asymptotic completeness for n body stark hamiltonians we prove asymptotic completeness for short and long rangen body stark hamiltonians with local singularities of at most coulomb type our results include the usual models for atoms and molecules

a time dependent theory of quantum resonances in this paper we further develop a general theory of metastable states resulting from perturbation of unstable eigenvalues we apply this theory to many body schrouml dinger operators and to the problem of quasiclassical tunneling

inverse spectral theory for random jacobi matrices we give necessary and sufficient conditions for a herglotz function to be thew function of a random stationary jacobi matrix

exponential decay of eigenfunctions in many body type scattering with second order perturbations abstract we show the exponential decay of eigenfunctions of second order geometric many body type hamiltonians at non threshold energies moreover in the case of first order and small second order perturbations we show that there are no eigenfunctions with positive energy

models algorithms and error estimation for computational viscoelasticity this article reviews numerical algorithms for problems in solid polymer viscoelasticity in both small and large deformation for the linear small strain case we review both the quasistatic and the dynamic problem and give recent results on a posteriori error estimation for the large strain case we focus on the formulation and computational modelling of constrained membrane inflation the application of which is to the thermoforming process

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

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

the quantum n body problem this selective review is written as an introduction to the mathematical theory of the schrodinger equation for n particles characteristic for these systems are the cluster properties of the potential in configuration space which are expressed in a simple geometric language the methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results the central theme is the interplay between the spectral theory of n body hamiltonians and the space time and phase space analysis of bound states and scattering states

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

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

absolutely continuous spectra of quantum tree graphs with weak disorder we consider the laplacian on a rooted metric tree graph with branching number k 2 and random edge lengths given by independent and identically distributed bounded variables our main result is the stability of the absolutely continuous spectrum for weak disorder a useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the weyl titchmarsh function to trees the proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder

an extension of kotani s theorem to random generalized sturm liouville operators we consider the random operator d m dx d dx q x wherem dx andq x are a stationary ergodic random measure and a random function respectively to this general case we extend kotani s theorem which asserts that the absolutely continuous spectrum is completely determined by the ljapounov indices our framework includes the case of stochastic jacobi matrices treated by simon

well posedness of a linearized fractional derivative fluid model the one dimensional fractional derivative maxwell model e g palade et al rheol acta 35 1996 265 of importance in modeling the linear viscoelastic response in the glass transition region has been generalized in palade et al internat j engrg sci 37 1999 315 to objective three dimensional constitutive equations ces for both fluids and solids regarding the rest state stability of the fluid ce in heibig and palade j math phys 49 2008 043101 we gave a proof for the existence of weak solutions to the corresponding boundary value problem the aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results

nonexponential decay laws in perturbation theory of near threshold eigenvalues we consider a two channel model of the form he hop00e0 e 0w12w210 on h hop c the operator hop is assumed to have the properties of a schrodinger operator in odd dimensions with a threshold at zero as the energy parameter e0 is tuned past the threshold we consider the survival probability 0 e ithe 0 2 where 0 is the eigenfunction corresponding to eigenvalue e0 for e 0 we find nonexponential decay laws for e small and e0 close to zero provided that the resolvent of hop is not at least lipschitz continuous at the threshold zero

the fermi golden rule and its form at thresholds in odd dimensions let h be a schrodinger operator on a hilbert space open image in new window such that zero is a nondegenerate threshold eigenvalue of h with eigenfunction 0 let w be a bounded selfadjoint operator satisfying 0 w 0 0 assume that the resolvent h z 1 has an asymptotic expansion around z 0 of the form typical for schrodinger operators on odd dimensional spaces let h h w for 0 and small we show under some additional assumptions that the eigenvalue at zero becomes a resonance for h in the time dependent sense introduced by a orth no analytic continuation is needed we show that the imaginary part of the resonance has a dependence on of the form 2 2 with the integer 1 and odd this shows how the fermi golden rule has to be modified in the case of perturbation of a threshold eigenvalue we give a number of explicit examples where we compute the location of the resonance to leading order in we also give results in the case where the eigenvalue is embedded in the continuum sharpening the existing ones

