on the spectral analysis of quantum field hamiltonians abstract we define c algebras on a fock space such that the hamiltonians of quantum field models with positive mass are affiliated to them we describe the quotient of such algebras with respect to the ideal of compact operators and deduce consequences in the spectral theory of these hamiltonians we compute their essential spectrum and give a systematic procedure for proving the mourre estimate

second order perturbation theory for embedded eigenvalues we study second order perturbation theory for embedded eigenvalues of an abstract class of self adjoint operators using an extension of the mourre theory under assumptions on the regularity of bound states with respect to a conjugate operator we prove upper semicontinuity of the point spectrum and establish the fermi golden rule criterion our results apply to massless pauli fierz hamiltonians for arbitrary coupling

eigenvalue accumulation for singular sturm liouville problems nonlinear in the spectral parameter abstract for certain singular sturm liouville equations whose coefficients depend continuously on the spectral parameter in an interval it is shown that accumulation nonaccumulation of eigenvalues at an endpoint of is essentially determined by oscillatory properties of the equation at the boundary as applications new results are obtained for the radial dirac operator and the klein gordon equation three other physical applications are also considered

eigenvalue accumulation for dirac operators with spherically symmetric potential we consider dirac operators h in with spherically symmetric potentials the main result is a criterion for eigenvalue accumulation and non accumulation at the endpoints 1 and 1 of the essential spectrum under rather weak assumptions on the potential this result is proved by showing an analogous criterion for the associated radial dirac operators h and by proving that for sufficiently large each h does not have any eigenvalues in the interval 1 0 and 0 1 respectively of the gap 1 1 of the essential spectrum for the latter properties of solutions of certain riccati differential equations depending on the parameter and the spectral parameter are used

correlation at low temperature ii asymptotics the present paper is a continuation of ref 4 where the truncated two point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature under a weak coupling assumption in this paper we compute the asymptotics of the correlation function as the temperature goes to zero this paper thus extends ref 3 in two directions the hamiltonian function is allowed to have several local minima other than a unique global minimum and we do not require translation invariance of the hamiltonian function we are in particular able to handle spin systems on a general lattice

a note on riesz bases of eigenvectors of certain holomorphic operator functions abstract operator valued functions of the form a a q with q a 1 compact valued and holomorphic on certain domains c are considered in separable hilbert space assuming that the resolvent of a is compact its eigenvalues are simple and the corresponding eigenvectors form a riesz basis for h of finite defect it is shown that under certain growth conditions on q a 1 the eigenvectors of a corresponding to a part of its spectrum also form a riesz basis of finite defect applications are given to operator valued functions of the form a a b d 1c and to spectral problems in l2 0 1 of the form f x p x f x q x f x f x with for example dirichlet boundary conditions

blow up for a three dimensional keller segel model with consumption of chemoattractant abstract we investigate blow up properties for the initial boundary value problem of a keller segel model with consumption of chemoattractant when the spatial dimension is three through a kinetic reformulation of the keller segel system we first derive some higher order estimates and obtain certain blow up criteria for the local classical solutions these blow up criteria generalize the results in 4 5 from the whole space r 3 to the case of bounded smooth domain r 3 lower global blow up estimate on n l is also obtained based on our higher order estimates moreover we prove local non degeneracy for blow up points

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

on the nature of fermi golden rule for open quantum systems we consider a general class of models consisting of a small quantum system s interacting with a reservoir r we compare three applications of 2nd order perturbation theory the fermi golden rule to the study of such models 1 the van hove weak coupling limit for the dynamics reduced to s 2 the fermi golden rule applied to the liouvillean an argument that was used in recent papers on the return to equilibrium 3 the fermi golden rule applied to the so called c liouvillean these three applications lead to three level shift operators as our main result we prove that if the reservoir r is thermal if it satisfies the kms condition then the level shift operator obtained in 1 often called the davies generator and the level shift operator constructed in 2 are connected by a similarity transformation we also show that the davies generator coincides with the level shift operator obtained in 3 for a general r

