extract the instantaneous amplitudes and frequencies it is demonstrated that the performance of the energy operator esa approach is vastly improved if the signal is first filtered through a bank of bandpass filters and at each instant analyzed via psi and the esa using the dominant local channel response moreover it is found that uniform worst case performance across the frequency spectrum is attained by using a constant q or multiscale wavelet like filter bank the elementary stochastic properties of psi and of the esa are developed first the performance of psi and the esa when applied to bandpass filtered versions of an am fm signal plus noise combination is then analyzed the predicted performance is greatly improved by filtering if the local signal frequencies occur in band these observations motivate the multiband energy operator and esa approach ensuring the in band analysis of local am pm energy in particular the multi bands must have the constant q or wavelet scaling property to ensure uniform performance across bands the theoretical predictions and the simulation results indicate that improved practical strategies are feasible for tracking and identifying am fm components in signals possessing pattern coherencies manifested as local concentrations of frequencies

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

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 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

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

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

spaces of tilings finite telescopic approximations and gap labeling the continuous hull of a repetitive tiling t in d with the finite pattern condition fpc inherits a minimal d lamination structure with flat leaves and a transversal open image in new window which is a cantor set this class of tiling includes the penrose the amman benkker ones in 2d as well as the icosahedral tilings in 3d we show that the continuous hull with its canonical d action can be seen as the projective limit of a suitable sequence of branched oriented and flat compact d manifolds as a consequence the longitudinal cohomology and the k theory of the corresponding c algebra open image in new window are obtained as direct limits of cohomology and k theory of ordinary manifolds moreover the space of invariant finite positive measures can be identified with a cone in the d th homology group canonically associated with the orientation of d at last the gap labeling theorem holds given an invariant ergodic probability measure on the hull the corresponding integrated density of states ids of any selfadjoint operators affiliated to open image in new window takes on values on spectral gaps in the module generated by the occurrence probabilities of finite patches in the tiling

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

spectral and scattering theory for symbolic potentials of order zero the spectral and scattering theory is investigated for a generalization to scattering metrics on two dimensional compact manifolds with boundary of the class of smooth potentials on r2 which are homogeneous of degree zero near infinity the most complete results require the additional assumption that the restriction of the potential to the circle s at infinity be morse generalized eigenfunctions associated to the essential spectrum at non critical energies are shown to originate both at minima and maxima although the latter are not germane to the l2 spectral theory asymptotic completeness is shown both in the traditional l2 sense and in the sense of tempered distributions this leads to a definition of the scattering matrix the structure of which will be described in a future publication

proof of the magnetic gap labelling conjecture for principal solenoidal tori abstract in this note we prove the magnetic spectral gap labelling conjecture as stated in 8 in all dimensions for principal solenoidal tori

attractors for two dimensional waves with homogeneous hamiltonians of degree 0 the density stratification in an incompressible fluid is responsible for the propagation of internal waves in domains with topography these waves exhibit interesting features in particular numerical and lab experiments show that in two dimensions for generic forcing frequencies these waves concentrate on attractors the goal of this paper is to analyze mathematically this behavior using tools from spectral theory and microlocal analysis the same results apply also to inertial waves in rotating fluids

a proof of the abstract limiting absorption principle by energy estimates abstract we give a proof in an abstract setting of various resolvent estimates including the limiting absorption principle starting from a mourre positive commutator estimate using standard energy estimates arguments

decay and non decay of the local energy for the wave equation on the de sitter schwarzschild metric we describe an expansion of the solution of the wave equation on the de sitter schwarzschild metric in terms of resonances the principal term in the expansion is due to a resonance at 0 the error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an varepsilon derivative loss in the angular directions

spectral and scattering theory for symmetric systems of nonconstant deficit abstract we study the spectral and scattering problems for symmetric systems of nonconstant deficit we prove i discreteness of eigenvalues ii absolute continuity of continuous spectrum iii principle of limiting absorption as an application to the scattering problems the decay of local energy and the completeness of wave operators are obtained the class of systems we consider here includes the linearized equation of magnetohydrodynamics as a typical example

sharp low frequency resolvent estimates on asymptotically conical manifolds on a class of asymptotically conical manifolds we prove two types of low frequency estimates for the resolvent of the laplace beltrami operator the first result is a uniform l 2 rightarrow l 2 bound for scal r 1 delta g z 1 scal r 1 when mbox re z is small with the optimal weight scal r 1 the second one is about powers of the resolvent for any integer n we prove uniform l 2 rightarrow l 2 bounds for scal epsilon r n epsilon 2 delta g z n scal epsilon r n when mbox re z belongs to a compact subset of 0 infty and 0 epsilon ll 1 these results are obtained by proving similar estimates on a pure cone with a long range perturbation of the metric at infinity