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

the mourre estimate for dispersive n body schrodinger operators we prove the mourre estimate for a certain class of dispersive n body schrodinger operators using this estimate we derive some properties of those operators such as the local finiteness of the finite spectrum and the absence of the singular continuous spectrum

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

rellich s theorem for spherically symmetric repulsive hamiltonians for spherically symmetric repulsive hamiltonians we prove rellich s theorem or identify the largest weighted space of agmon hormander type where the generalized eigenfunctions are absent the proof is intensively dependent on commutator arguments our novelty here is a use of conjugate operator associated with some radial flow not with dilations and not with translations our method is simple and elementary and does not employ any advanced tools such as the operational calculus or the fourier analysis

threshold properties of matrix valued schrodinger operators ii resonances we present some results on the perturbation of eigenvalues embedded at a threshold for a matrix valued hamiltonian with three dimensional dilation analytic schrodinger operators as entries and with a small off diagonal perturbation the main result describes how a threshold eigenvalue generates resonances that is poles of the meromorphic continuation of the perturbed hamiltonian

resonance theory for schrodinger operators resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods a general theory is developed with new and weaker conditions allowing for perturbations of threshold eigenvalues and relaxed fermi golden rule the exponential decay rate of resonances is addressed its uniqueness in the time dependent picture is shown in certain cases the relation to the existence of meromorphic continuation of the properly weighted green s function to time dependent resonance is further elucidated by giving an equivalent time dependent asymptotic expansion of the solutions of the schrodinger equation

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

exponential lower bounds to solutions of the schrodinger equation lower bounds for the spherical average for a large class of generalizedn body schrodinger operators h we show that ife inf ess h and is an eigenfunction ofh with eigenvaluee then begin array 20 c lim r to infty end array r 1 ln left int limits s n 1 psi r omega 2 d omega right 1 2 alpha 0 with 0 2 e a threshold similar results are given fore

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

exponential bounds and semifiniteness of point spectrum for n body schrodinger operators for a large class ofn body schrodinger operatorsh we prove that eigenvalues ofh cannot accumulate from above at any threshold ofh our proof relies onl2 exponential upper bounds for eigenfunctions ofh with explicit constants obtained by modifying methods of froese and herbst

localization for off diagonal disorder and for continuous schrodinger operators we extend the proof of localization by delyon levy and souillard to accommodate the anderson model with off diagonal disorder and the continuous schrodinger equation with a random potential

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

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

l 2 exponential lower bounds to solutions of the schrodinger equation we study decay properties of solutions of the schrodinger equation v e typical of our results is one which shows that ifv o x 1 2 at infinity or ifv is a homogeneousn body potential for example atomic or molecular then ife 0 and alpha sqrt e e alpha left x right psi notin l 2 left mathbb r n right we also construct examples to show that previous essential spectrum dependent upper bounds can be far from optimal if is not the ground state

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

on inverse scattering for then body schrodinger equation abstract we consider the generalized n body schrodinger operator h q x q x a a v a x a x r d x a x a r d x a x a with short range regular interactions we obtain in particular the following results 1 we give new formulas and proofs relating high energy asymptotics in a weak sense of the scattering operators with the same cluster decomposition a for and and the x ray transform pi defined on the set of all lines in x a with nonsingular directions of the effective potentials i x a x a x a these results significantly clarify some of those given in the literature 2 we describe completely ker p and give a method for reconstruction of i mod ker p from pi 3 we prove pointwise high energy asymptotics for the two cluster two cluster scattering amplitudes f with the fourier transform i in the leading term for the case when the cluster decomposition for and is the same 4 we give several additional results for the case of perturbed stratified medium when q x v a x a v b x b x a x 1 x b x 1 x d x and each v c x c rapidly decreases as x c c a b our results including proofs in some cases significantly simplify methods of high energy inverse scattering for the n body schrodinger operator given earlier by wang and by enss and weder