positive commutators in non equilibrium quantum statistical mechanics the method of positive commutators developed for zero temperature problems over the last twenty years has been an essential tool in the spectral analysis of hamiltonians in quantum mechanics we extend this method to positive temperatures i e to non equilibrium quantum statistical mechanics we use the positive commutator technique to give an alternative proof of a fundamental property of a certain class of large quantum systems called return to equilibrium this property says that equilibrium states are asymptotically stable if a system is slightly perturbed from its equilibrium state then it converges back to that equilibrium state as time goes to infinity

on essential spectra of operator matrices and their feshbach maps abstract a connection between the essential spectrum of certain operator matrices and essential spectra of the corresponding feshbach maps is discussed and applied to some concrete rational operator valued functions

essential spectra of some matrix operators and application to two group transport operators with general boundary conditions abstract in this article we investigate the essential spectra of a 2 2 block operator matrix on a banach space furthermore we apply the obtained results to determine the essential spectra of two group transport operators with general boundary conditions in the banach space l p a a 1 1 l p a a 1 1 a 0

a comparison of techniques for overcoming non uniqueness of boundary integral equations for the collocation partition of unity method in two dimensional acoustic scattering the partition of unity method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements the non uniqueness of solution obtained from the conventional boundary integral equation cbie is well known the cbie derived through green s identities suffers from a problem of non uniqueness at certain characteristic frequencies two of the standard methods of overcoming this problem are the so called combined helmholtz integral equation formulation chief method and that of burton and miller the latter method introduces a hypersingular integral which may be treated in various ways in this paper we present the collocation partition of unity boundary element method pubem for the helmholtz problem and compare the performance of chief against a burton miller formulation regularised using the approach of li and huang

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

renormalization group analysis of spectral problems in quantum field theory abstract in this paper we present a self contained and detailed exposition of the new renormalization group technique proposed in 1 2 its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator the partition function or correlation functions we apply this renormalization transformation to a hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing

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

existence and uniqueness theorem on mild solutions to the keller segel system coupled with the navier stokes fluid abstract we consider the keller segel system coupled with the navier stokes fluid in the whole space and prove the existence of global mild solutions with the small initial data in the scaling invariant space our method is based on the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data as a byproduct we show the asymptotic stability of solutions as the time goes to infinity since we may deal with the initial data in the weak l p spaces the existence of self similar solutions provided the initial data are small homogeneous functions

a collocation method for high frequency scattering by convex polygons we consider the problem of scattering of a time harmonic acoustic incident plane wave by a sound soft convex polygon for standard boundary or finite element methods with a piecewise polynomial approximation space the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave recently chandler wilde and langdon proposed a novel galerkin boundary element method for this problem for which by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency here we propose a related collocation method using the same approximation space for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the galerkin scheme but with a substantially reduced computational cost

level shift operators for open quantum systems level shift operators describe the second order displacement of eigenvalues under perturbation they play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures we exhibit intrinsic properties of level shift operators properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems they determine the geometry of resonances bifurcating from eigenvalues of positive temperature hamiltonians and they relate the gibbs state the kernel of level shift operators and zero energy resonances we show that degeneracy of energy levels of the small part of the open quantum system causes the fermi golden rule condition to be violated and we analyze ergodic properties of such systems

how unstable is spatial homogeneity in keller segel systems a new critical mass phenomenon in two and higher dimensional parabolic elliptic cases the parabolic elliptic keller segel system begin aligned left begin array ll u t delta u nabla cdot u nabla v 0 delta v mu u quad mu frac 1 omega int omega u end array right qquad qquad star end aligned is considered under homogeneous neumann boundary conditions in the ball omega b r 0 subset mathbb r n the main objective is to reveal that in the context of radially symmetric solutions this problem exhibits an apparently novel type of critical mass phenomenon it is shown namely that for any choice of n ge 2 and r 0 there exists a positive number m c m c n r with the following properties in consequence precisely at mass levels above m c the constant steady states of star possess the extreme instability property of repelling arbitrary concentration increasing perturbations in such a drastic sense that corresponding trajectories collapse in finite time