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

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

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

on amplitude and frequency demodulation using energy operators it is shown that the nonlinear energy tracking signal operator psi x dx dt sup 2 xd sup 2 x dt sup 2 and its discrete time counterpart can estimate the am and fm modulating signals specifically psi can approximately estimate the amplitude envelope of am signals and the instantaneous frequency of fm signals bounds are derived for the approximation errors which are negligible under general realistic conditions these results coupled with the simplicity of psi establish the usefulness of the energy operator for am and fm signal demodulation these ideas are then extended to a more general class of signals that are sine waves with a time varying amplitude and frequency and thus contain both an am and an fm component for such signals it is shown that psi can approximately track the product of their amplitude envelope and their instantaneous frequency the theoretical analysis is done for both continuous and discrete time signals

on the spectral properties of hamiltonians without conservation of the particle number we consider quantum systems with variable but finite number of particles for such systems we develop geometric and commutator techniques we use these techniques to find the location of the spectrum to prove absence of singular continuous spectrum and identify accumulation points of the discrete spectrum the fact that the total number of particles is bounded allows us to give relatively elementary proofs of these basic results for an important class of many body systems with nonconserved number of particles

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

a generalized demodulation approach to time frequency projections for multicomponent signals in this paper we introduce a flexible approach for the time frequency analysis of multicomponent signals involving the use of analytic vectors and demodulation the demodulated analytic signal is projected onto the time frequency plane so that as closely as possible each component contributes exclusively to a different tile in a wavelet packet tiling of the time frequency plane and at each time instant the contribution to each tile definitely comes from no more than one component a single reverse demodulation is then applied to all projected components the resulting instantaneous frequency of each component in each tile is not constrained to a set polynomial form in time and is readily calculated as is the corresponding hilbert energy spectrum two examples illustrate the method in order better to understand the effect of additive noise the approximate variance of the estimated instantaneous frequency in any tile has been formulated by starting with pure noise and studying its evolving covariance structure through each step of the algorithm the validity and practical utility of the resulting expression for the variance of the estimated instantaneous frequency is demonstrated via a simulation experiment

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

time delay and dilation properties in scattering theory we show the identity of the eisenbud wigner time delay and that of lavine for potentials v v1 v2 v1 1 x 1 e x v1 1 x 1 e v2 1 x 2 e for x finally we show the additivity of time delay for the same class of potentials

the decay of unstable noncommutative solitons we study the classical decay of unstable scalar solitons in noncommutative field theory in 2 1 dimensions this can but does not have to be viewed as a toy model for the decay of d branes in string theory in the limit that the noncommutativity parameter is infinite the gradient term is absent there are no propagating modes and the soliton does not decay at all if is large but finite the rotationally symmetric decay channel can be described as a highly excited nonlinear oscillator weakly coupled to a continuum of linear modes this system is closely akin to those studied in the context of discrete breathers we here diagonalize the linear problem and compute the decay rate to first order using a version of fermi s golden rule leaving a more rigorous treatment for future work

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

radiation condition bounds on manifolds with ends abstract we study spectral theory for the schrodinger operator on manifolds possessing an escape function a particular class of examples are manifolds with euclidean and or hyperbolic ends certain exterior domains for possibly unbounded obstacles are included we prove rellich s theorem the limiting absorption principle radiation condition bounds and the sommerfeld uniqueness result striving to extending and refining previously known spectral results on manifolds the proofs are given by an extensive use of commutator arguments these arguments have a classical spirit essentially not involving energy cutoffs or microlocal analysis and require presumably minimum regularity and decay properties of perturbations this paper has interest of its own right but it also serves as a basis for the stationary scattering theory developed fully in the sequel 19