counterexamples to the kotani last conjecture for continuum schrodinger operators via character automorphic hardy spaces abstract the kotani last conjecture states that every ergodic operator in one space dimension with non empty absolutely continuous spectrum must have almost periodic coefficients this statement makes sense in a variety of settings for example discrete schrodinger operators jacobi matrices cmv matrices and continuum schrodinger operators in the main body of this paper we show how to construct counterexamples to the kotani last conjecture for continuum schrodinger operators by adapting the approach developed by volberg and yuditskii to construct counterexamples to the kotani last conjecture for jacobi matrices this approach relates the reflectionless operators associated with the prescribed spectrum to a family of character automorphic hardy spaces and then relates the shift action on the space of operators to the resulting action on the associated characters the key to our approach is an explicit correspondence between the space of continuum reflectionless schrodinger operators associated with a given set and the space of reflectionless jacobi matrices associated with a derived set once this correspondence is established we can rely to a large extent on the previous work of volberg and yuditskii to produce the resulting action on the space of characters we analyze this action and identify situations where we can observe absolute continuity without almost periodicity in the appendix we show how to implement this strategy and obtain analogous results for extended cmv matrices

stark effect in multielectron systems non existence of bound states we prove non existence of bound states for a class ofn body systems in homogeneous electric fields this class includes atoms and born oppenheimer molecules this result in conjunction with a stability result of hs implies existence of resonances for such systems

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

decay of eigenfunctions of elliptic pde s i we study exponential decay of eigenfunctions of self adjoint higher order elliptic operators on rd we show that the possible global critical decay rates are determined algebraically in addition we show absence of super exponentially decaying eigenfunctions and a refined exponential upper bound

perturbation theory for the decay rate of eigenfunctions in the generalized n body problem simple examples are known where eigenfunctions decay faster than the usual upper bounds would lead one to believe we develop aspects of the perturbation theory of the decay rate of eigenfunctions as measured by radial exponential weights we show that generically in a baire category sense eigenfunction decay rates are governed by the lowest threshold

a viscohyperelastic finite element model for rubber abstract a recently developed finite deformation internal solid theory for viscoelastic deformations of rubber is reviewed and an axisymmetric finite element implementation of the theory is presented the theoretical model is a three dimensional total lagrangian finite deformation generalization of the maxwell internal solid theory for viscoelasticity it utilizes time dependent reference shapes for internal hyperelastic solids to approximate the viscous effects found in rubber and is called a viscohyperelastic model the material constants for the sum of the internal solids hyperelastic energy functions can be found by performing the classical tensile biaxial and shear tests as step strain tests a nearly incompressible finite element version of the theory is presented the viscous equations of motion are solved with a predictor corrector integration scheme time dependent pressure loads are applied to the interior of both a thick walled rubber cylinder and sphere to numerically demonstrate the viscohyperelastic finite element algorithm

criteria for the kato smoothness with respect to a dispersive n body schrodinger operator the operators of the form h d va ax which are a natural generalization of n body schrodinger operators are studied certain criteria are presented that allow one to verify if a given pseudodifferential operator is kato smooth with respect to h

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

localization in one dimensional random media a scattering theoretic approach we use scattering theoretic methods to prove exponential localization for random displacement models in one dimension the operators we consider model both quantum and classical wave propagation our main tools are the reflection and transmission coefficients for compactly supported single site perturbations we show that randomly displaced non reflectionless single sites lead to localization