smooth feshbach map and operator theoretic renormalization group methods abstract a new variant of the isospectral feshbach map defined on operators in hilbert space is presented it is constructed with the help of a smooth partition of unity instead of projections and is therefore called smooth feshbach map it is an effective tool in spectral and singular perturbation theory as an illustration of its power a novel operator theoretic renormalization group method is described and applied to analyze a general class of hamiltonians on fock space the main advantage of the new renormalization group method over its predecessors is its technical simplicity which it owes to the use of the smooth feshbach map

efficient evaluation of highly oscillatory acoustic scattering surface integrals we consider the approximation of some highly oscillatory weakly singular surface integrals arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three dimensional convex obstacles described globally in spherical coordinates as the frequency of the incident wave increases the performance of standard quadrature schemes deteriorates naive application of asymptotic schemes also fails due to the weak singularity we propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system for the case of a spherical scatterer we demonstrate via error analysis and numerical results that provided the observation point is sufficiently far from the shadow boundary a high level of accuracy can be achieved with a minimal computational cost

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

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

product integration collocation methods for noncompact integral operator equations we discuss the numerical solution of a class of second kind integral equations in which the integral operator is not compact such equations arise for example when boundary integral methods are applied to potential problems in a two dimensional domain with corners in the boundary we are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes provided some simple modifications are made to the underlying basis functions these are sufficient to ensure stability but do not damage the rate of convergence numerical experiments show that such modifications are necessary in certain circumstances

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

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

commutators c0 semigroups and resolvent estimates abstract we study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self adjoint operator h in the framework of the conjugate operator method initiated by mourre we allow the conjugate operator a to be the generator of a c 0 semigroup finer estimates require a to be maximal symmetric and we consider situations where the first commutator h i a is not comparable to h the applications include the spectral theory of zero mass quantum field models

global stability of homogeneous steady states in scaling invariant spaces for a keller segel navier stokes system abstract in this paper we study the global stability of homogeneous equilibria in keller segel navier stokes equations in scaling invariant spaces we prove that for any given 0 m 1 1 with 1 being the first eigenvalue of neumann laplacian the initial boundary value problem of the keller segel navier stokes system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to m m 0 in the norm of l d 2 w 1 d l d and satisfies a natural average mass condition our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup our result suggests a new observation that nontrivial classical solution for keller segel navier stokes equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large correspondingly

accumulation of discrete eigenvalues of the radial dirac operator abstract for bounded potentials which behave like cx at infinity we investigate whether discrete eigenvalues of the radial dirac operatorh accumulate at 1 or not it is well known that 2 is the critical exponent we show thatc 1 8 1 2 is the critical coupling constant in the case 2 our approach is to transform the radial dirac equation into a sturm liouville equation nonlinear in the spectral parameter and to apply a new general result on accumulation of eigenvalues of such equations

quantum electrodynamics of confined nonrelativistic particles abstract we consider a system of finitely many nonrelativistic quantum mechanical electrons bound to static nuclei the electrons are minimally coupled to the quantized electromagnetic field but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives and the interactions between the radiation field and electrons localized very far from the nuclei are turned off for a class of hamiltonians we prove exponential localization of bound states establish the existence of a ground state and derive sufficient conditions for its uniqueness furthermore we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field to this end we develop a novel renormalization transformation which acts directly on the space of hamiltonians

on a characterization of the essential spectra of some matrix operators and application to two group transport operators in this paper we investigate the essential approximate point spectrum and the essential defect spectrum of a 2 2 block operator matrix on a banach space furthermore we apply the obtained results to two group transport operators in the banach space l p a a 1 1 l p a a 1 1 a 0 p 1

on the existence and uniqueness of ground states of a generalized spin boson model a generalization of the standard spin boson model is considered the hamiltonianh of the model with a coupling parameter racts in the tensor product h fbof a hilbert space h and the boson symmetric fock space fboverl2 r the existence and uniqueness of ground states ofh are investigated the degeneracy of the ground states is also discussed the results obtained arenonperturbative the methods used are those of constructive quantum field theory and the min max principle an exact asymptotic formula for the ground state energy ofh as is also established

level shift operator and second order perturbation theory we give approximate formulas for spectrum and the corresponding spectral projections of perturbed linear operators the main tool is the so called level shift operator which expresses the effects of second order perturbation theory on the point spectrum

a continuous analogue of sturm sequences in the context of sturm liouville equation a shooting method is presented for finding the nth eigenvalue and eigenfunction of a sturm liouville equation in which the eigenvalue occurs nonlinearly the method is analyzed in two ways by applying the sturm comparison and oscillation theorems to the continuous problem and by applying sturm sequences to a discretization the method works for general separated boundary conditions and provides an a posteriors error estimate for the approximate eigenvalue a related method which involves critical lengths in the invariant imbedding method is shown to be incorrect for general boundary conditions