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

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

spectral analysis for optical fibres and stratified fluids i the limiting absorption principle the wave equation t2 h 0 withh c2 x y x y z x y 1 z z x y x rk y rm describes the propagation of acoustical waves in stratified fluids m 1 k 2 as well as electromagnetic waves in layered dielectric media m 1 k 2 and in optical fibres m 2 k 1 and p x y 1 the density p x y and signal speed c x y are possibly long range perturbations of a density p0 y and speed c0 y i e p x y p0 y c x y c0 y 0 as z positive commutator methods are used to show that the spectrum of h is absolutely continuous except possibly for a sequence of isolated eigenvalues of finite multiplicity that can accumulate only at zero and away from those eigenvalues a limiting absorption principle for h is established it is proven that the resolvent h k 1 of h has a norm limit as an operator between l2 rk m 1 z2 2 c2 1 dnz and l2 rk m 1 z2 2 c2 1 dnz for 12 as k c r approaches the real axis either from above or from below

time frequency analysis of time varying modulated signals based on improved energy separation by iterative generalized demodulation energy separation algorithm is good at tracking instantaneous changes in frequency and amplitude of modulated signals but it is subject to the constraints of mono component and narrow band in most cases time varying modulated vibration signals of machinery consist of multiple components and have so complicated instantaneous frequency trajectories on time frequency plane that they overlap in frequency domain for such signals conventional filters fail to obtain mono components of narrow band and their rectangular decomposition of time frequency plane may split instantaneous frequency trajectories thus resulting in information loss regarding the advantage of generalized demodulation method in decomposing multi component signals into mono components an iterative generalized demodulation method is used as a preprocessing tool to separate signals into mono components so as to satisfy the requirements by energy separation algorithm by this improvement energy separation algorithm can be generalized to a broad range of signals as long as the instantaneous frequency trajectories of signal components do not intersect on time frequency plane due to the good adaptability of energy separation algorithm to instantaneous changes in signals and the mono component decomposition nature of generalized demodulation the derived time frequency energy distribution has fine resolution and is free from cross term interferences the good performance of the proposed time frequency analysis is illustrated by analyses of a simulated signal and the on site recorded nonstationary vibration signal of a hydroturbine rotor during a shut down transient process showing that it has potential to analyze time varying modulated signals of multi components 2010 elsevier ltd all rights reserved

the limiting absorption principle for the discrete wigner von neumann operator abstract we apply weighted mourre commutator theory to prove the limiting absorption principle for the discrete schrodinger operator perturbed by the sum of a wigner von neumann and long range type potential in particular this implies a new result concerning the absolutely continuous spectrum for these operators even for the one dimensional operator we show that methods of classical mourre theory based on differential inequalities and on the generator of dilation cannot apply to the aforementioned schrodinger operators

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

regularity of eigenstates in regular mourre theory abstract the present paper gives an abstract method to prove that possibly embedded eigenstates of a self adjoint operator h lie in the domain of the kth power of a conjugate operator a conjugate means here that h and a have a positive commutator locally near the relevant eigenvalue in the sense of mourre the only requirement is c k 1 a regularity of h regarding integer k our result is optimal under a natural boundedness assumption of the multiple commutators we prove that the eigenstate dilated by exp i a is analytic in a strip around the real axis in particular the eigenstate is an analytic vector with respect to a natural applications are dilation analytic systems satisfying a mourre estimate where our result can be viewed as an abstract version of a theorem due to balslev and combes 1971 3 as a new application we consider the massive spin boson model

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

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

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

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

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

spectral theory of a class of block jacobi matrices and applications abstract we develop a spectral analysis of a class of block jacobi operators based on the conjugate operator method of mourre we give several applications including scalar jacobi operators with periodic coefficients a class of difference operators on cylindrical domains such as discrete wave propagators and certain fourth order difference operators

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

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

limiting amplitude principle for acoustic propagators in perturbed stratified fluids abstract by use of the commutator method we study the resolvent estimate at low frequencies of acoustic operators c x 2 in perturbed stratified fluids and prove the principle of limiting amplitude for such operators

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