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

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

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

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

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

absence of embedded eigenvalues for riemannian laplacians in this paper we study absence of embedded eigenvalues for schrodinger operators on non compact connected riemannian manifolds a principal example is given by a manifold with an end possibly more than one in which geodesic coordinates are naturally defined in this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end another condition is an upper bound of the trace of this quantity while a third one is a bound of the derivatives of part of the trace some oscillatory behaviour of the trace is allowed in addition to geometric bounds we need conditions on the potential a regularity property of the domain of the schrodinger operator and the unique continuation property examples include ends endowed with asymptotic euclidean or hyperbolic metrics

numerical models of steady rolling for non linear viscoelastic structures in finite deformations a lagrangian formulation of constitutive laws for a viscoelastic material based on irreversible thermodynamics is first presented these laws are expressed by a non linear differential equation governing the evolution of an internal variable then equations describing the steady rolling of an axisymmetric viscoelastic structure are obtained from the conservation laws of continuum mechanics a finite element approximation and a solution technique of the algebraic system is proposed the eiimination of the internal variable allows one to keep an elastic like algorithm with an independent solution of the viscoelastic equation numerical tests show the feasibility and the efficiency of the proposed methods in large three dimensional situations

computational modelling of thermoforming processes in the case of finite viscoelastic materials abstract in this paper we describe the computational simulation of the inflation phase of a thermoforming process under which a thin polymer sheet is deformed into a mould under the action of applied pressure it is assumed that the sheet undergoes finite viscoelastic deformation which is treated using a hyperelastic model containing internal variables the simplification is adopted that the sheet can be treated as a membrane and also that there is a total sticking contact condition when the sheet comes in contact with the mould the computational model uses finite elements in space and incorporates mesh adaptivity based on a residual estimator in order to simulate the deformation accurately and efficiently the internal variables satisfy an ordinary differential equation in time which is solved using a predictor corrector scheme the constitutive model is a generalisation of that of le tallec and rahier p le tallec c rahier numerical models of steady rolling for non linear viscoelastic structures in finite deformations int j numer methods engrg 37 1994 1159 1186 in the simulation it is demonstrated how effectively the estimator works in controlling the meshes for some demanding mould shapes

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

multidimensional inverse scattering for the nonlinear klein gordon equation with a potential in this paper we solve the multidimensional inverse scattering problem for the nonlinear klein gordon equation on rn n 2 2 t2u x t u x t u x t v0 x u x t j 1 vj x u 2 j0 j u x t 0 we prove that the small amplitude limit of the scattering operator determines uniquely all the vj j 0 1 our proof gives as well a method for the reconstruction of the vj j 0 1

a posteriori error estimates and adaptivity for finite element solutions in finite elasticity methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems in contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context in the paper a brief outline of continuum theory of finite elasticity is first given using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally the proposed error estimator is physically interpreted in the energy sense we then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences

space time picture of semiclassical resonances this paper addresses the theory of quasiclassical resonances for schrodinger operators with potentials smooth outside some possible local singularities we introduce the notion of quasiresonance similar to that of quasimode but incorporating a condition revealing its scattering nature and describe its space time behaviour the definition is given in terms of the original schrodinger operator and uses a description of its frequency set the result on the space time behaviour justifies the intuitive picture of resonances as metastable states or bound states with finite life times we demonstrate how quasiresonances arise in several natural situations

spectral analysis of n body stark hamiltonians we prove that the spectrum for a large class ofn body stark hamiltonians is purely absolutely continuous we need slow decay at infinity and local singularities of at most coulomb type in particular our results include the usual models for atoms and molecules

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

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

non linear wave and schrodinger equations i instability of periodic and quasiperiodic solutions we investigate stability of periodic and quasiperiodic solutions of linear wave and schrodinger equations under non linear perturbations we show in the case of the wave equations that such solutions are unstable for generic perturbations for the schrodinger equations periodic solutions are stable while the quasiperiodic ones are not we extend these results to periodic solutions of non linear equations

inverse problem for polynomials orthogonal on the unit circle polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum are investigated it is shown that the coefficients are in fact quasi periodic the measures associated with these quasi periodic coefficients are exhibited and necessary and sufficient conditions relating quasi periodicity and spectral measures of this type are given analogs for polynomials orthogonal on subsets of the real line are also presented

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies

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