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

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

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

diophantine tori and spectral asymptotics for nonselfadjoint operators we study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint h pseudodifferential operators in dimension 2 assuming that the classical flow of the unperturbed part possesses several invariant lagrangian tori enjoying a diophantine property we get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases in the first case we assume that the strength e of the perturbation is o hd for some d 0 and is bounded from below by a fixed positive power of h in the second case e is assumed to be sufficiently small but independent of h and we describe the eigenvalues completely in a fixed h independent domain in the complex spectral plane

numerical evaluation of the two dimensional partition of unity boundary integrals for helmholtz problems there has been considerable attention given in recent years to the problem of extending finite and boundary element based analysis of helmholtz problems to higher frequencies one approach is the partition of unity method which has been applied successfully to boundary integral solutions of helmholtz problems providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy these benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions in this paper we adapt the numerical steepest descent method to evaluate these integrals for two dimensional problems the approach is successful in reducing the computational effort for most integrals encountered the paper includes some numerical features that are important for successful practical implementation of the algorithm

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

high frequency scattering by convex curvilinear polygons we consider the scattering of a time harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with lipschitz boundary for standard boundary or finite element methods with a piecewise polynomial approximation space the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave here we propose a novel galerkin boundary element method with a hybrid approximation space consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes a uniform mesh on illuminated sides and graded meshes refined towards the corners of the polygon on illuminated and shadow sides numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases

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

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

spectral theory of massless pauli fierz models we study the spectral theory of massless pauli fierz models using an extension of the mourre method we prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant in addition we show that the expectation value of the number operator is finite on all eigenvectors

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

positive commutators fermi golden rule and the spectrum of zero temperature pauli fierz hamiltonians abstract we perform the spectral analysis of a zero temperature pauli fierz system for small coupling constants under the hypothesis of fermi golden rule we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy we rely on a positive commutator approach introduced by skibsted and pursued by georgescu gerard moller we complete some results obtained so far by derezinski jaksic on one side and by bach frohlich sigal soffer on the other side

stationary van hove limit the weak coupling van hove limit of one parameter groups of contractions is studied by the stationary approach we show that the resolvent of the properly renormalized and rescaled generator of a contractive semigroup has a limit as the coupling constant goes to zero this limit is the resolvent of the generator of a certain contractive semigroup our results can be viewed as a stationary counterpart to the well known results about the weak coupling limit obtained by the time dependent approach due to davies we compare both approaches

a high frequency hp boundary element method for scattering by convex polygons in this paper we propose and analyze a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by sound soft convex polygons in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution we demonstrate both theoretically and via numerical examples exponential convergence with respect to the order of the polynomials moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern in which the dependence on the frequency of all constants is explicit importantly these estimates prove that to achieve any desired accuracy in the computation of these quantities it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases in contrast to the at least linear growth required by conventional methods

essential self adjointness of translation invariant quantum field models for arbitrary coupling constants the hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants the hamiltonian has a translation invariant part by means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self adjoint extension of the translation invariant part is shown with a non perturbative approach it is proved that the hamiltonian is essentially self adjoint on a domain a typical example is the pauli fierz model with spin 1 2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants

a hybrid numerical asymptotic method for scattering problems we develop a hybrid numerical asymptotic method for the helmholtz equation the method is a galerkin finite element method in which the space of trial solutions is spanned by asymptotically derived basis functions the basis functions are very efficient in representing the solution because each is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase is determined a priori by solving the eiconal equation using the ray method while the smooth amplitude is represented by piecewise polynomials the number of unknowns necessary to achieve a given accuracy with this new basis is dramatically smaller than the number necessary with a standard method and it is virtually independent of the wavenumber k we apply the method to the problems of scattering from a parabola and from a circle and compare the results with those of a standard finite element method

singular dirac systems and sturm liouville problems nonlinear in the spectral parameter abstract for the dirac operator with spherically symmetric potential v 0 r we investigate the problem of whether the boundary points of the essential spectrum are accumulation points of discrete eigenvalues or not our main result shows that the accumulation of such eigenvalues is essentially determined by the asymptotic behaviour of v at 0 and we obtain this result by using a levinson type theorem for asymptotically diagonal systems depending on some parameter a comparison theorem for the principal solutions of singular dirac systems and some criteria on the eigenvalue accumulation respectively non accumulation of nonlinear singular sturm liouville problems

on the convergence of collocation methods for boundary integral equations on polygons the integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions convergence proofs and error estimates for these approximation methods have been only available in the following cases fredholm integral equations of the second kind 4 7 one dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves 2 3 17 26 29 and some special results on the classical neumann integral equation of potential theory for polygonal plane domains 5 8 9 here we give convergence proofs for collocation with piecewise linear trial functions for neumann s integral equation and symm s integral equation on plane curves with corners we derive asymptotic error estimates in sobolev norms and analyze the effect of graded meshes 0 introduction in this paper we give convergence proofs and asymptotic error estimates in sobolev norms for collocation with piecewise linear spline trial func tions applied to two basic integral equations of potential theory on plane polygons namely the integral equation of the second kind with the double layer potential neumann s integral equation and the integral equation of the first kind with the simple layer potential symm s integral equation we use an idea of arnold and wendland 2 namely considering dirac delta functions the test functions in the collocation method as second derivatives of piecewise linear functions therefore similar results as presented here should be possible for splines of higher odd order corresponding results for even order splines are not yet available thus for one of the simplest methods of numerically solving dirichlet s problem on a plane domain with corners the midpoint collocation with piecewise constant trial functions for the first kind integral equation with the simple layer potential convergence is still an open problem the method of fourier series that yields the convergence proof in the case of a smooth boundary 27 cannot be applied in the presence of corners we apply the method of local mellin transformation that has previously been used to derive error estimates for galerkin methods for a wide class of operators including those occurring in boundary element methods in acoustics electromag

acoustic isogeometric boundary element analysis an isogeometric boundary element method based on t splines is used to simulate acoustic phenomena we restrict our developments to low frequency problems to establish the fundamental properties of the proposed approach using t splines the computer aided design cad and boundary element analysis are integrated without recourse to geometry clean up or mesh generation a regularized burton miller formulation is used resulting in integrals which are at most weakly singular we employ a collocation based approach to generate the linear system of equations the method is verified against closed form solutions and direct comparisons are made with conventional lagrangian discretizations it is demonstrated that the superior accuracy of the isogeometric approach emanates from the exact geometric description encapsulated in the t spline the method is then applied to a real world application to illustrate the potential for integrated engineering design and analysis

multiplicity of ground states in quantum field models applications of asymptotic fields abstract ground states of hamiltonian h of quantum field models are investigated the infimum of the spectrum of h is in the edge of its essential spectrum by means of the asymptotic field theory we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite from which we give an upper bound of the multiplicity of ground states of h typical examples are massless gsb models and the pauli fierz model with spin 1 2

a galerkin boundary element method for high frequency scattering by convex polygons in this paper we consider the problem of time harmonic acoustic scattering in two dimensions by convex polygons standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave here we present a novel galerkin boundary element method which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon we prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency numerical results demonstrate the same logarithmic dependence on the frequency for the galerkin method solution our boundary element method is a discretization of a well known second kind combined layer potential integral equation we provide a proof that this equation and its adjoint are well posed and equivalent to the boundary value problem in a sobolev space setting for general lipschitz domains

ground states of the massless derezinski gerard model we consider the massless derezinski gerard model introduced by derezinski and gerard in 1999 we give a sufficient condition for the existence of a ground state of the massless derezinski gerard model without the assumption that the hamiltonian of particles has compact resolvent